# Heteroclinic traveling waves of 2D parabolic Allen-Cahn systems

Ramon Oliver-Bonafoux Sorbonne Université, Laboratoire Jacques-Louis Lions. 4
Place Jussieu, 75005 Paris (France). Email<EMAIL_ADDRESS>
###### Abstract

In this paper we show the existence of traveling waves
$w:[0,+\infty)\times{\mathbb{R}}^{2}\to{\mathbb{R}}^{k}$ ($k\geq 2$) for the
parabolic Allen-Cahn system

$\partial_{t}w-\Delta w=-\nabla_{u}V(w)\mbox{ in
}[0,+\infty)\times{\mathbb{R}}^{2},$ (0.1)

satisfying some heteroclinic conditions at infinity. The potential $V$ is a
non-negative and smooth multi-well potential, which means that its null set is
finite and contains at least two elements. The traveling wave $w$ propagates
along the horizontal axis according to a speed $c^{\star}>0$ and a profile
$\mathfrak{U}$. The profile $\mathfrak{U}$ joins as $x_{1}\to\pm\infty$ (in a
suitable sense) two locally minimizing 1D heteroclinics which have different
energies and the speed $c^{\star}$ satisfies certain uniqueness properties.
The proof of variational and, in particular, it requires the assumption of an
upper bound, depending on $V$, on the difference between the energies of the
1D heteroclinics.

##  1 Introduction

Consider the parabolic system of equations

$\partial_{t}w-\Delta w=-\nabla_{u}V(w)\mbox{ in
}[0,+\infty)\times{\mathbb{R}}^{2},$ (1.1)

where $V:{\mathbb{R}}^{k}\to{\mathbb{R}}$ is a smooth, non-negative, multi-
well potential (see assumptions (H1), (H2), (H3) later) and
$w:[0,+\infty)\times{\mathbb{R}}^{2}\to{\mathbb{R}}^{k}$, with $k\geq 2$. We
seek for traveling wave solutions to (1.1). That is, we impose on $w$

$\forall(t,x_{1},x_{2})\in[0,+\infty)\times{\mathbb{R}}^{2},\hskip
5.69054ptw(t,x_{1},x_{2})=\mathfrak{U}(x_{1}-c^{\star}t,x_{2}),$ (1.2)

where $\mathfrak{U}:{\mathbb{R}}^{2}\to{\mathbb{R}}^{k}$ is the profile of the
wave and $c^{\star}>0$ is the speed of propagation of the wave, which occurs
in the $x_{1}$-direction. Both the profile and the speed are the unknowns of
the problem. Replacing in (1.1), we find that the profile $\mathfrak{U}$ and
$c^{\star}$ must satisfy the elliptic system

$-c^{\star}\partial_{x_{1}}\mathfrak{U}-\Delta\mathfrak{U}=-\nabla_{u}V(\mathfrak{U})\mbox{
in }{\mathbb{R}}^{2}.$ (1.3)

The system (1.1) can be seen as a reaction-diffusion system. Since the early
works, motivated by questions from population dynamics, of Fisher [41] and
Kolmogorov, Petrovsky and Piskunov [50], devoted to a scalar reaction-
diffusion equation in one space dimension known today as the Fisher-KPP
equation, traveling and stationary waves are known to play a major role in the
dynamics of reaction-diffusion problems. For instance, in [40, 39], Fife and
McLeod proved stability results for the equations considered in [41, 50].
Regarding higher dimensional problems (but always in the scalar case),
existence results for traveling waves were obtained by Aronson and Weinberger
[10] for equations with ${\mathbb{R}}^{N}$ as space domain and by Berestycki,
Larrouturou and Lions [17], Berestycki and Nirenberg [20] for unbounded
cylinders of the type ${\mathbb{R}}\times\omega$, with
$\omega\subset{\mathbb{R}}^{N-1}$ a bounded domain. We also mention that
asymptotic stability results (for a suitable class of perturbations) for
traveling waves in the scalar Allen-Cahn equation in ${\mathbb{R}}^{N}$ were
obtained by Matano, Nara and Taniguchi [55].

All the papers mentioned above are devoted to scalar equations and they rely
on the application of the maximum principle and its related tools. As it is
well-known, the maximum principle does not apply in general to systems of
equations, meaning that other techniques are needed in order to study the
existence of traveling waves (and their properties in case they exist) for
systems. Different, more general, approaches had been taken in order to
circumvent the lack of the maximum principle when dealing with parabolic
systems. We refer to the books by Smoller [68] and Volpert, Volpert and
Volpert [70]. One of these approaches consists on the use of variational
methods. In the context of reaction-diffusion equations, this approach seems
to appear for the first time in Heinze’s PhD thesis [46] (even though the
existence of a variational framework for reaction diffusion problems was known
since [40, 39]) and subsequently carried on also by Muratov [58], Lucia,
Muratov and Novaga [52], Alikakos and Katzourakis [7] (see also Alikakos,
Fusco and Smyrnelis [6]), Risler [65, 66, 64] and, more recently, by Chen,
Chien and Huang [38]. In the latter, the authors consider a parabolic Allen-
Cahn system in a two dimensional strip ${\mathbb{R}}\times(-l,l)$ and find
traveling waves which join a well and an approximation of an heteroclinic
orbit in $(-l,l)$, for a class of symmetric triple-well potentials. Lastly, we
mention that variational methods have also been applied to scalar reaction-
diffusion equations, see for instance Bouhours and Nadin [31] for the case of
heterogeneous equations as well as Lucia, Muratov and Novaga [53]. In this
paper we shall also take a variational approach for dealing with the following
question:

Question:  Assuming that there exist two heteroclinic orbits, joining two
fixed wells, with different energy (defined in (2.2)) levels, does there exist
a solution $(c,\mathfrak{U})$ to (1.3) such that $\mathfrak{U}$ joins the two
heteroclinic orbits at infinity, uniformly in $x_{1}$?

Heteroclinic orbits are curves $\mathfrak{q}:{\mathbb{R}}\to{\mathbb{R}}^{k}$
which solve the equation

$\mathfrak{q}^{\prime\prime}=\nabla_{u}V(\mathfrak{q})\mbox{ in }{\mathbb{R}}$
(1.4)

and join two different wells of $\Sigma$ at $\pm\infty$. Moreover, one asks
that the 1D energy (i. e., the functional associated to the previous equation,
see (2.2)) is finite. We show that, under the proper assumptions, the question
we posed has an affirmative answer. Our motivation comes from two different
sides:

  1. 1.

Stationary heteroclinic-type solutions of (1.1) have been known to exist in
several situations for a long time. Indeed, for a class of symmetric
potentials, Alama, Bronsard and Gui in [2] showed the existence of a
stationary wave (that is, a solution to (1.3) with $c=0$) in the situation
such that two heteroclinics with equal energy levels exist and are global
minimizers of the 1D energy. Their analysis was later extended to potentials
without symmetry in several papers, which in some cases obtained similar
results by means of different techniques. See Fusco [42], Monteil and
Santambrogio [57], Schatzman [67], Smyrnelis [69]. A key observation is that
this problem can be seen as a heteroclinic orbit problem for a potential (the
1D energy, see (2.2)) defined in the infinite-dimensional space
$L^{2}({\mathbb{R}},{\mathbb{R}}^{k})$. Therefore, it is natural to aim a
solving a connecting orbit problem for potentials defined in, say, Hilbert
spaces and then deduce the original problem as a particular case. This is the
approach taken in [57] (in the metric space setting) and in [69] (in the
Hilbert space setting).

  2. 2.

Alikakos and Katzourakis [7], showed the existence of traveling waves for a
class of 1D parabolic systems of gradient type. Essentially, they assume that
the potential possesses two local minima (one of them global) at different
levels. Hence, their potential is not of multi-well type in general. The
profile of the traveling waves connects the two local minima at infinity and
the determination of the speed becomes also part of the problem.

The results of this paper follow by suitably merging the ideas of the previous
items. More precisely, we formulate and provide solutions for a heteroclinic
traveling problem as that in [7] for potentials defined in an abstract Hilbert
space. Then, we recover as a particular case the existence of a traveling wave
solution for (1.1) with heteroclinic behavior at infinity,

### Acknowledgments

I wish to thank my PhD advisor Fabrice Bethuel for bringing this problem into
my attention and for many useful comments and remarks during the elaboration
of this paper.  This program has received funding from the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie grant agreement No 754362.

###### Contents

  1. 1 Introduction
  2. 2 The main results: Statements and discussions
    1. 2.1 Basic assumptions and definitions
    2. 2.2 Existence
    3. 2.3 Conditions at infinity
    4. 2.4 Min-max characterization of the speed
    5. 2.5 Definition of the upper bounds
    6. 2.6 Methods and ideas of the proofs
    7. 2.7 Examples of potentials verifying the assumptions
  3. 3 Discussion on the previous literature and open problems
    1. 3.1 Related reaction-diffusion models and the question of stability
    2. 3.2 The heteroclinic stationary wave for 2D Allen-Cahn systems
    3. 3.3 Traveling waves for 1D parabolic systems of gradient type
    4. 3.4 Link with the singular limit problem
  4. 4 The abstract setting
    1. 4.1 Main definitions and notations
    2. 4.2 Statement of the abstract results
  5. 5 Proof of the abstract results
    1. 5.1 Scheme of the proofs
    2. 5.2 Preliminaries
    3. 5.3 The infima of $\mathbf{E}_{c}$ in $X_{T}$ are well defined
    4. 5.4 General continuity and semi-continuity results
    5. 5.5 Existence of an infimum for $\mathbf{E}_{c}$ in $X_{T}$
    6. 5.6 The comparison result
    7. 5.7 Existence of the unconstrained solutions
    8. 5.8 Uniqueness of the speed
    9. 5.9 Proof of Theorem 5 completed
    10. 5.10 Asymptotic behavior of the constrained solutions at $-\infty$
    11. 5.11 Proof of Theorem 6 completed
    12. 5.12 Proof of Theorem 7 completed
  6. 6 Proofs of the main results completed
    1. 6.1 Proving the link between the main setting and the abstract setting
    2. 6.2 Proof of Theorem 1 completed
    3. 6.3 Proof of Theorem 2 completed
    4. 6.4 Proof of Theorem 3
    5. 6.5 Proof of Theorem 4 completed

##  2 The main results: Statements and discussions

We now state the results of this paper. In Theorem 1, which is the main
result, existence of a traveling wave solution with speed $c^{\star}$ and
profile $\mathfrak{U}$ is established as well as the uniqueness (in some
sense) of $c^{\star}$ and the $L^{2}$ exponential convergence of
$\mathfrak{U}$ at the limit $x_{1}\to+\infty$. For proving such a result, we
use the bound assumption (H6). In Theorem 2, we show that under the additional
assumption (H7)) the condition at infinity as $x_{1}\to-\infty$ can be
strengthened with respect to that given in Theorem 1 and in particular we show
that the solution converges at $-\infty$ at an exponential rate. In Theorem 3,
we show that under the previous assumptions we have uniform convergence of the
solution in the $x_{1}$ and the $x_{2}$ direction. Assumption (H7) is also
used for proving Theorem 4, which gives further properties on the speed
$c^{\star}$. We conclude this section by describing the outline and main ideas
of our proofs (subsection 2.6) as well as giving examples of potentials that
verify the assumptions of this paper (subsection 2.7).

###  2.1 Basic assumptions and definitions

Before stating the results, we recall some standard assumptions, definitions
and results and we introduce some notation. The multi-well potentials $V$
considered in this paper satisfy the following:

######  (H1).

$V\in\mathcal{C}^{2}_{\mathrm{loc}}({\mathbb{R}}^{k})$ and $V\geq 0$ in
${\mathbb{R}}^{k}$. Moreover, $V(u)=0$ if and only if $u\in\Sigma$, where, for
some $l\geq 2$

$\Sigma:=\\{\sigma_{1},\ldots,\sigma_{l}\\}.$ (2.1)

######  (H2).

There exist $\alpha_{0},\beta_{0},R_{0}>0$ such that for all
$u\in{\mathbb{R}}^{k}$ with $\lvert u\rvert\geq R_{0}$ it holds
$\langle\nabla_{u}V(u),u\rangle\geq\alpha_{0}\lvert u\rvert^{2}$ and
$V(u)\geq\beta_{0}$.

######  (H3).

For all $\sigma\in\Sigma$, the matrix $D^{2}V(\sigma)$ is positive definite.

As we advanced before, one considers the 1D energy functional

$E(q):=\int_{\mathbb{R}}e(q)(t)dt:=\int_{\mathbb{R}}\left[\frac{1}{2}\lvert
q^{\prime}(t)\rvert^{2}+V(q(t))\right]dt,\hskip 5.69054ptq\in
H^{1}_{\mathrm{loc}}({\mathbb{R}},{\mathbb{R}}^{k}).$ (2.2)

Given a pair of wells $(\sigma_{i},\sigma_{j})\in\Sigma^{2}$, as done for
instance in Rabinowitz [61] we define

$X(\sigma_{i},\sigma_{j}):=\left\\{q\in
H^{1}_{\mathrm{loc}}({\mathbb{R}},{\mathbb{R}}^{k}):E(q)<+\infty\mbox{ and
}\lim_{t\to-\infty}q(t)=\sigma_{i},\lim_{t\to+\infty}q(t)=\sigma_{j}\right\\},$
(2.3)

the set of curves in ${\mathbb{R}}^{k}$ connecting $\sigma_{i}$ and
$\sigma_{j}$. The space $X(\sigma_{i},\sigma_{j})$ is a metric space when it
is endowed with the $L^{2}$ and the $H^{1}$ distances, since $q-\tilde{q}\in
H^{1}({\mathbb{R}},{\mathbb{R}}^{k})$ whenever $q$ and $\tilde{q}$ belong to
$X(\sigma_{i},\sigma_{j})$. If $\mathfrak{q}$ is a critical point of the
energy $E$ in $X(\sigma_{i},\sigma_{j})$, we say that $\mathfrak{q}$ is an
homoclinic orbit when $\sigma_{i}=\sigma_{j}$ and that $\mathfrak{q}$ is an
heteroclinic orbit when $\sigma_{i}\not=\sigma_{j}$. Define as well the
corresponding infimum value

$\mathfrak{m}_{\sigma_{i}\sigma_{j}}:=\inf\\{E(q):q\in
X(\sigma_{i},\sigma_{j})\\}.$ (2.4)

If $\sigma^{-}$ and $\sigma^{+}$ are two distinct wells in $\Sigma$, it turns
out that $\mathfrak{m}_{\sigma^{-}\sigma^{+}}$ is not attained in general. We
need to add the following assumption:

######  (H4).

We have that

$\forall\sigma\in\Sigma\setminus\\{\sigma^{-},\sigma^{+}\\},\hskip
5.69054pt\mathfrak{m}_{\sigma^{-}\sigma^{+}}<\mathfrak{m}_{\sigma^{-}\sigma}+\mathfrak{m}_{\sigma\sigma^{+}}.$
(2.5)

Notice that one can always find a pair $(\sigma^{-},\sigma^{+})\in\Sigma^{2}$
such that (H4) holds. Assuming that (H1), (H2), (H3) and (H4) hold, it is well
known that there exists a minimizer of $E$ in $X(\sigma^{-},\sigma^{+})$.
Moreover, we have the compactness of minimizing sequences as follows: For any
$(q_{n})_{n\in{\mathbb{N}}}$ in $X(\sigma^{-},\sigma^{+})$ such that
$E(q_{n})\to\mathfrak{m}$, there exists $(\tau_{n})_{n\in{\mathbb{N}}}$ in
${\mathbb{R}}$ and $\mathfrak{q}\in X(\sigma^{-},\sigma^{+})$ such that
$E(\mathfrak{q})=\mathfrak{m}$ and, up to subsequences

$\lVert
q_{n}(\cdot+\tau_{n})-\mathfrak{q}\rVert_{H^{1}({\mathbb{R}},{\mathbb{R}}^{k})}\to
0\mbox{ as }n\to+\infty.$ (2.6)

As we said before, this result is well-known. The earlier references are
Bolotin [29], Bolotin and Kozlov [30], Bertotti and Montecchiari [21] and
Rabinowitz [62, 63], sometimes in a slightly different setting. Proofs and
applications of the compactness property (2.6) are also given in Alama,
Bronsard and Gui [2], Alama et al [1] and Schatzman [67].

We fix the two wells $\sigma^{-}$ and $\sigma^{+}$ for the rest of the paper
as well as $\mathfrak{m}:=\mathfrak{m}_{\sigma^{-}\sigma^{+}}$. According to
the previous discussion, we have that the set

$\mathcal{F}:=\\{\mathfrak{q}:\mathfrak{q}\in X(\sigma^{-},\sigma^{+})\mbox{
and }E(\mathfrak{q})=\mathfrak{m}\\},$ (2.7)

is not empty. We term the elements of $\mathcal{F}$ as globally minimizing
heteroclinics between $\sigma^{-}$ and $\sigma^{+}$. The term heteroclinics
comes from the fact that $\sigma^{-}$ and $\sigma^{+}$ are different. An
important fact is that, due to the translation invariance of $E$ and
$X(\sigma^{-},\sigma^{+})$, we have that if $\mathfrak{q}\in\mathcal{F}$, then
for all $\tau\in{\mathbb{R}}$ it holds
$\mathfrak{q}(\cdot+\tau)\in\mathcal{F}$.

###  2.2 Existence

The assumptions (H1), (H2), (H3) and (H4) stated before are classical. In
order to obtain our results, we shall supplement them with the following one,
which is more specific to the setting of this paper:

######  (H5).

Assume that (H1), (H2), (H3) and (H4) hold for the potential $V$. We keep the
previous notations. We assume the following:

  1. 1.

It holds that
$\mathcal{F}^{-}:=\mathcal{F}=\\{\mathfrak{q}^{-}(\cdot+\tau):\tau\in{\mathbb{R}}\\}$
for some $\mathfrak{q}^{-}\in X(\sigma^{-},\sigma^{+})$, where $\mathcal{F}$
was defined in (2.7). We also set $\mathfrak{m}^{-}:=\mathfrak{m}$.

  2. 2.

There exists $\mathfrak{m}^{+}>\mathfrak{m}^{-}$ and $\mathfrak{q}^{+}\in
X(\sigma^{-},\sigma^{+})$ such that $E(\mathfrak{q}^{+})=\mathfrak{m}^{+}$ and
$\mathfrak{q}^{+}$ is a local minimizer of $E$ with respect to the $H^{1}$
norm. We denote
$\mathcal{F}^{+}:=\\{\mathfrak{q}^{+}(\cdot+\tau):\tau\in{\mathbb{R}}\\}$.

  3. 3.

We have the spectral nondegeneracy assumption due to Schatzman ([67]): For all
$q\in X(\sigma^{-},\sigma^{+})$, let $A(q)$ be the unbounded linear operator
in $L^{2}({\mathbb{R}},{\mathbb{R}}^{k})$ with domain
$H^{2}({\mathbb{R}},{\mathbb{R}}^{k})$ defined as

$A(q):v\to-v^{\prime\prime}+D^{2}V(q)v,$ (2.8)

then, it holds that for any
$\mathfrak{q}\in\mathcal{F}^{-}\cup\mathcal{F}^{+}$ we have
$\mathrm{Ker}(A(\mathfrak{q}))=\\{\mathfrak{q}^{\prime}\\}$. The fact that
$\mathfrak{q}^{\prime}\in H^{2}({\mathbb{R}},{\mathbb{R}}^{k})$ follows from
the identity
$\mathfrak{q}^{\prime\prime\prime}=D^{2}V(\mathfrak{q})\mathfrak{q}^{\prime}$.

Notice that if we had $\mathfrak{m}^{+}=\mathfrak{m}^{-}$ we would be in the
framework of Alama, Bronsard and Gui [2], for which the 2D solution connecting
$\mathfrak{q}^{-}$ and $\mathfrak{q}^{+}$ is stationary. Essentially,
conditions 1. and 2. in (H5) imply that $\mathfrak{q}^{-}$ is a globally
minimizing heteroclinic and $\mathfrak{q}^{+}$ is a locally (but not globally)
minimizing heteroclinic. Regarding assumption 3., introduced in [67], it must
be seen as a generalized non-degeneracy assumption for the minima. They are
still degenerate critical points because every critical point of $E$ is
degenerate due to the invariance by translations. Nevertheless, the assumption
3. implies that they are non-degenerate up to invariance by translations. As
shown in [67], such a condition is generic in the sense that given a potential
satisfying 1. and 2. one can always find a potential which verifies 1. 2.
(with the same minimizers) and 3. and it is arbitrarily close to the given
potential111One can think about the analogy between this property and the
classical results for Morse functions (i. e., functions without degenerate
critical points), which state that such type of functions are generic.. The
most important consequence of this assumption, as proven in [67], is the
existence of two constants $\rho_{0}^{+}>0$ and $\rho_{0}^{-}$ such that

$\displaystyle\forall q\in
L^{2}_{\mathrm{loc}}({\mathbb{R}},{\mathbb{R}}^{k}),$ $\displaystyle\hskip
5.69054pt\mathrm{dist}_{L^{2}({\mathbb{R}},{\mathbb{R}}^{k})}(q,\mathcal{F}^{\pm})\leq\rho_{0}^{\pm}$
(2.9) $\displaystyle\Rightarrow\exists!\tau^{\pm}(q)\in{\mathbb{R}}:\lVert
q-\mathfrak{q}^{\pm}(\cdot+\tau^{\pm}(q))\rVert_{L^{2}({\mathbb{R}},{\mathbb{R}}^{k})}=\mathrm{dist}_{L^{2}({\mathbb{R}},{\mathbb{R}}^{k})}(q,\mathcal{F}^{\pm})$
(2.10)

and for some constant $\beta^{\pm}$ we have

$\displaystyle\forall q\in X(\sigma^{-},\sigma^{+}),$ $\displaystyle\hskip
5.69054pt\mathrm{dist}_{H^{1}({\mathbb{R}},{\mathbb{R}}^{k})}(q,\mathcal{F}^{\pm})\leq\rho_{0}^{\pm}\Rightarrow\mathrm{dist}_{H^{1}({\mathbb{R}},{\mathbb{R}}^{k})}(q,\mathcal{F}^{\pm})^{2}\leq\beta^{\pm}(E(q)-\mathfrak{m}^{\pm}).$
(2.11)

In fact, in [67] this is only proven for global minimizers but the proof
readily extends to local ones as well. Notice that (2.9) and (2.11) state that
the energy is quadratic around $\mathcal{F}^{-}$ and $\mathcal{F}^{+}$, which
is the infinite-dimensional analogue of (H3), taking into account the
degeneracy generated by the group of translations. We will define for $r>0$
the sets

$\mathcal{F}^{\pm}_{r}:=\\{q\in
L^{2}_{\mathrm{loc}}({\mathbb{R}},{\mathbb{R}}^{k}):\mathrm{dist}_{L^{2}({\mathbb{R}},{\mathbb{R}}^{k})}(q,\mathcal{F}^{\pm})\leq
r\\},$ (2.12)

so that we can assume without loss of generality that
$\mathcal{F}^{+}_{\rho_{0}^{+}}\cap\mathcal{F}^{-}_{\rho^{-}_{0}}=\emptyset$.
See Figure 2.1 for an explanatory design of (H5). Let us now assume that
$\mathfrak{m}^{+}-\mathfrak{m}^{-}$ is bounded above as follows:

Figure 2.1: Situation described by (H5). The curves correspond to the traces
of $\mathfrak{q}^{-}$ and $\mathfrak{q}^{+}$ as indicated. The shadowed
regions correspond to the traces of the functions in
$\mathcal{F}^{-}_{\rho_{0}^{-}}$ and $\mathcal{F}^{+}_{\rho_{0}^{+}}$, which
are a neighborhood of $\mathcal{F}^{-}$ and $\mathcal{F}^{+}$ respectively.

######  (H6).

Assume that (H5) holds and, moreover,

$0<\mathfrak{m}^{+}-\mathfrak{m}^{-}<E_{\max},$ (2.13)

where $E_{\max}$ will be defined later in (2.46). Moreover, assume that

$\\{q\in
X(\sigma^{-},\sigma^{+}):E(q)<\mathfrak{m}^{+}\\}\subset\mathcal{F}^{-}_{\rho^{-}_{0}/2},$
(2.14)

with $\mathcal{F}^{-}_{\rho^{-}_{0}/2}$ as in (2.12).

See also Figure 2.2. Essentially, (H6) requires that
$\mathfrak{m}^{-}-\mathfrak{m}^{+}$ is not too large and the bound is given by
a constant $E_{\max}$ that can be computed through the constants produced in
(2.9) and (2.11) as a consequence of (H5). If (H6) holds, then we are able to
answer the question that we posed at the beginning of the paper in a positive
way. More precisely, recall the equation of the profile:

$-c\partial_{x_{1}}\mathfrak{U}-\Delta\mathfrak{U}=-\nabla_{u}V(\mathfrak{U})\mbox{
in }{\mathbb{R}}^{2}$ (2.15)

and consider the conditions at infinity

$\exists L^{-}\in{\mathbb{R}},\forall x_{1}\leq L^{-},\hskip
5.69054pt\mathfrak{U}(x_{1},\cdot)\in\mathcal{F}^{-}_{\rho^{-}_{0}/2},$ (2.16)
$\exists L^{+}\in{\mathbb{R}},\forall x_{1}\geq L^{+},\hskip
5.69054pt\mathfrak{U}(x_{1},\cdot)\in\mathcal{F}^{+}_{\rho^{+}_{0}/2}$ (2.17)

As stated before, our proof is variational, which implies that the profile
$\mathfrak{U}$ can be characterized as a critical point of a functional. The
variational framework is as follows: assume that (H6) holds and set

$\displaystyle S:=\\{U\in
H^{1}_{\mathrm{loc}}({\mathbb{R}},L^{2}({\mathbb{R}},{\mathbb{R}}^{k})):\exists
L\geq 1,$ $\displaystyle\forall x_{1}\geq L,\hskip
5.69054ptU(x_{1},\cdot)\in\mathcal{F}^{+}_{\rho_{0}^{+}/2}$ (2.18)
$\displaystyle\forall x_{1}\leq-L,\hskip
5.69054ptU(x_{1},\cdot)\in\mathcal{F}^{-}_{\rho^{-}_{0}/2}\\}.$ (2.19)

For $U\in S$ and $c>0$ we define the energy

$E_{2,c}(U):=\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\frac{\lvert\partial_{x_{1}}U(x_{1},x_{2})\rvert^{2}}{2}dx_{2}+(E(U(x_{1},\cdot))-\mathfrak{m}^{+})\right)e^{cx_{1}}dx_{1}.$
(2.20)

Formally, critical points of $E_{2,c}$ give rise to solutions of (2.15). If
$U\in S$, we can define the translated function
$U^{\tau}:=U(\cdot+\tau,\cdot)$ for $\tau\in{\mathbb{R}}$. Then, for all $c>0$
we have

$E_{2,c}(U^{\tau})=e^{-c\tau}E_{2,c}(U)$ (2.21)

which implies that

$\forall c>0,\hskip 5.69054pt\inf_{U\in S}E_{2,c}(U)\in\\{-\infty,0\\}.$
(2.22)

We have by now introduced the notations which allow us to state the main
result of this paper:

######  Theorem 1 (Main Theorem).

Assume that (H6) holds. Then, we have:

  1. 1.

Existence. There exist $c^{\star}>0$ and
$\mathfrak{U}\in\mathcal{C}^{2,\alpha}({\mathbb{R}}^{2},{\mathbb{R}}^{k})\cap
S$, $\alpha\in(0,1)$, which fulfill (2.15). The profile $\mathfrak{U}$
satisfies the conditions at infinity (2.16) and (2.17) as well as the
variational characterization

$E_{2,c^{\star}}(\mathfrak{U})=0=\inf_{U\in S}E_{2,c^{\star}}(U).$ (2.23)

  2. 2.

Uniqueness of the speed. The speed $c^{\star}$ is unique in the following
sense: Assume that $\overline{c^{\star}}>0$ is such that

$\inf_{U\in S}E_{2,\overline{c^{\star}}}(U)=0$ (2.24)

and that $\overline{\mathfrak{U}}\in S$ is such that
$(\overline{c^{\star}},\overline{\mathfrak{U}})$ solves (2.15) and
$E_{2,\overline{c^{\star}}}(\overline{\mathfrak{U}})<+\infty$. Then,
$\overline{c^{\star}}=c^{\star}$.

  3. 3.

Exponential convergence. The convergence of $\mathfrak{U}$ at $+\infty$ is
exponential with respect to the $L^{2}$-norm. More precisely, there exists
$\mathfrak{M}^{+}>0$ and $\tau^{+}\in{\mathbb{R}}$ such that for all
$x_{1}\in{\mathbb{R}}$

$\lVert\mathfrak{U}(x_{1},\cdot)-\mathfrak{q}^{+}(\cdot+\tau^{+})\rVert_{L^{2}({\mathbb{R}},{\mathbb{R}}^{k})}\leq\mathfrak{M}^{+}e^{-c^{\star}t}.$
(2.25)

######  Remark 2.1.

The existence part of Theorem 1 states that there exists a solution
$(c^{\star},\mathfrak{U})$ such that $\mathfrak{U}$ is a global minimizer of
$E_{c^{\star}}$ in $S$. We also have that the speed $c^{\star}$ is unique for
some class of solutions, namely for finite energy solutions and speeds for
which the corresponding energy is bounded below in $S$. In particular,
$c^{\star}$ is unique among the class of globally minimzing profiles. In other
words, if $c>0$ is such that the infimum of $E_{c}$ in $S$ is attained, then
$c=c^{\star}$. This is analogous to what it was shown in Alikakos and
Katzourakis [7]. As explained in the introduction, the main drawback of our
approach is the existence assumption (H6). In particular, the definition of
the upper bound $E_{\max}$ is technical and it is possible that in several
situations it could be small. Nevertheless, in subsection 2.7 we show that
there exists examples of potentials for which (H6) holds.

Figure 2.2: Representation of (H6). While the larger shadowed region
corresponds to $\mathcal{F}^{-}_{\rho_{0}^{-}/2}$, the smaller one which is
contained inside represents the set $\\{q\in
X(\sigma^{-},\sigma^{+}):E(q)<\mathfrak{m}^{+}\\}$. Moreover, the value
$\mathfrak{m}^{+}-\mathfrak{m}^{-}$ must be smaller than $E_{\max}$, defined
in (2.46).

###  2.3 Conditions at infinity

The solutions given by Theorem 1 satisfy the conditions at infinity (2.16) and
(2.17). As we can see, condition (2.16) is more imprecise than expected, as it
only states that $\mathfrak{U}(x_{1},\cdot)$ is not far from $\mathcal{F}^{-}$
with respect to the $L^{2}$ distance when $x_{1}$ is close enough to
$-\infty$. In particular, we cannot ensure that $\mathfrak{U}$ is really
heteroclinic, in the sense of connecting two stable states as
$x_{1}\to\pm\infty$. Therefore, it is reasonable to wonder if we can establish
a behavior of the type

$\exists\tau^{-}\in{\mathbb{R}},\hskip
5.69054pt\lVert\mathfrak{U}(x_{1},\cdot)-\mathfrak{q}^{-}(\cdot+\tau^{-})\rVert_{L^{2}({\mathbb{R}},{\mathbb{R}}^{k})}\to
0\mbox{ as }x_{1}\to-\infty,$ (2.26)

so that $\mathfrak{U}$ is an actual heteroclinic. While (2.26) was established
in [7], their argument does not seem to apply to the infinite-dimensional
setting, which means that new ideas are needed. We have been able to show that
(2.26) holds under the following additional assumption:

######  (H7).

We have that assumption (H6) holds and, additionally:

$\mathfrak{m}^{+}-\mathfrak{m}^{-}<\frac{(\mu^{-}\mathfrak{d}_{0})^{2}}{2},$
(2.27)

where the constants $\mathfrak{d}_{0}$ and $\mu^{-}$ are defined later in
(2.39) and (2.41) respectively.

Assumption (H7) is not too restrictive (at least with respect to the
assumptions we already have), since an upper bound on
$\mathfrak{m}^{+}-\mathfrak{m}^{-}$ is already imposed in (H6), meaning that
at worst one only needs to lower it. The definition of the constant
$\mathfrak{d}_{0}$ is essentially technical depends only on the distance
between the sets $\mathcal{F}^{-}$ and $\mathcal{F}^{+}$, while $\mu^{-}$
depends only on local information around $\mathcal{F}^{-}$. Anyway, the result
given by (H7) writes as follows:

######  Theorem 2.

Assume that (H6) and (H7) hold. Let $(c^{\star},\mathfrak{U})$ be the solution
given by Theorem 1. Then, $\mathfrak{U}$ satisfies the stronger condition
(2.26). Moreover, it holds that $c^{\star}<\mu^{-}$, $\mu^{-}$ to be defined
later in (2.41), and there exists $\mathfrak{M}^{-}>0$ such that for all
$x_{1}\in{\mathbb{R}}$

$\lVert\mathfrak{U}(x_{1},\cdot)-\mathfrak{q}^{-}(\cdot+\tau^{-})\rVert_{L^{2}({\mathbb{R}},{\mathbb{R}}^{k})}\leq\mathfrak{M}^{-}e^{(\mu^{-}-c^{\star})x_{1}}$
(2.28)

The natural question is whether (2.25) in Theorem 1 and (2.28) in Theorem 2
can be improved. In particular, whether the $L^{2}$-norm can be replaced by
the $H^{1}$-norm. We conjecture that the answer to this question is positive,
but we do not have a proof of this fact. However, as one can check in
Smyrnelis [69] and Fusco [42], such a fact holds for the balanced 2D
heteroclinic solution. They obtain these properties by combining standard
elliptic estimates with some properties which are intrinsic to minimal
solutions of the elliptic system (1.3) with $c^{\star}=0$. See the results of
section 4 in Alikakos, Fusco and Smyrnelis [6], mainly based on Alikakos and
Fusco [4]. The main obstacle is that even if one was able to extend their
analysis to the case $c^{\star}>0$, a crucial hypothesis of in their results
is that solutions are minimal with respect to compactly supported
perturbations, but the solution of Theorem 1 is only locally minimizing (due
to the fact that $\mathfrak{q}^{+}$ is a local minimizer of the 1D energy).
Therefore, we leave this question open. Nevertheless, besides the
$L^{2}$-convergence rates (2.25) and (2.28) on can prove uniform convergence
both in the $x_{1}$ and the $x_{2}$ direction:

######  Theorem 3.

Assume that (H6) holds. Let $(c^{\star},\mathfrak{U})$ be the solution given
by Theorem 1. Then, we have that

$\lim_{x_{1}\to+\infty}\lVert\mathfrak{U}(x_{1},\cdot)-\mathfrak{q}^{+}(\cdot+\tau^{+})\rVert_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{k})}=0$
(2.29)

and for all $L\in{\mathbb{R}}$ we have

$\lim_{x_{2}\to\pm\infty}\lVert\mathfrak{U}(\cdot,x_{2})-\sigma^{\pm}\rVert_{L^{\infty}([L,+\infty),{\mathbb{R}}^{k})}=0.$
(2.30)

If, moreover, (H7) holds, then we have

$\lim_{x_{1}\to-\infty}\lVert\mathfrak{U}(x_{1},\cdot)-\mathfrak{q}^{-}(\cdot+\tau^{-})\rVert_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{k})}=0$
(2.31)

and (2.30) can be improved into

$\lim_{x_{2}\to\pm\infty}\lVert\mathfrak{U}(\cdot,x_{2})-\sigma^{\pm}\rVert_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{k})}=0.$
(2.32)

###  2.4 Min-max characterization of the speed

We provide here a min-max characterization of the speed $c^{\star}$ and other
related properties which are summarized in Theorem 4. The idea of providing a
variational characterization for the speed of traveling waves in reaction-
diffusion systems can be traced back to Heinze [46], Heinze, Papanicolau and
Stevens [47] and it was used later in several other papers [7, 31, 52, 53,
58].

######  Theorem 4.

Assume that (H6) and (H7) hold. Let $(c^{\star},\mathfrak{U})$ be the solution
given by Theorem 1. Then for any $\tilde{\mathfrak{U}}\in S_{i}$ such that

$E_{2,c^{\star}}(\tilde{\mathfrak{U}})=0$ (2.33)

we have that $(c^{\star},\tilde{\mathfrak{U}})$ solves (2.15) and

$c^{\star}=\frac{\mathfrak{m}^{+}-\mathfrak{m}^{-}}{\int_{{\mathbb{R}}^{2}}\lvert\partial_{x_{1}}\tilde{\mathfrak{U}}(x_{1},x_{2})\rvert^{2}dx_{2}dx_{1}}.$
(2.34)

In particular, the quantity
$\int_{{\mathbb{R}}^{2}}\lvert\partial_{x_{1}}\tilde{\mathfrak{U}}(x_{1},x_{2})\rvert^{2}dx_{2}dx_{1}$
is well-defined and constant among the set of minimizers of $E_{2,c}$ in $S$.
Moreover, it holds

$c^{\star}=\sup\\{c>0:\inf_{U\in S}E_{2,c}(U)=-\infty\\}=\inf\\{c>0:\inf_{U\in
S}E_{2,c}(U)=0\\}$ (2.35)

and we have the bound

$c^{\star}\leq\frac{\sqrt{2(\mathfrak{m}^{+}-\mathfrak{m}^{-})}}{\mathfrak{d}_{0}}<\min\left\\{\frac{\sqrt{2E_{\max}}}{\mathfrak{d}_{0}},\mu^{-}\right\\}$
(2.36)

where $\mathfrak{d}_{0}$, $\mu^{-}$ and $E_{\max}$ will be defined later in
(2.39), (2.41) and (2.46) respectively and the second inequality follows from
the bound on $\mathfrak{m}^{+}-\mathfrak{m}^{-}$ given by (H6) and (H7).

######  Remark 2.2.

Notice that the conditions at infinity imply that any $U\in S$ is such that

$\int_{{\mathbb{R}}^{2}}\frac{\lvert\partial_{x_{1}}U(x_{1},x_{2})\rvert^{2}}{2}dx_{2}dx_{1}>0$
(2.37)

As it can be seen, Theorem 4 shows that the speed $c^{\star}$ is characterized
by the explicit formula (2.34), which nevertheless requires knowledge on a
profile $\tilde{\mathfrak{U}}$. However, one also has the variational
characterization (2.35), which does not involve any information on the
profiles. Indeed, one only needs to be able to compute the infimum of the
energies with $c>0$ as a parameter. Moreover, notice that combining (2.35)
with the uniqueness part of Theorem 1, we obtain that if
$\overline{c}>c^{\star}$ and $(\overline{c},\overline{\mathfrak{U}})$, with
$\overline{\mathfrak{U}}\in S$, solves (2.15), then
$E_{2,\overline{c}}(\overline{\mathfrak{U}})=+\infty$, which is actually a
contradiction. On the contrary, if we take $\overline{c}<c^{\star}$, then
(2.35) implies that $\inf_{U\in S}E_{2,\overline{c}}(U)=-\infty$, meaning that
Theorem 1 does not apply and nothing else can be said.

###  2.5 Definition of the upper bounds

We will now define some important numerical constants which are necessary in
order to formulate assumptions (H6) and (H7). Assume first that (H5) holds.
Let $\rho_{0}^{\pm}$ as in (2.9) and (2.11). Recall that we chose
$\rho_{0}^{+}$ and $\rho_{0}^{-}$ such that

$\mathcal{F}^{+}_{\rho_{0}^{+}}\cap\mathcal{F}^{-}_{\rho_{0}^{-}}=\emptyset$
(2.38)

and, since those two sets (see the definition in (2.12)) are $L^{2}$-closed
due to the local compactness of the sets $\mathcal{F}^{-}$ and
$\mathcal{F}^{+}$, we have that

$\mathfrak{d}_{0}:=\mathrm{dist}_{L^{2}({\mathbb{R}},{\mathbb{R}}^{k})}(\mathcal{F}^{+}_{\rho_{0}^{+}/2},\mathcal{F}^{-}_{\rho_{0}^{-}/2})$
(2.39)

is positive. Therefore, as we advanced before, one can see that the constant
$\mathfrak{d}_{0}$ depends only on the distance between the two families of
minimizing heteroclincs. Next, under (H5), recall the constants $\beta^{\pm}$
from (2.11). Set

$\overline{\beta^{\pm}}:=\frac{1}{2}(\beta^{\pm})^{2}((\beta^{\pm})^{2}+(\beta^{\pm}+1)^{2})>0$
(2.40)

and, subsequently

$\mu^{-}:=\frac{1}{\beta^{-}+\overline{\beta^{-}}}>0$ (2.41)

which is the constant appearing in (H7). Of course, the nature of the
definition given in (2.41) obeys to technical considerations. But $\mu^{-}$
should be thought as a constant depending only on the local behavior of the
energy around $\mathcal{F}^{-}$ and, in particular, independent on the
behavior of the energy near $\mathcal{F}^{+}$. Now let for
$r\in(0,\rho^{\pm}_{0}]$

$\mathfrak{e}^{\pm}_{r}:=\inf\\{E(q):q\in
X(\sigma^{-},\sigma^{+}),\mathrm{dist}_{L^{2}({\mathbb{R}},{\mathbb{R}}^{k})}(q,\mathcal{F}^{\pm})\in[r,\rho^{\pm}_{0}]\\}.$
(2.42)

Known results which follow from compactness of minimizing sequences (see for
instance Schatzman [67]) imply that $\mathfrak{e}^{\pm}_{r}>0$. Moreover, we
also have that for $r\in(0,\rho^{\pm}_{0}]$ there exists $\nu^{\pm}(r)>0$ such
that

$\forall q\in\mathcal{F}^{\pm}_{\rho_{0}^{\pm}/2},\hskip
5.69054ptE(q)-\mathfrak{m}^{\pm}\leq\nu^{\pm}(r)\Rightarrow\mathrm{dist}_{H^{1}({\mathbb{R}},{\mathbb{R}}^{k})}(q,\mathcal{F}^{\pm})\leq
r.$ (2.43)

This leads to define the constants

$\delta_{0}^{-}:=\min\left\\{\sqrt{e^{-1}\frac{\rho_{0}^{-}}{4}\sqrt{2(\mathfrak{e}^{-}_{\rho_{0}^{-}/4}-\mathfrak{m}^{-})}},\frac{\rho_{0}^{-}}{4}\right\\}>0,$
(2.44) $\mathfrak{r}^{-}:=\frac{\rho_{0}^{-}}{\beta^{-}+1}>0$ (2.45)

and

$E_{\max}:=\frac{1}{(\beta^{-})^{2}(\beta^{-}+1)}\min\left\\{\frac{(\delta_{0}^{-})^{2}}{4},\mathfrak{e}^{-}_{\delta_{0}^{-}}-\mathfrak{m}^{-},\nu^{-}(\mathfrak{r}^{-}),\nu^{-}(\delta_{0}^{-})\right\\}>0$
(2.46)

which is the constant appearing in (H6). Again, the definition on $E_{\max}$
is essentially due to technical reasons, but it must be thought as a constant
which only depends on local information around $\mathcal{F}^{-}$.

###  2.6 Methods and ideas of the proofs

The main result of this paper is Theorem 1, which establishes the existence of
a solution $(c^{\star},\mathfrak{U})$, with the profile $\mathfrak{U}$
satisfying the heteroclinic asymptotic conditions (2.16), (2.17). We also
prove an exponential rate of convergence for the profile at $+\infty$ (with
respect to the $L^{2}$-norm). We finally show that the speed $c^{\star}$ has
some uniqueness properties. Important properties on the profile and the speed,
as well as improvements on the results under additional assumptions, are also
established in Theorems 2, 3 and 4.

As already stated, the proof of our results follows by bringing together two
different lines of research, see items 1. and 2. in the introduction. More
precisely, in the spirit of [57, 67], we adapt the result of Alikakos and
Katzourakis [7] (actually, we rather follow more closely the simplified
version given in Alikakos, Fusco and Smyrnelis [6]) to potentials defined in
an abstract, possibly infinite-dimensional, Hilbert space and possessing two
local minima at different levels. This abstract setting is established in
section 4 and the main abstract results are Theorems 5, 6 and 7. The proof of
these results is found in section 5. Assumption (H5) guarantees that our main
results (Theorems 1-4) are a particular case of the abstract results.
Naturally, the advantage of proving the results in an abstract framework is
that one can apply them to several problems different than the original one.
In our case, the results in this paper apply to the 1D system

$\partial_{t}w-\partial_{x}^{2}w=-\nabla_{u}W(w)\mbox{ in
}[0,+\infty)\times{\mathbb{R}},$ (2.47)

where $W$ is a smooth potential bounded below possessing two local and non-
degenerate minima at different levels. As said before, this is formally the
system considered in [7], but the results of this paper allow to somewhat
relax the non-degeneracy assumption used in [7]. More details, as well as
other extensions, are given in our companion paper [59].

Generalizing the result from [7] for curves taking values in a more general,
possibly infinite-dimensional, Hilbert space raises several additional
difficulties. A detailed outline of our proof is given in subsection 5.1, but
let us here try to motivate the main difficulties of the problem we are
facing.

As pointed out before, the approach in [7] is variational. A family of
weighted energy functionals (essentially those introduced in Fife and McLeod
[40, 39]) depending on a speed parameter $c>0$ is considered. In order to make
the functionals well defined in the space of curves that connect the minima,
the global minimum of the potential must be negative and the local one must be
zero, which is always true up to an additive constant. As a consequence, one
deals with an energy density which changes sign, which is in contrast with the
equal depth (balanced) case, for which the energy is always non-negative.
Recall that finite energy 1D connecting heteroclinics between two wells at the
same level must be stationary. Another difficulty of the heteroclinic
traveling wave existence problem comes from the fact that not only the profile
but also the speed of the wave is an unknown as well.

The method used in [7] is an adaptation of that introduced in Alikakos and
Fusco [5] for the equal depth case. This method consists on considering
families of solutions with prescribed behavior outside an interval of length
$2T$ (namely, they are forced to stay close to the respective minimum) and
minimizing the weighted energy functionals seeing the speed as parameter.
Since compactness is restored due to the constrains, the problem has a
solution for each $c>0$ and $T\geq 1$. These ideas can be adapted to our
setting without major difficulty. The next step consists on determining the
solution speed $c^{\star}$ and then showing that for $c^{\star}$ and a
suitable $T$ the corresponding constrained minimizer does not meet the
constraints, meaning that it is an actual solution. Since the energy
functionals change sign, one needs to show that the constrained minimizers do
not oscillate between positive and negative regions of the energy (which would
produce compensations) inside arbitrarily long intervals as $T$ goes to
infinity. In order to show that, the authors in [7] assume that the local
minima are isolated and that in the negative region of the functional one has
strict radial monotonicity with respect to the global minimum. Subsequently,
they use this property in combination with the ODE system and minimality
arguments in order to exclude oscillations.

For several reasons, the previous idea does not seem available in our setting
without substantial modifications. Despite the fact that, as we show in [59],
one can adapt the assumption of [7] for potentials in infinite-dimensional
spaces with possibly degenerate minima, in this case we have trouble showing
that our original problem can be put as a particular case of the abstract one.
In other words, it does not seem reasonable to expect that such adaptation of
the radial monotonicity assumption of [7] would be met in our original
problem. Indeed, one would need to prove some kind of radial monotonicity for
the energy $E$ (see (2.2)), in some suitable subset. We think that this might
be too restrictive and we cannot prove it even for simple explicit examples.
The difficulty comes from the fact that, while in the finite-dimensional case
one can directly modify the potential, the level sets of $E$ depend on a
rather indirect way on the potential $V$ and they are infinite-dimensional
manifolds. Therefore, the most important difficulty of our problem is to
replace the radial monotonicity assumption of [7] by another one which can be
met in our situation and which allows to obtain a similar type of conclusion
(namely, exclude oscillatory behavior for the constrained minimizers in
arbitrarily large intervals). We have been able to provide one assumption,
(H6), which plays this role. It consists on imposing an upper bound on the
difference between the energy levels. This upper bound is (the abstract
version of) the constant $\mathcal{E}_{\max}$, defined in (2.46), subsection
2.5. It enables us to exclude oscillations on the minimizers because the
(renormalized) energy is positive outside the region in which the solution is
constrained. Once oscillations are excluded, we conclude the proof as in [7].

The main drawback of our proof is that the computation of the upper bound in
(H6) is not straightforward and it obeys technical considerations, as the
definition of $\mathcal{E}_{\max}$ (2.46) in subsection 2.5 shows. In
particular, we cannot exclude the possibility that $\mathcal{E}_{\max}$ is
small. However, in subsection 2.7 we give a general method in order to obtain
potentials for which the corresponding energy functional satisfies the bound
assumption (H6). Essentially, one considers a potential for which two
different globally minimizing heteroclinics exist (which implies $k\geq 2$)
and then modifies it in a suitable manner. It would be also interesting to
know whether our assumption (H6) is only technical or rather there is some
kind of obstruction for existence when the difference between the energy of
the heteroclinics is too large. We think that the answer to this question is
possibly related to the loss of compactness for the 1D energy functional (see
[60] and the references therein) and hence we conjecture that (H6) is not only
technical (although it is likely non-optimal) and that in its absence some
counterexamples might be found. It is also reasonable to conjecture that the
removal of (H6) would imply the existence of traveling waves with more
complicated behavior at infinity, for example approaching chains of connecting
orbits (heteroclinic or homoclinic) stable in some suitable sense.

At the final stage of the proof, one needs to ensure that the solution
obtained presents the suitable heteroclinic behavior at infinity. Moreover, we
want to obtain more refined convergence results, see Theorems 1 and 2. This
asymptotic analysis is delicate, as finite energy functions do not necessarily
converge at all $-\infty$ and they converge exponentially at $+\infty$ but
only with respect to the $L^{2}$-norm. Moreover, one needs to distinguish
between $L^{2}$ convergence (which is weaker and does not imply convergence of
the energy) and $H^{1}$ convergence. We deal with all these difficulties by
using minimality of the solution (inspired for instance by [69]), which allows
us to obtain the convergence at $-\infty$, exponentially with respect to the
$L^{2}$-norm. Assumption (H7) is needed in order to obtain the convergence at
$-\infty$. Moreover, working in the main setting, we obtain Theorem 3, which
shows that convergence is not only $L^{2}$ but also $L^{\infty}$, and not only
according to $x_{1}$ but also $x_{2}$, the limit in this case being the wells
$\sigma^{\pm}$.

We will use extensively the fact that the energy has good properties at least
on a neighborhood of the 1D minimizing heteroclinics. Essentially, those are
the results and assumptions made by Schatzman [67], which in our case are
given in (H5) and the discussion that follows. To our knowledge, these
properties have not been shown to hold for the 2D heteroclinic solutions of
[2, 67] and one could expect that some of them do not hold. This represents,
in our opinion, a (momentary) obstruction to establishing the existence of 3D
heteroclinic traveling waves connecting two 2D heteroclinics. Moreover, notice
that the fact that $V$ is multi-well implies that 1D heteroclinic traveling
waves do not exist in general, unless one imposes the existence of a local
minimum at a level higher than 0 and some other properties are verified.

###  2.7 Examples of potentials verifying the assumptions

In order to conclude this section, we exhibit a rather general and elementary
method in order to produce examples of potentials for which the assumptions we
make in this paper are satisfied. As we advanced before, the idea is to modify
a given multi-well potential $V_{0}:{\mathbb{R}}^{k}\to{\mathbb{R}}$
satisfying (H1), (H2) and (H3) such that the associated energy possesses two
minimizing heteroclinics (up to translations) for two given wells
$\sigma^{-}$, $\sigma^{+}$ in a finite set $\Sigma$. We also assume that the
strict triangle’s inequality (H4) is met for $V_{0}$ with respect to
$(\sigma^{-},\sigma^{+})$. Furthermore, we assume that the generic Schatzman’s
spectral assumption ([67]) is satisfied for those heteroclinics, meaning that
the constants defined in subsection 2.5 (with the obvious modifications) also
make sense here. That is, one can think of any potential $V_{0}$ satisfying
the assumptions of Schatzman’s paper [67]. For the reader’s convenience, we
shall give here some explicit examples of such potentials which we found on
the literature.

The first of the examples we give was found by Antonopoulos and Smyrnelis, see
Remark 3.6 in [9]. Consider the case $k=2$. Let $V_{GL}$ be the Ginzburg-
Landau potential

$V_{GL}:u=(u_{1},u_{2})\in{\mathbb{R}}^{2}\to\frac{(1-\lvert
u\rvert^{2})^{2}}{4}\in{\mathbb{R}}$ (2.48)

and consider the corresponding energy

$E_{GL}(q):=\int_{\mathbb{R}}\left[\frac{\lvert
q^{\prime}(t)\rvert^{2}}{2}+\frac{(1-\lvert
q(t)\rvert^{2})^{2}}{4}\right]dt,\hskip 5.69054ptq\in
H^{1}_{\mathrm{loc}}({\mathbb{R}},{\mathbb{R}}^{2}).$ (2.49)

The idea is to perturb $V_{GL}$ in order to obtain a double-well potential
with zero set $\\{(-1,0),(1,0)\\}$ and symmetric with respect to the axis
$\\{u_{2}=0\\}$. Such a potential will possess two heteroclinics provided that
any curve with trace in $\\{u_{2}=0\\}$ can be beaten by a competitor with a
trace that is not contained in this set. Notice that for all
$(u_{1},0)\in{\mathbb{R}}\times\\{0\\}$ we have that
$V_{GL}(u_{1},0)=(1-u_{1}^{2})^{2}/4$, which is the standard scalar double-
well potential. As it is well known, the (unique) heteroclinic for such
potential is given by the odd function
$\mathfrak{q}_{AC}:t\in{\mathbb{R}}\to\tanh(t/\sqrt{2})\in{\mathbb{R}}$.
Therefore, each curve $q=(q_{1},q_{2})$ in
$H^{1}_{\mathrm{loc}}({\mathbb{R}},{\mathbb{R}}^{2})$ with $q_{2}=0$ and
$\lim_{t\to\pm\infty}q_{1}(t)=\pm 1$ verifies

$E_{GL}(q)\geq E_{GL}(\mathfrak{q}_{AC})=\frac{2\sqrt{2}}{3}.$ (2.50)

For $T>0$, define

$q_{T}(t):=\begin{cases}(-1,0)&\mbox{ if }t\leq-T-1,\\\
((t+T)+(t+T+1)\mathfrak{q}_{AC}(-T),0)&\mbox{ if }-T-1\leq t\leq-T,\\\
-\mathfrak{q}_{AC}(T)(\cos(\pi(t+T)/(2T)),\sin(\pi(t+T)/(2T)))&\mbox{ if
}-T\leq t\leq T,\\\ ((t-T)-(t-T-1)\mathfrak{q}_{AC}(T),0)&\mbox{ if }T\leq
t\leq T+1,\\\ (1,0)&\mbox{ if }T+1\leq t.\end{cases}$ (2.51)

A modification of the computations made in [9] shows that

$\lim_{T\to+\infty}E_{GL}(q_{T})=0$ (2.52)

meaning that by (2.50) there exists $\overline{T}>0$ such that
$E_{GL}(q_{\overline{T}})<E_{GL}(\mathfrak{q}_{AC})$. Then, given
$\varepsilon:=(1-\lvert\mathfrak{q}_{AC}(\overline{T})\rvert^{2})/4$, consider
$\phi\in\mathcal{C}^{\infty}({\mathbb{R}},[0,+\infty))$ such that

$\phi(t)=\begin{cases}0&\mbox{ if
}t\leq\lvert\mathfrak{q}_{AC}(\overline{T})\rvert^{2}+\varepsilon\\\ 1&\mbox{
if }t\geq 1-\varepsilon\end{cases}$ (2.53)

and define $\tilde{V}_{0}:u=(u_{1},u_{2})\in{\mathbb{R}}^{2}\to
V_{GL}(u)+u_{2}^{2}\phi(\lvert u\rvert^{2})$. Let $\tilde{E}_{0}$ be the
corresponding energy. Notice that $\tilde{V}_{0}$ is a double-well potential
verifying (H1), (H2) and (H3). By definition, we have that if
$q=(q_{1},q_{2})\in H^{1}([-R,R],{\mathbb{R}}^{2})$ is such that $q_{2}=0$,
then $\tilde{E}_{0}(q)=E_{GL}(q)$. Moreover, we also have
$\tilde{E}_{0}(q_{\overline{T}})=E_{GL}(q_{\overline{T}})<E_{GL}(\mathfrak{q}_{AC})$.
As a consequence, the minimizer
$\mathfrak{q}=(\mathfrak{q}_{1},\mathfrak{q}_{2})$ of $\tilde{E}_{0}$ in the
class of curves in $H^{1}_{\mathrm{loc}}({\mathbb{R}},{\mathbb{R}}^{k})$ which
tend to $(\pm 1,0)$ at $\pm\infty$ satisfies $\mathfrak{q}_{2}\not=0$, which
means that $\hat{\mathfrak{q}}:=(\mathfrak{q}_{1},-\mathfrak{q}_{2})$ is also
a minimizer due to the symmetry of $\tilde{V}_{0}$ and $\hat{\mathfrak{q}}$ is
not a translation of $\mathfrak{q}$. Therefore, $\tilde{V}_{0}$ possesses two
geometrically distinct globally minimizing heteroclinics. In order to find our
example of potential, we need that such heteroclinics are non-degenerate in
the sense asked by Schatzman in [67], see our 3. in (H5). However, as shown in
her Theorem 4.3 such assumption is generic, i. e., we can find $V_{0}$
arbitrarily close to $\tilde{V}_{0}$ which is still a double-well potential
with wells $(-1,0)$, $(1,0)$ and with $\mathfrak{q}$ and $\hat{\mathfrak{q}}$
non-degenerate globally minimizing heteroclinics which satisfy the spectral
assumptions.

Another example, this time in dimension $k=3$, is provided by Zuñiga and
Sternberg [72]. They consider the potential

$\tilde{V}_{0}:u=(u_{1},u_{2},u_{3})\to
u_{1}^{2}(1-u_{1}^{2})^{2}+\left(u_{2}^{2}-\frac{1}{2}(1-u_{1}^{2})^{2}\right)^{2}+\left(u_{3}^{2}-\frac{1}{2}(1-u_{1}^{2})^{2}\right)^{2}\in{\mathbb{R}},$
(2.54)

which vanishes exactly on the points

$\displaystyle(-1,0,0),(1,0,0),$ (2.55)
$\displaystyle\left(0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right),\left(0,-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right),\left(0,\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right),\left(0,-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right).$
(2.56)

By explicit computations, they show that the potential $\tilde{V}_{0}$
satisfies (H1), (H2), (H3) and (H4) with $\sigma^{\pm}:=(\pm 1,0,0)$ and,
moreover, that the infimum of the corresponding energy $\tilde{E_{0}}$ in
$X(\sigma^{-},\sigma^{+})$ is not attained by a curve with trace contained in
$\\{u_{2}=u_{3}=0\\}$. Using the reflections $(0,u_{2},0)\to(0,-u_{2},0)$ and
$(0,0,u_{3})\to(0,0,-u_{3})$, one deduces the multiplicity up to translations
of the globally minimizing heteroclinics for $\tilde{E}_{0}$ in
$X(\sigma^{-},\sigma^{+})$. As above, one can obtain $V_{0}$ arbitrarily close
to $\tilde{V}_{0}$ such that the globally minimizing heteroclinics satisfy the
spectral assumption.

Let us now return to the initial problem, and let $V_{0}$ be any potential
satisfying the previous assumptions. In order to obtain a potential which
satisfies the requirements of our setting, the idea is to make arbitrarily
small smooth perturbations of $V_{0}$ around the trace of one of the
heteroclinics, in such a way that its energy increases but a locally
minimizing heteroclinic still exists (at least for small perturbations), which
must necessarily have larger energy. One then chooses a perturbation which is
not too large so that the upper bound on the difference of the energies is
met. The idea is pictured in Figure 2.3. We now show how to rigorously
implement this idea. Let $\mathfrak{q}^{-}$ and $\mathfrak{q}^{+}$ in
$X(\sigma^{-},\sigma^{+})$ be different up to translations and such that

$E_{0}(\mathfrak{q}^{-})=E_{0}(\mathfrak{q}^{+})=\mathfrak{m}_{0}:=\inf_{q\in
X(\sigma^{-},\sigma^{+})}E_{0}(q),$ (2.57)

where, for $q\in H^{1}_{\mathrm{loc}}({\mathbb{R}},{\mathbb{R}}^{k})$

$E_{0}(q):=\int_{\mathbb{R}}\left[\frac{\lvert
q^{\prime}(t)\rvert^{2}}{2}+V_{0}(q(t))\right]dt.$ (2.58)

Recall that there exist $\rho_{0}^{\pm}$ such that

$\forall q\in X(\sigma^{-},\sigma^{+}),\hskip
5.69054pt\mathrm{dist}_{H^{1}({\mathbb{R}},{\mathbb{R}}^{k})}(q,\mathcal{F}^{\pm})\leq\rho_{0}^{\pm}\Rightarrow\mathrm{dist}_{H^{1}({\mathbb{R}},{\mathbb{R}}^{k})}(q,\mathcal{F}^{\pm})^{2}\leq\beta^{\pm}(E_{0}(q)-\mathfrak{m}_{0})$
(2.59)

where

$\mathcal{F}^{\pm}:=\\{\mathfrak{q}^{\pm}(\cdot+\tau):\tau\in{\mathbb{R}}\\}.$
(2.60)

Let $t_{0}\in{\mathbb{R}}$ be such that
$\mathrm{dist}(\mathfrak{q}^{+}(t_{0}),\Sigma)=\max_{t\in{\mathbb{R}}}\mathrm{dist}(\mathfrak{q}^{+}(t),\Sigma)$
for some $\mathfrak{q}^{+}\in\mathcal{F}^{+}$ and set
$u_{0}:=\mathfrak{q}^{+}(t_{0})$. Let

$r:=\min\\{\rho_{0}^{+}/2,\mathrm{dist}(\mathfrak{q}^{+}(t_{0}),\Sigma)/2\\}>0.$
(2.61)

Define $\chi\in\mathcal{C}^{\infty}_{c}({\mathbb{R}}^{k})$ be such that
$0\leq\chi\leq 1$, $\chi=1$ on $B(u_{0},r)$ and $\mathrm{supp}(\chi)\subset
B(u_{0},2r)$. For each $\delta>0$, consider the potential
$V_{\delta}:=V+\delta\chi\geq 0$. Define

$E_{\delta}(q):=\int_{\mathbb{R}}\left[\frac{\lvert
q^{\prime}(t)\rvert^{2}}{2}+V_{\delta}(q(t))\right]dt$ (2.62)

Notice that, by the choice of $\chi$, $V_{\delta}$ vanishes exactly in
$\Sigma$. Let now be $q\in X(\sigma^{-},\sigma^{+})$ such that
$\mathrm{dist}_{H^{1}({\mathbb{R}},{\mathbb{R}}^{k})}(q,\mathcal{F}^{+})\leq\rho_{0}^{+}/2$.
We have that

$\mathfrak{m}_{0}+\frac{1}{\beta^{+}}\mathrm{dist}_{H^{1}({\mathbb{R}},{\mathbb{R}}^{k})}(q,\mathcal{F}^{+})^{2}\leq
E_{0}(q)<E_{0}(q)+\delta\int_{\mathbb{R}}\chi(q)=E_{\delta}(q)$ (2.63)

and notice that for $q\in\mathcal{F}^{+}$ we have that
$E_{\delta}(q)=\mathfrak{m}_{0}+\delta\mathfrak{i}$ with

$\mathfrak{i}:=\int_{\mathbb{R}}\chi(\mathfrak{q}^{+}(t))dt>0.$ (2.64)

A contradiction argument shows that

$\mathfrak{m}^{+}_{\delta}:=\inf\\{E_{\delta}(q):\mathrm{dist}_{H^{1}({\mathbb{R}},{\mathbb{R}}^{k})}(q,\mathcal{F}^{+})\leq\rho_{0}^{+}/2\\}>\mathfrak{m}_{0}$
(2.65)

and we have $\mathfrak{m}^{+}_{\delta}\leq
E_{\delta}(\mathfrak{q}^{+})=\mathfrak{m}_{0}+\delta\mathfrak{i}$. Since the
cut-off function is supported away from $\Sigma$, we can show by the usual
concentration-compactness arguments that there exists
$\mathfrak{q}^{+}_{\delta}\in X(\sigma^{-},\sigma^{+})$ such that
$\mathrm{dist}_{H^{1}({\mathbb{R}},{\mathbb{R}}^{k})}(\mathfrak{q}^{+}_{\delta},\mathcal{F}^{+})\leq\rho_{0}^{+}/2$
and $E_{\delta}(\mathfrak{q}^{+}_{\delta})=\mathfrak{m}^{+}_{\delta}$. If we
show that
$\mathrm{dist}_{H^{1}({\mathbb{R}},{\mathbb{R}}^{k})}(\mathfrak{q}^{+}_{\delta},\mathcal{F}^{+})<\rho_{0}^{+}/2$,
then the constraints of the minimization problem are not saturated and
$\mathfrak{q}^{+}_{\delta}$ is an actual critical point. Notice that if $q\in
X(\sigma^{-},\sigma^{+})$ is such that
$\mathrm{dist}_{H^{1}({\mathbb{R}},{\mathbb{R}}^{k})}(q,\mathcal{F}^{+})=\rho_{0}^{+}/2$,
then by (2.63) we obtain
$E_{0}(q)\geq\mathfrak{m}_{0}+(\rho_{0}^{+})^{2}/(4\beta^{+})>\mathfrak{m}_{0}$.
Then, if we take $\delta<\delta_{1}$ with

$\delta_{1}:=\frac{(\rho_{0}^{+})^{2}}{4\beta^{+}\mathfrak{i}}>0,$ (2.66)

it holds
$E_{\delta}(q)>E_{0}(q)\geq\mathfrak{m}_{0}+\delta\mathfrak{i}\geq\mathfrak{m}^{+}_{\delta}$,
so that $q$ cannot be a minimum. Therefore, for such $\delta$ items 1. and 2.
in (H5) are satisfied for $E_{\delta}$ with minimizing heteroclinics
$\mathfrak{q}^{-}$ and $\mathfrak{q}^{+}_{\delta}$, with the obvious
modifications on the notations. Regarding item 3., which is the spectral
assumption of Schatzman [67], it is a generic assumption, meaning that,
arguing as it is done in her Theorem 4.3, we find that $V_{\delta}$ can be
modified with an arbitrary small perturbation away from the traces of
$\mathfrak{q}^{+}_{\delta}$ and $\mathfrak{q}^{-}$ so that 3. holds. As a
consequence, we can assume that (H5) holds for all $\delta\in(0,\delta_{1})$.
Regarding (H6), compute the constant $E_{\max}$ as in (2.46), which by the
choice of $r$ and $\chi$ does not depend on $\delta$, and set

$\delta_{2}:=\frac{E_{\max}}{\mathfrak{i}}>0,$ (2.67)

so that for all $\delta\in(0,\delta_{2})$ we have
$\mathfrak{m}^{+}_{\delta}-\mathfrak{m}_{0}<E_{\max}$. Define now
$\mathcal{F}_{\rho_{0}^{-}/2}^{-}$ as in (2.12). The choice of $r$ and $\chi$
implies that $\mathcal{F}_{\rho_{0}^{-}/2}^{-}$ does not depend on $\delta$,
meaning that we can find $\delta_{3}$ such that for all
$\delta\in(0,\delta_{3})$ it holds

$\\{q\in
X(\sigma^{-},\sigma^{+}):E_{\delta}(q)<\mathfrak{m}_{\delta}^{+}\\}\subset\mathcal{F}_{\rho^{-}_{0}/2}^{-},$
(2.68)

meaning that (H6) holds for $E_{\delta}$ provided that
$\delta\in(0,\delta_{\max})$ with
$\delta_{\max}:=\min\\{\delta_{1},\delta_{2},\delta_{3}\\}>0$. As a
consequence, we have found a family
$\\{V_{\delta}\\}_{\delta\in(0,\delta_{\max})}$ of potentials which are in the
framework of Theorem 1 and we obtain a heteroclinic traveling wave with limits
$\mathfrak{q}^{-}$ and $\mathfrak{q}^{+}$. Moreover, recall that if we put
when $\delta=0$ we recover the classical potentials considered in [2, 42, 57,
67, 69], meaning that in this setting one can prove convergence results of the
traveling waves toward stationary waves as $\delta\to 0^{+}$. Moreover, we see
that is possible to decrease the value of $\delta$ even more so that the
convergence assumption (H7) holds and Theorems 2 and 4 also apply.

Figure 2.3: Representation of the cut-off function $\chi$ used in order to
produce the family of perturbed functionals $V_{\delta}$. We also draw the
corresponding local minimizer $\mathfrak{q}^{+}_{\delta}$ (discontinuous
curve).

##  3 Discussion on the previous literature and open problems

###  3.1 Related reaction-diffusion models and the question of stability

As we said in the introduction, the problem of existence of traveling waves
for reaction-diffusion systems as well as their qualitative properties has
been widely studied since the early works of Fisher [41], Kolmogorov,
Petrovsky and Piskunov [50] regarding the equation today known as the Fisher-
KPP equation. From the modeling perspective, while the aim of these authors
was to describe of the dynamics of a given population, reaction-diffusion
systems have also been proposed as models in other domains of the natural and
social sciences. For example, applications in chemistry were given by
Zeldovich [71] and Kanel [49] (see also Berestycki, Nicolaenko and Scheurer
[19]) and the same Allen-Cahn model that we consider in this paper was
proposed by Allen and Cahn [8], following Cahn and Hilliard [34], for
describing phase transition problems in material physics. It is also worth
mentioning that for the most classical studies for traveling waves in
reaction-diffusion problems the profile tends at infinity to two (possibly
equal) constant stable states. However, other type of stable states (in
particular, non constant) can be considered as conditions at infinity (as we
do here). Moreover, the notion of traveling wave can be generalized in order
to contain and describe similar structures. We refer to the papers by
Berestycki and Hamel [14, 13] and the references therein.

Going back to the model Fisher-KPP equation, it can be written as follows:

$\partial_{t}w-\partial_{x}^{2}w=f(w),\mbox{ in
}[0,+\infty)\times{\mathbb{R}}$ (3.1)

where $f:{\mathbb{R}}\to{\mathbb{R}}$ is such that $f(0)=0$, $f(1)=0$, $f>0$
in $(0,1)$, $f<0$ in $(-\infty,0)$ and $f(u)<f^{\prime}(0)u$ for $u>0$.
Traveling waves for this equation are solutions of the type

$w(t,x)=U(x-ct)$ (3.2)

with $c>0$ and $U:{\mathbb{R}}\to{\mathbb{R}}$ satisfies

$\lim_{x\to-\infty}U(x)=0\mbox{ and }\lim_{x\to+\infty}U(x)=1.$ (3.3)

From the point of view of modelling, traveling waves intend, for instance, to
describe the invasion from a stable state to another one. An important feature
of the Fisher-KPP equation is the existence of an important speed parameter,
$c_{KPP}>0$, usually called the invasion speed which can be explicitly
computed as follows:

$c_{KPP}:=2\sqrt{f^{\prime}(0)}.$ (3.4)

The previous problem, and related ones, is studied by means of the maximum
principle and comparison results. Using these methods, one proves existence
and uniqueness of a traveling wave with fixed speed $c>0$ if and only if
$c\geq c_{KPP}$. This seems to be an important contrast with respect to the
model that we consider here. Indeed, recall that our Theorem 1 states that the
threshold speed $c^{\star}$ is, in particular, unique among the class of
profiles which are minimizers in our variational setting. In fact, the same
phenomenon is observed in earlier papers which also establish existence of
traveling waves for reaction-diffusion systems by a variational procedure:
Muratov [58], Lucia, Muratov and Novaga [52], Alikakos and Katzourakis [7],
Chen, Chien and Huang [38]. Nevertheless, we point out that our results (the
same as the ones we cite) do not exclude the possibility of other type of
traveling wave solutions with speed different than $c^{\star}$. In particular,
there could exist traveling waves with heteroclinic profiles which are
obtained from a different variational setting than ours, or even from non-
variational methods.

The analysis in [41, 50] was substantially extended in subsequent works. Fife
and McLeod [40, 39] established stability properties for traveling waves in
the Fisher-KPP equation. Generalizations bringing into consideration higher-
dimensional equations (in space) were also made. For instance, Aronson and
Weinberger [10] (see also Hamel and Nadirashvili [44] and the references
therein) considered the case of ${\mathbb{R}}^{N}$ as space domain. We also
mention the work of Berestycki, Larrouturou and Lions [17] Berestycki and
Nirenberg [20] for the case of a cylinder ${\mathbb{R}}\times\omega$, with
$\omega\subset{\mathbb{R}}^{N-1}$ a bounded domain. For the case of periodic
domains, see Berestycki and Hamel [12], Berestycki, Hamel and Nadirashvili
[15]. The case of more general domains is adressed in Berestycki, Hamel and
Nadirashvili [16]. For the non-local problem see Berestycki et. al. [18].

The family of non-linear functions $f$ which are admissible for the Fisher-KPP
model does not contain non-linearities of Allen-Cahn type. Indeed, such non-
linearities are written as $f=-V_{AC}^{\prime}$ where $V_{AC}$ is a non-
negative double-well potential, the prototypical case being

$V_{AC}:u\in{\mathbb{R}}\to\frac{(1-u^{2})^{2}}{4}\in{\mathbb{R}},$ (3.5)

which does not satisfy the assumptions for required the equations of Fisher-
KPP type, written above. For the scalar Allen-Cahn equation, we mention the
result due to Matano, Nara and Taniguchi regarding the stability for traveling
waves with ${\mathbb{R}}^{N}$ as space domain. In this case, the waves
propagate according to one direction and connect the stable states $\pm 1$ at
infinity. The case of traveling waves that connect one stable state with one
unstable, non-constant periodic 1D solution was studied by Hamel and
Roquejoffre [45]. The non-local case was adressed in Bates et. al. [11].

Many results are available for Allen-Cahn systems, but mostly in one space
dimension. In this case, the (negative) gradient flow structure implies that
for initial data of finite energy which connects two different wells, the
corresponding solution at long time the solution should look as a chain of
glued 1D connecting orbits. More generally, if the initial condition connects
at infinity two local minima at possibly different levels (and a suitable
weighted energy is finite), then traveling waves should also appear in the
asymptotic pattern. Proofs of these facts, even in a more general framework,
can be found in Risler [65, 66, 64]. Moreover, one can aim at obtaining
quantitative results which describe more precisely the previous qualitative
behavior and also introduce the problem of considering the system as a
singular perturbation. That is, one considers a coefficient $\varepsilon^{-2}$
multiplying the non-linear term and passes to the limit $\varepsilon\to
0^{+}$. In this direction, it has been shown that the fronts (that is, the
regions in which the solution is far from the set of wells) of the solution of
the gradient flow problem move at slow motion. The first rigorous proofs of
this fact was given by Carr and Pego [36, 35], Fusco and Hale [43], for the
Allen-Cahn equation. This analysis was later extended to multi-well systems by
Bethuel, Orlandi and Smets [25] for multi-well systems. Bethuel and Smets [26,
27] obtained results regarding the motion law and the long-time interaction
between stationary solutions in the multi-well scalar case, allowing also for
degenerate wells, but for the moment their work has not been extended to
systems.

Regarding Allen-Cahn systems in higher dimensions, besides the classical
articles regarding the stationary wave and this paper, we are only aware of
the recent work of Chen, Chien and Huang [38]. In the latter, the authors
consider the strip ${\mathbb{R}}\times(-l,l)$ as space domain and, for a class
of symmetric triple-well potentials on the plane (similar to that from
Bronsard, Gui and Schatzman [32]), show the existence of traveling wave
solutions connecting at infinity a well and an approximation in $(-l,l)$ of a
globally minimizing heteroclinic, which they assume to be unique. Their proof
follows by a suitable application of the variational device of Muratov [58],
which differs from that by Alikakos and Katzourakis [7] mainly on the fact
that a different constrained minimization problem is considered.

A discussion concerning the mathematical methods used for addressing these
problems is in order. As it is well known, while the maximum principle and the
comparison theorems play a key role in the study of most scalar reaction-
diffusion equations (such as the Fisher-KPP equation), those tools are no
longer available for systems except in some particular classes, for instance
when dealing with the so-called monotone systems, see Volpert, Volpert and
Volpert [70]. As a consequence, for more general classes of systems one needs
other (more general) tools. Several approaches were developed, for example the
use of Leray-Schauder degree [70] or Conley theory as discussed in Smoller
[68]. We refer to the reader to the sources given in [68, 70]. While the
gradient structure of some reaction-diffusion equations enables the
application of variational methods (see the already cited references [5, 31,
38, 46, 53, 52, 58, 65, 66, 64]), these methods have not been extensively used
in this context. This is some kind of contrast with respect to the case of
dispersive equations where, since the seminal work of Cazenave and Lions [37],
a large amount of results regarding the existence and orbital stability of
traveling waves and solitons has been produced. For instance, this has been
done for Gross-Pitaevskii equations and systems, which are in some sense the
dispersive counterpart of the parabolic problems of Allen-Cahn type. See
Bethuel, Gravejat and Saut [23], Bethuel et. al. [24] for the orbital
stability of traveling waves for the 1D Gross-Pitaevskii equation, Maris [54]
for the Gross-Pitaevskii equation in ${\mathbb{R}}^{N}$, $N\geq 3$. The
orbital stability of stationary waves (of heteroclinic type) for two coupled
Gross-Pitaevskii equations was proven by Alama et. al. [1].

In order to conclude this section, we mention that the question of the local
stability for the parabolic system of this paper is wide open. Even for the
minimizing stationary wave obtained by Alama, Bronsard and Gui [2], stability
properties have not been studied to our knowledge. Of course, the question is
also open for the traveling wave solutions that we obtain here.

###  3.2 The heteroclinic stationary wave for 2D Allen-Cahn systems

As pointed out before, the profile of the traveling wave solution that we
obtain of this paper behaves at infinity as the stationary waves obtained in
[2]. We briefly recall here how the existence of these solutions is shown,
which we hope will make the links with our problem clearer. Consider the
elliptic system

$-\Delta\mathfrak{U}=\nabla_{u}V(\mathfrak{U})\mbox{ in }{\mathbb{R}}^{2},$
(3.6)

which corresponds to stationary solutions of (1.1). The main result obtained
in [2] states that if one assumes the existence of two distinct globally
minimizing heteroclinics up to translations, $\mathfrak{q}_{-}$ and
$\mathfrak{q}_{+}$, and adds a symmetry assumption on the potential, there
exists a solution $\mathfrak{U}$ to (3.6) satisfying the conditions at
infinity

$\begin{cases}\mathfrak{U}(x_{1},x_{2})\to\sigma^{\pm}&\mbox{ as
}x_{2}\to\pm\infty,\mbox{ uniformly in }x_{1},\\\
\mathfrak{U}(x_{1},x_{2})\to\mathfrak{q}_{\pm}(\cdot+\tau^{\pm})&\mbox{ as
}x_{1}\to\pm\infty,\mbox{ uniformly in }x_{2},\end{cases}$ (3.7)

for some translation parameters $(\tau^{-},\tau^{+})\in{\mathbb{R}}^{2}$ (in
the symmetric case $\tau^{-}=\tau^{+}=0$). The symmetry assumption was later
removed by Schatzman in [67], so that the parameters $(\tau^{-},\tau^{+})$ are
part of the solution as well. While the existence proofs in [2] and [67] are
(roughly speaking) addressed by addressing directly a functional associated to
(3.6), a more general approach was carried out successfully in a more recent
paper by Monteil and Santambrogio [57], later by Smyrnelis in [69]. Their
approach follows from the key observation (for more details see Alessio and
Montecchiari [3] and the references therein) that $\mathfrak{U}$ can be seen
as a curve in the space

$\displaystyle
Y(\mathfrak{q}_{-},\mathfrak{q}_{+}):=\left\\{U:H^{1}_{\mathrm{loc}}({\mathbb{R}},(X(\sigma^{-},\sigma^{+}),\right.$
$\displaystyle\left.\lVert\cdot\rVert_{L^{2}({\mathbb{R}},{\mathbb{R}}^{k})})):\right.$
(3.8)
$\displaystyle\left.\lim_{x_{1}\to\pm\infty}\inf_{\tau\in{\mathbb{R}}}\lVert
U(x_{1})-\mathfrak{q}_{\pm}(\cdot+\tau)\rVert_{L^{2}({\mathbb{R}},{\mathbb{R}}^{k})}=0\right\\}.$
(3.9)

The previous leads to consider the functional

$E_{2}:U\in Y(\mathfrak{q}_{-},\mathfrak{q}_{+})\to
E_{2}(U):=\int_{{\mathbb{R}}}\left[\frac{\lVert
U^{\prime}(x_{1})\rVert_{L^{2}({\mathbb{R}},{\mathbb{R}}^{k})}^{2}}{2}+\mathcal{V}(U(x_{1}))\right]dx_{1}\in{\mathbb{R}},$
(3.10)

where $\mathcal{V}:X(\sigma^{-},\sigma^{+})\to{\mathbb{R}}$ is the normalized
energy

$\mathcal{V}:q\in X(\sigma^{-},\sigma^{+})\to
E(q)-\mathfrak{m}\in{\mathbb{R}},$ (3.11)

so that $\mathcal{V}$ is a non-negative functional in
$X(\sigma^{-},\sigma^{+})$ with zero set equal to $\mathcal{F}$. Therefore,
$\mathcal{V}$ can be thought as a multi-well potential (modulo translations)
in an infinite dimensional space. Identifying $U$ with a function in
$H^{1}_{\mathrm{loc}}({\mathbb{R}}^{2},{\mathbb{R}}^{k})$ in the obvious way,
we rewrite from (3.10)

$\displaystyle E_{2}(U)=\int_{\mathbb{R}}\left[\int_{\mathbb{R}}\right.$
$\displaystyle\frac{\lvert\partial_{x_{1}}U(x_{1},x_{2})\rvert^{2}}{2}dx_{2}$
(3.12)
$\displaystyle\left.+\left[\int_{\mathbb{R}}\left(\frac{\lvert\partial_{x_{2}}U(x_{1},x_{2})\rvert^{2}}{2}+V(U(x_{1},x_{2}))\right)dx_{2}-\mathfrak{m}\right]\right]dx_{1},$
(3.13)

so, formally, critical points $\mathfrak{U}$ of $E_{2}$ in
$Y(\mathfrak{q}_{-},\mathfrak{q}_{+})$ are solutions to (3.6) satisfying the
conditions at infinity

$\lim_{x_{1}\to\pm\infty}\inf_{\tau\in{\mathbb{R}}}\lVert\mathfrak{U}(x_{1})-\mathfrak{q}(\cdot+\tau)\rVert_{L^{2}({\mathbb{R}},{\mathbb{R}}^{k})}=0$
(3.14)

In particular, the solution found in [2] is a global minimizer of $E_{2}$ in
$Y(\mathfrak{q}_{-},\mathfrak{q}_{+})$. For the case of global minimizers, one
shows that the stronger condition (3.7) holds, which we suspect might not be
true for other critical points. From this starting point, the authors in [57]
generalize the one-dimensional problem (2.4) seeing it as a problem of finding
geodesics in a metric space (more general than ${\mathbb{R}}^{k}$), which can
be chosen to be equal to $Y(\mathfrak{q}_{-},\mathfrak{q}_{+})$. Subsequently,
they are able to deduce the results of [2] as particular cases. These type of
ideas inspired us for proving the results of this paper, which we do in the
framework of abstract Hilbert spaces similar to that in [69].

###  3.3 Traveling waves for 1D parabolic systems of gradient type

For the reader’s convenience, we provide some more details on the result which
was proven by Alikakos and Katzourakis [7] and how it links to our problem.
Consider the 1D parabolic system

$\partial_{t}\mathfrak{w}-\partial^{2}_{x}\mathfrak{w}=-\nabla_{u}W(\mathfrak{w})\mbox{
in }[0,+\infty)\times{\mathbb{R}}.$ (3.15)

Here $W:{\mathbb{R}}^{k}\to{\mathbb{R}}$ is an unbalanced double-well
potential. The existence of traveling wave solutions for (3.15) has been
adressed by several authors, see for instance Risler [65], Lucia, Muratov,
Novaga [52] as well as Alikakos and Katzourakis [7]. As we mentioned earlier,
in this paper we look closely to the proof given in [7] (see also the book by
Alikakos, Fusco and Smyrnelis [6]). In order to be more precise, one looks for
a pair $(c^{\star},\mathfrak{u})$ such that the function

$\mathfrak{w}:(t,x)\in[0,+\infty)\times{\mathbb{R}}\to\mathfrak{u}(x-ct)\in{\mathbb{R}}^{k}$
(3.16)

solves (3.15) and the profile $\mathfrak{u}$ joins at infinity two local
minimzers of $W$ at different levels. The solution $\mathfrak{w}$ is then a
traveling wave solution. More precisely, the profile $\mathfrak{u}$ solves the
system

$-c^{\star}\mathfrak{u}^{\prime}-\mathfrak{u}^{\prime\prime}=-\nabla_{u}W(\mathfrak{u})\mbox{
in }{\mathbb{R}},$ (3.17)

and it satisfies at infinity

$\lim_{t\to\pm\infty}\mathfrak{u}(t)=a^{\pm}$ (3.18)

where $a^{-}\in{\mathbb{R}}^{k}$ is a global minimum of $W$ with $W(a^{-})<0$
and $a^{+}$ is a local minimum of $W$ and $W(a^{+})=0$. Moreover, in [7] it is
also shown that the speed $c^{\star}$ is unique (a property to which our
Proposition 5.3 is analogous), while the profile does not need to be.

The approach in [7, 6] is variational and uses some previous ideas from
Muratov [58]. More precisely, they study the family of weighted functionals
introduced by Fife and McLeod [40, 39]

$E_{c}(q):=\int_{\mathbb{R}}\left(\frac{\lvert
q^{\prime}(t)\rvert^{2}}{2}+W(q(t))\right)e^{ct}dt,$ (3.19)

where $q$ belongs to a suitable subspace in
$H^{1}_{\mathrm{loc}}({\mathbb{R}},{\mathbb{R}}^{k})$ such that
$\lim_{t\to\pm\infty}q(t)=a^{\pm}$. We formally check that critical points of
$E_{c^{\star}}$ solve (3.17). The strategy of the proof in [7] was introduced
before in Alikakos and Fusco [5] and it can be summarized as follows: First,
one solves a family of constrained minimization problems for $E_{c}$, where
$c>0$ is at this point thought just as parameter. Once these problems have
been solved one needs to find proper speed $c^{\star}$. Finally, one needs to
“remove” the constraints, that is, to show that for $c^{\star}$ one can find a
constrained minimizer which does not saturate the constraints, meaning that it
is an actual solution to (3.17). These last two steps are accomplished by
showing that constrained minimizers exhibit a suitable asymptotic behavior
(more precisely, that they do not present a oscillatory behavior in
arbitrarily large regions) .

Therefore, the idea is to follow Monteil and Santambrogio [57], Smyrnelis [69]
and adapt the result of Alikakos and Katzourakis for infinite-dimensional ODE
systems, in which curves take values on an abstract Hilbert space. More
precisely, we consider for $c>0$ the functional

$E_{2,c}(U):=\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\frac{\lvert\partial_{x_{1}}U(x_{1},x_{2})\rvert^{2}}{2}dx_{2}+(E(U(x_{1},\cdot))-\mathfrak{m}^{+})\right)e^{cx_{1}}dx_{1}$
(3.20)

which we already introduced before. We then see $U$ (with the proper
identifications) as a mapping $U:x_{1}\in{\mathbb{R}}\to U(x_{1},\cdot)\in
L^{2}({\mathbb{R}},{\mathbb{R}}^{k})$. For $v\in
L^{2}({\mathbb{R}},{\mathbb{R}}^{k})$, we set

$\mathcal{W}(v):=\begin{cases}E(U(x_{1},\cdot))-\mathfrak{m}^{+}&\mbox{ if
}v\in H^{1}({\mathbb{R}},{\mathbb{R}}^{k}),\\\ +\infty&\mbox{ otherwise}.\\\
\end{cases}$ (3.21)

Then, under assumption (H5) we have that $\mathcal{W}$ is an unbalanced double
well potential in $L^{2}({\mathbb{R}},{\mathbb{R}}^{k})$ and $E_{2,c}$ can be
rewritten as

$E_{2,c}(U)=\int_{{\mathbb{R}}}\left[\frac{\lVert
U^{\prime}(x_{1})\rVert_{L^{2}({\mathbb{R}},{\mathbb{R}}^{k})}}{2}+\mathcal{W}(U(x_{1}))\right]e^{cx_{1}}dx_{1},$
(3.22)

which is as $E_{c}$ but for curves taking values in
$L^{2}({\mathbb{R}},{\mathbb{R}}^{k})$ instead of ${\mathbb{R}}^{k}$.
Therefore, the main issue here is to adapt the result of [7] for curves which
take values in a possibly infinite-dimensional Hilbert space $\mathscr{L}$ (to
be thought as $L^{2}({\mathbb{R}},{\mathbb{R}}^{k})$) and possessing a proper
subspace $\mathscr{H}$ (to be thought as
$H^{1}({\mathbb{R}},{\mathbb{R}}^{k})$) satisfying suitable properties with
respect to $\mathcal{W}$. A difficulty arises, since the minima of
$\mathcal{W}$ are non-isolated due to the invariance by translations of $E$.
Nevertheless, this difficulty can be circumvented, and otherwise we could
always restrict to potentials which are symmetric as Alama, Bronsard and Gui
[2] and working in the resulting space of equivariant curves, in which
invariance by translations disappears. The major difficulty comes from the
fact that in [7] the authors impose some non-degeneracy and radial
monotonicity assumptions which prevent the constrained minimizers for
exhibiting a degenerate, oscillatory behavior. This assumption can be, in some
sense, weakened in order to allow degenerate minima (we prove this in [59]),
but we cannot prove that $\mathcal{W}$ fulfills them even for simple examples
and we think it can be too restrictive. The reason is that the geometry of the
level sets $\mathcal{W}$ is difficult to understand, as it depends indirectly
on $V$. For this reason, a new type of assumption, which in our case is (H6),
is needed to replace the one from [7].

Finally, we point out that since our results are proven in an abstract
setting, they also apply to 1D systems as (3.15) by seeing this time
$\mathscr{L}$ and $\mathscr{H}$ as ${\mathbb{R}}^{k}$. Moreover, in the
companion paper [59] we show that the abstract approach allows to modify the
non-degeneracy assumptions on the minimizers used in [7] and consider classes
of potentials with some kind of degenerate minima.

###  3.4 Link with the singular limit problem

The asymptotic behavior as $\varepsilon\to 0$ for families of solutions
$(u_{\varepsilon})_{\varepsilon>0}$ of

$\partial_{t}u_{\varepsilon}-\Delta
u_{\varepsilon}=-\varepsilon^{-2}\nabla_{u}V(u_{\varepsilon}),\hskip
5.69054ptu_{\varepsilon}:[0,T]\times\Omega\to{\mathbb{R}}^{k}$ (3.23)

has been extensively studied for bounded domains
$\Omega\subset{\mathbb{R}}^{N}$ and $T>0$. Concerning the scalar case $k=1$,
Ilmanen showed in [48] that the equation above converges to Brakke’s motion by
mean curvature as $\varepsilon\to 0$. Regarding the vectorial case, while
analogous results are established under several additional assumptions, little
is proven regarding the general picture. We refer to Bronsard and Reitich [33]
as well as the more recent Laux and Simon [51] and the references therein. A
state of the art regarding the elliptic problem can be found in Bethuel [22].
We will now briefly comment on how the results obtained in this paper can be
linked to (and hopefully shed some light into) to asymptotic problem
introduced above. For $\varepsilon>0$, consider $(c^{\star},\mathfrak{U})$ the
solution given by Theorem 1 and let

$w_{\varepsilon}(t,x_{1},x_{2}):=\mathfrak{U}_{\varepsilon}(x_{1}-c_{\varepsilon}t,x_{2}),\mbox{
for }(t,(x_{1},x_{2}))\in[0,+\infty)\times{\mathbb{R}}^{2}$ (3.24)

where for $(\tilde{x}_{1},\tilde{x}_{2})\in{\mathbb{R}}^{2}$

$\mathfrak{U}_{\varepsilon}(\tilde{x}_{1},\tilde{x}_{2}):=\mathfrak{U}(\varepsilon^{-1}\tilde{x}_{1},\varepsilon^{-1}\tilde{x}_{2})$
(3.25)

and

$c_{\varepsilon}:=\varepsilon^{-1}c.$ (3.26)

Combining (3.24), (3.25) and (3.26) we have that for
$(t,x_{1},x_{2})\in[0,+\infty)\times{\mathbb{R}}^{2}$

$w_{\varepsilon}(t,x_{1},x_{2})=\mathfrak{U}(\varepsilon^{-1}x_{1}-\varepsilon^{-2}ct,\varepsilon^{-1}x_{2})$
(3.27)

and recall that by Theorem 1 we have

$-c\partial_{x_{1}}\mathfrak{U}-\Delta\mathfrak{U}=-\nabla_{u}V(\mathfrak{U})\mbox{
in }{\mathbb{R}}^{2}$ (3.28)

which implies that for all $\varepsilon>0$

$\partial_{t}w_{\varepsilon}-\Delta
w_{\varepsilon}=\varepsilon^{-2}(-c\partial_{x_{1}}\mathfrak{U}-\Delta\mathfrak{U})=-\varepsilon^{-2}\nabla_{u}V(\mathfrak{U})=-\varepsilon^{-2}\nabla_{u}V(w_{\varepsilon})\mbox{
in }[0,+\infty)\times{\mathbb{R}}^{2}.$ (3.29)

Therefore, for any bounded domain $\Omega\subset{\mathbb{R}}^{2}$, $T>0$ and
$\varepsilon>0$, $w_{\varepsilon}$ solves (3.23). That is, to sum up,
$w_{\varepsilon}$ is a traveling wave for the re-scaled potential
$V_{\varepsilon}:=\varepsilon^{-2}V$, with profile
$\mathfrak{U}_{\varepsilon}$ as in (3.25) and with speed $c_{\varepsilon}$ as
in (3.26). Notice that $c_{\varepsilon}\to+\infty$ as $\varepsilon\to 0$.
Regarding the asymptotics of $(w_{\varepsilon})_{\varepsilon>0}$, let
$\Omega=(-1,1)^{2}\subset{\mathbb{R}}^{2}$ for simplification and consider a
time interval $[0,T]$, $T>0$. Assume also that (H7) holds, so that by Theorem
2 we have

$\lim_{x_{1}\to\pm\infty}\lVert\mathfrak{U}(x_{1},\cdot)-\mathfrak{q}^{\pm}(\cdot+\tau^{\pm})\rVert_{H^{1}({\mathbb{R}},{\mathbb{R}}^{k})}=0.$
(3.30)

In combination with (3.27), (3.30) implies that for all $x^{+}\in(0,1)$

$\lim_{\varepsilon\to 0}\sup_{x_{1}\in[x^{+},1)}\lVert
w_{\varepsilon}(0,x_{1},\cdot)-\mathfrak{q}^{+}(\varepsilon^{-1}(\cdot+\tau^{+}))\rVert_{L^{\infty}((-1,1),{\mathbb{R}}^{k})}=0$
(3.31)

and for all $x^{-}\in(-1,0)$

$\lim_{\varepsilon\to 0}\sup_{x_{1}\in(-1,x^{-}]}\lVert
w_{\varepsilon}(0,x_{1},\cdot)-\mathfrak{q}^{-}(\varepsilon^{-1}(\cdot+\tau^{-}))\rVert_{L^{\infty}((-1,1),{\mathbb{R}}^{k})}=0.$
(3.32)

Therefore, we find a phase transition on the line
$\\{(x_{1},x_{2})\in{\mathbb{R}}^{2}:x_{1}=0\\}$, as it happens in the
elliptic case with the rescaling of the stationary wave. On the contrary, for
any $t>0$ we have

$\lim_{\varepsilon\to 0}\sup_{x_{1}\in(-1,1)}\lVert
w_{\varepsilon}(t,x_{1},\cdot)-\mathfrak{q}^{-}(\varepsilon^{-1}(\cdot+\tau^{-}))\rVert_{L^{\infty}((-1,1),{\mathbb{R}}^{k})}=0.$
(3.33)

That is, for positive time and small $\varepsilon$, the rescaled solutions
tend to look like the globally minimizing heteroclinic at the limit
$x_{1}\to\pm\infty$, in contrast with what is observed for $t=0$. In terms of
the interfacial density, the previous means that an initial condition with
non-constant density gives a solution with constant density for $t>0$. This
phenomenon is probably explained by some kind of parabolic regularization
effects. To conclude this paragraph, notice that the considerations presented
here are obtained by direct scaling computations. That is, they do not depend
on the way $\mathfrak{U}$ is obtained. It is only required that $\mathfrak{U}$
solves (3.28) with conditions (3.30). In particular, the assumption (H6) is
not relevant here and the same would apply to profiles obtained by different
means under some other type of assumptions.

##  4 The abstract setting

###  4.1 Main definitions and notations

As we advanced in the introduction, instead of proving directly Theorems 1, 2
and 4 , we will prove a set of more general results which will allow us to
deduce the original ones as particular cases. In particular, we introduce an
abstract setting similar to the ones considered in [57] and specially [69].
The proof of the main abstract results, Theorems 5, 6 and 7 below, are thus
the core of the paper. The passage between the abstract and the original
setting is established in section 6, which in turn proves Theorems 1, 2 and 4.

As we said before, the abstract results should be thought as an extension of
the work by Alikakos and Katzourakis [7] to curves taking values in a more
general Hilbert space and with minimum sets instead of isolated minimum
points. In fact, we essentially perform an adaptation of their strategy of
proof, which turns out to carry on to our setting. That is, our approach will
consist on establishing existence of a pair $(c,\mathbf{U})$ in
$(0,+\infty)\times X$ which fulfills

$\mathbf{U}^{\prime\prime}-D_{\mathscr{L}}\mathcal{E}(\mathbf{U})=-c\mathbf{U}^{\prime}\mbox{
in }{\mathbb{R}}$ (4.1)

and satisfies the conditions at infinity

$\exists T^{-}\in{\mathbb{R}}:\forall t\leq T^{-},\hskip
5.69054pt\mathbf{U}(t)\in\mathscr{F}^{-}_{r_{0}^{-}/2},$ (4.2) $\exists
T^{+}\in{\mathbb{R}}:\forall t\geq T^{+},\hskip
5.69054pt\mathbf{U}(t)\in\mathscr{F}^{+}_{r_{0}^{+}/2}.$ (4.3)

Notice that this problem can also be thought as a heteroclinic connection
problem on Hilbert spaces for a second order potential system with friction
term. Such a problem could have its own interest besides the main application
to the existence of traveling waves that we give here. Of course, analogous
considerations can be also applied to the results in [7] as well as our
companion paper [59].

The nature of the objects introduced above will be made precise along this
paragraph. Let $\mathscr{L}$ be a Hilbert space with inner product
$\langle\cdot,\cdot\rangle_{\mathscr{L}}$ and induced norm
$\lVert\cdot\rVert_{\mathscr{L}}$. Let $\mathscr{H}\subset\mathscr{L}$ a
Hilbert space with inner product $\langle\cdot,\cdot\rangle_{\mathscr{H}}$. In
the original setting, $\mathscr{L}$ is $L^{2}({\mathbb{R}},{\mathbb{R}}^{k})$
and $\mathscr{H}$ is $H^{1}({\mathbb{R}},{\mathbb{R}}^{k})$, both endowed with
their natural inner products. We will take
$\mathcal{E}:\mathscr{L}\to(-\infty,+\infty]$ an unbalanced potential. In the
setting of Theorem 1, $\mathcal{E}$ will _essentially_ coincide with
$E-\mathfrak{m}^{+}$ in $H^{1}({\mathbb{R}},{\mathbb{R}}^{k})$ and with
$+\infty$ elsewhere222this statement is not exact as the energy $E$ is not
defined in $H^{1}({\mathbb{R}},{\mathbb{R}}^{k})$, but on an affine space
based on $H^{1}({\mathbb{R}},{\mathbb{R}}^{k})$. However, we can trivially
obtain a functional defined on $H^{1}({\mathbb{R}},{\mathbb{R}}^{k})$ from
$E$. See subsection 6.1.. Here we just impose a set of abstract assumptions on
$\mathcal{E}$. Most of those assumptions follow bi combining ideas in [7] with
ideas in Schatzman [67] and Smyrnelis [69]. We will begin by fixing two sets
$\mathscr{F}^{-}$ and $\mathscr{F}^{+}$ in $\mathscr{L}$. For $r>0$, we define

$\mathscr{F}^{\pm}_{r}:=\left\\{v\in\mathscr{L}:\inf_{\mathbf{v}\in\mathscr{F}^{\pm}}\lVert
v-\mathbf{v}\rVert_{\mathscr{L}}\leq r\right\\},$ (4.4)

and

$\mathscr{F}^{\pm}_{\mathscr{H},r}:=\left\\{v\in\mathscr{H}:\inf_{\mathbf{v}\in\mathscr{F}^{\pm}}\lVert
v-\mathbf{v}\rVert_{\mathscr{H}}\leq r\right\\},$ (4.5)

that is, the closed balls in $\mathscr{L}$ and $\mathscr{H}$ respectively,
with radius $r>0$ and center $\mathscr{F}^{\pm}$. The main assumption reads as
follows:

######  (H1’).

The potential $\mathcal{E}$ is weakly lower semicontinuous in $\mathscr{L}$.
The sets $\mathscr{F}^{-}$ and $\mathscr{F}^{+}$ are closed in $\mathscr{L}$.
There exists a constant $a<0$ such that

$\forall v\in\mathscr{L},\forall\mathbf{v}^{-}\in\mathscr{F}^{-},\hskip
5.69054pt\mathcal{E}(v)\geq\mathcal{E}(\mathbf{v}^{-})=a$ (4.6)

and each $\mathbf{v}^{+}\in\mathscr{F}^{+}$ is a local minimizer satisfying
$\mathcal{E}(\mathbf{v}^{+})=0$. Moreover, there exist two positive constants
$r_{0}^{-}$, $r_{0}^{+}$ such that
$\mathscr{F}^{+}_{r_{0}^{-}}\cap\mathscr{F}^{-}_{r_{0}^{+}}=\emptyset$ (see
(4.4)). There also exist $C^{\pm}>1$ such that

$\forall v\in\mathscr{F}^{\pm}_{r_{0}^{\pm}},\hskip
5.69054pt(C^{\pm})^{-1}\mathrm{dist}_{\mathscr{L}}(v,\mathscr{F}^{\pm})^{2}\leq\mathcal{E}(v)-\min\\{\pm(-a),0\\}.$
(4.7)

Moreover, for any $v\in\mathscr{F}_{r_{0}^{\pm}}^{\pm}$, there exists a unique
$\mathbf{v}^{\pm}(v)\in\mathscr{F}^{\pm}$ such that

$\lVert
v-\mathbf{v}^{\pm}(v)\rVert_{\mathscr{L}}=\inf_{\mathbf{v}^{\pm}\in\mathscr{F}^{\pm}}\lVert
v-\mathbf{v}^{\pm}\rVert_{\mathscr{L}}.$ (4.8)

Moreover, the projection maps

$P^{\pm}:v\in\mathscr{F}^{\pm}_{r_{0}^{\pm}}\to\mathbf{v}^{\pm}(v)\in\mathscr{F}^{\pm}$
(4.9)

are $C^{2}$ with respect to the $\mathscr{L}$-norm.

Hypothesis (H1’) defines $\mathcal{E}$ as an unbalanced double well potential
with respect to $\mathscr{F}^{-}$ and $\mathscr{F}^{+}$ and gives local
information of the minimizing sets. Compare with (H5) and the remarks that
follow. We have the following immediate consequence, which will be useful in
the sequel:

######  Lemma 4.1.

Assume that (H1’) holds. If we define for $r\in(0,r_{0}^{\pm}]$ we define

$\kappa^{\pm}_{r}:=\inf\\{\mathcal{E}(v):\mathrm{dist}_{\mathscr{L}}(v,\mathscr{F}^{\pm})\in[r,r_{0}^{\pm}]\\}$
(4.10)

then we have $\kappa^{\pm}_{r}>\min\\{\pm(-a),0\\}$. Moreover,

$\forall v\in\mathscr{F}^{+}_{r_{0}^{+}/2},\hskip 5.69054pt\mathcal{E}(v)\geq
0.$ (4.11)

###### Proof.

It follows directly from (4.7) in (H1’). ∎

We now impose the following regarding the relationship between $\mathscr{L}$
and $\mathscr{H}$:

######  (H2’).

We have that $\mathscr{H}=\\{v\in\mathscr{L}:\mathcal{E}(v)<+\infty\\}$ and
$\lVert\cdot\rVert_{\mathscr{L}}\leq\lVert\cdot\rVert_{\mathscr{H}}$. In
particular, $\mathscr{F}^{\pm}\subset\mathscr{H}$. Moreover, $\mathcal{E}$ is
a $C^{1}$ functional on $(\mathscr{H},\lVert\cdot\rVert_{\mathscr{H}})$ with
differential $D\mathcal{E}:v\in\mathscr{H}\to
D\mathcal{E}(v)\in\mathscr{H}^{\prime}$, where $\mathscr{H}^{\prime}$ is the
(topological) dual of $\mathscr{H}$. Furthermore, there exists an even smaller
space $\tilde{\mathscr{H}}$ with an inner product
$\langle\cdot,\cdot\rangle_{\tilde{\mathscr{H}}}$ and associated norm
$\lVert\cdot\rVert_{\tilde{\mathscr{H}}}\geq\lVert\cdot\rVert_{\mathscr{H}}$
such that we can find a continuous correspondence

$D_{\mathscr{L}}\mathcal{E}:v\in(\tilde{\mathscr{H}},\lVert\cdot\rVert_{\tilde{\mathscr{H}}})\to
D_{\mathscr{L}}\mathcal{E}(v)\in(\mathscr{L},\lVert\cdot\rVert_{\mathscr{L}})$
(4.12)

such that

$\forall v\in\tilde{\mathscr{H}},\forall w\in\mathscr{H},\hskip
5.69054ptD_{\mathscr{L}}\mathcal{E}(v)(w)=D\mathcal{E}(v)(w).$ (4.13)

Notice that in the context of Theorem 1 assumption (H2’) is easily verified.
The space $\tilde{\mathscr{H}}$ will be chosen
$H^{2}({\mathbb{R}},{\mathbb{R}}^{k})$ and (4.13) is no other that integration
by parts. The notation $D_{\mathscr{L}}\mathcal{E}$ is chosen to emphasize the
formal $\mathscr{L}$-gradient flow structure of the corresponding abstract
evolution equation. We now continue by imposing a compactness assumption on
$\mathscr{F}^{\pm}$:

######  (H3’).

$\mathscr{L}$-bounded subsets of $\mathscr{F}^{\pm}$ are compact with respect
to $\mathscr{H}$-convergence.333hence, they are in particular compact with
respect to $\mathscr{L}$-convergence

Assumption (H3’) readily implies the following:

######  Lemma 4.2.

Assume that (H1’) and (H3’) hold. Then, the sets
$\mathscr{F}^{\pm}_{r_{0}^{\pm}/2}$ defined in (4.4) are closed in
$\mathscr{L}$.

Assumption (H3’) is necessary in order to establish the conditions at
infinity. In the main context, it is a straightforward consequence of the
compactness of the minimizing sequences. Subsequently, we impose the
following:

######  (H4’).

Assume that (H1’) holds. For $\mathscr{F}^{\pm}$, one of the two following
alternatives holds:

  1. 1.

$\mathscr{F}^{\pm}$ is $\mathscr{L}$-bounded.

  2. 2.

For all
$(v,\mathbf{v}^{\pm})\in\mathscr{F}^{\pm}_{r_{0}^{\pm}}\times\mathscr{F}^{\pm}$,
there exists an associated map
$\hat{P}^{\pm}_{(v,\mathbf{v}^{\pm})}:\mathscr{L}\to\mathscr{L}$ such that

$P^{\pm}(\hat{P}^{\pm}_{(v,\mathbf{v}^{\pm})}(v))=\mathbf{v}^{\pm}$ (4.14)

and

$\mathrm{dist}_{\mathscr{L}}(\hat{P}^{\pm}_{(v,\mathbf{v}^{\pm})}(v),\mathscr{F}^{\pm})=\mathrm{dist}_{\mathscr{L}}(v,\mathscr{F}^{\pm}).$
(4.15)

Moreover, $\hat{P}^{\pm}_{(v,\mathbf{v}^{\pm})}:\mathscr{L}\to\mathscr{L}$ is
differentiable and

$\forall(w_{1},w_{2})\in\mathscr{L}^{2},\hskip 5.69054pt\lVert
D(\hat{P}^{\pm}_{(v,\mathbf{v}^{\pm})})(w_{1},w_{2})\rVert_{\mathscr{L}}=\lVert
w_{2}\rVert_{\mathscr{L}}$ (4.16)
$\mathcal{E}(\hat{P}_{(v,\mathbf{v}^{\pm})}(v))=\mathcal{E}(v).$ (4.17)

Essentially, in 2. we impose that the projections $P^{\pm}$ from (H1’) are, in
some sense, invertible. Again, this is straigtforward in the concrete setting,
as the projections $P^{\pm}$ consist on performing a translation. We now
impose an assumption for the sets $\mathscr{F}^{\pm}_{\mathscr{H},r_{0}}$:

######  (H5’).

For any $v\in\mathscr{F}_{\mathscr{H},r_{0}^{\pm}}^{\pm}$, as defined in
(4.5), there exists a unique
$\mathbf{v}_{\mathscr{H}}^{\pm}(v)\in\mathscr{F}^{\pm}$ such that

$\lVert
v-\mathbf{v}_{\mathscr{H}}^{\pm}(v)\rVert_{\mathscr{L}}=\inf_{\mathbf{v}^{\pm}\in\mathscr{F}^{\pm}}\lVert
v-\mathbf{v}^{\pm}\rVert_{\mathscr{L}}.$ (4.18)

Moreover, the projection maps

$P^{\pm}_{\mathscr{H}}:v\in\mathscr{F}^{\pm}_{\mathscr{H},r_{0}^{\pm}}\to\mathbf{v}_{\mathscr{H}}^{\pm}(v)\in\mathscr{F}^{\pm}$
(4.19)

are $C^{1}$ with respect to the $\mathscr{H}$-norm. Moreover, if $C^{\pm}>1$
is the constant from (H1’), we have

$\forall v\in\mathscr{F}^{\pm}_{\mathscr{H},r_{0}^{\pm}},\hskip
5.69054pt\lVert P^{\pm}(v)-P^{\pm}_{\mathscr{H}}(v)\rVert_{\mathscr{H}}\leq
C^{\pm}\lVert v-P^{\pm}_{\mathscr{H}}(v)\rVert_{\mathscr{H}}.$ (4.20)

Furthermore, for each $r^{\pm}\in(0,r_{0}^{\pm}]$ there exist constants
$\beta^{\pm}(r^{\pm})>0$ such that in case that
$v\in\mathscr{F}^{\pm}_{r_{0}^{\pm}}$ satisfies

$\mathcal{E}(v)\leq\min\\{\pm(-a),0\\}+\beta^{\pm}(r^{\pm}),$ (4.21)

then $v\in\mathscr{F}_{\mathscr{H},r}^{\pm}$. Finally, we have the following

$\forall v\in\mathscr{F}_{\mathscr{H},r_{0}^{\pm}}^{\pm},\hskip
5.69054pt(C^{\pm})^{-2}\lVert
v-P^{\pm}_{\mathscr{H}}(v)\rVert_{\mathscr{H}}^{2}\leq\mathcal{E}(v)-\min\\{\pm(-a),0\\}\leq(C^{\pm})^{2}\lVert
v-P^{\pm}_{\mathscr{H}}(v)\rVert_{\mathscr{H}}^{2}.$ (4.22)

Assumption (H5’) is made in order to ensure the suitable local properties
around $\mathscr{F}^{\pm}$ in $\mathscr{H}$. In the main setting, those are
known results which follow essentially from the spectral assumption by
Schatzman [67]. Before introducing the last assumptions, we need some
additional notation. For $U\in H^{1}_{\mathrm{loc}}({\mathbb{R}},\mathscr{L})$
and $c>0$, we (formally) define

$\mathbf{E}_{c}(U):=\int_{\mathbb{R}}\mathbf{e}_{c}(U)(t)dt:=\int_{{\mathbb{R}}}\left[\frac{\lVert
U^{\prime}(t)\rVert_{\mathscr{L}}^{2}}{2}+\mathcal{E}(U(t))\right]e^{ct}dt.$
(4.23)

More generally, for $I\subset{\mathbb{R}}$ a non-empty interval and $U\in
H^{1}_{\mathrm{loc}}(I,\mathscr{L})$, put

$\mathbf{E}_{c}(U;I):=\int_{I}\mathbf{e}_{c}(U)(t)dt.$ (4.24)

Notice that the integrals defined in (4.23) and (4.24) might not even make
sense in general due to the fact that $\mathcal{E}$ has a sign. Nevertheless,
we can define the notion of local minimizer of $\mathbf{E}_{c}(\cdot;I)$ as
follows:

######  Definition 4.1.

Assume that (H1’) and (H2’) hold. Let $I\subset{\mathbb{R}}$ be a bounded,
non-empty interval. Assume that $U\in H^{1}_{\mathrm{loc}}(I,\mathscr{L})$ is
such that $E_{c}(U;I)$ is well-defined and finite. Assume also that there
exists $C>0$ such that for any
$\phi\in\mathcal{C}^{1}_{c}(\mathrm{int}(I),(\mathscr{H},\lVert\cdot\rVert_{\mathscr{H}}))$
such that

$\max_{t\in I}\lVert\phi(t)\rVert_{\mathscr{H}}<C,$ (4.25)

the quantity $\mathbf{E}_{c}(U+\phi;I)$ is well-defined and larger than
$\mathbf{E}_{c}(U;I)$. Then, we say that $U$ is a _local minimizer_ of
$\mathbf{E}_{c}(\cdot;I)$.

We assume the following property for local minimizers:

######  (H6’).

Assume that (H1’) and (H2’) hold. There exists a map
$\mathfrak{P}:\mathscr{L}\to\mathscr{L}$ such that

$\forall v\in\mathscr{L},\hskip
5.69054pt\mathcal{E}(\mathfrak{P}(v))\leq\mathcal{E}(v)\mbox{ and
}\mathcal{E}(\mathfrak{P}(v))=\mathcal{E}(v)\Leftrightarrow\mathfrak{P}(v)=v,$
(4.26) $\forall(v_{1},v_{2})\in\mathscr{L}^{2},\hskip
5.69054pt\lVert\mathfrak{P}(v_{1})-\mathfrak{P}(v_{2})\rVert_{\mathscr{L}}\leq\lVert
v_{1}-v_{2}\rVert_{\mathscr{L}},$ (4.27)

and

$\mathfrak{P}|_{\mathscr{F}^{\pm}}=\mathrm{Id}|_{\mathscr{F}^{\pm}}.$ (4.28)

Let $I\subset{\mathbb{R}}$, possibly unbounded and non-empty. Let $c>0$. If
$\mathbf{W}\in H^{1}_{\mathrm{loc}}(I,\mathscr{L})$ is a local minimizer of
$\mathbf{E}_{c}(\cdot;I)$ in the sense of Definition 4.1, which, additionally,
is such that for all $t\in I$, $\mathbf{W}(t)=\mathfrak{P}(\mathbf{W}(t))$,
then $\mathbf{W}\in\mathcal{A}(I)$ where for any open set
$O\subset{\mathbb{R}}$, $\mathcal{A}(O)$ is defined as

$\mathcal{A}(O):=\mathcal{C}^{2}_{\mathrm{loc}}(O,\mathscr{L})\cap\mathcal{C}^{1}_{\mathrm{loc}}(O,(\mathscr{H},\lVert\cdot\rVert_{\mathscr{H}}))\cap\mathcal{C}^{0}_{\mathrm{loc}}(O,(\tilde{\mathscr{H}},\lVert\cdot\rVert_{\tilde{\mathscr{H}}}))$
(4.29)

and $\mathbf{W}$ solves

$\mathbf{W}^{\prime\prime}-D_{\mathscr{L}}\mathcal{E}(\mathbf{W})=-c\mathbf{W}^{\prime}\mbox{
in }I,$ (4.30)

where $D_{\mathscr{L}}\mathcal{E}$ was introduced in (4.12).

In the context of Theorem 1, (H6’) is a consequence of classical elliptic
regularity results as well as properties on the energy functional. The purpose
of the projection $\mathfrak{P}$ is technical, and in the main setting it will
mean that constrained minimizers are bounded with respect to the $L^{\infty}$
norm. Before stating the abstract result, we introduce the following constants
(assuming that all the previous assumptions hold) which are obviously
analogous with those introduced in subsection 2.5:

$\eta_{0}^{-}:=\min\left\\{\sqrt{e^{-1}\frac{r_{0}^{-}}{4}\sqrt{2(\kappa^{-}_{r_{0}^{-}/4}-a)}},\frac{r_{0}^{-}}{4}\right\\}>0,$
(4.31) $\hat{r}^{-}:=\frac{r_{0}^{-}}{C^{-}+1}>0$ (4.32)
$\mathcal{E}_{\max}^{-}:=\frac{1}{(C^{-})^{2}(C^{-}+1)}\min\left\\{\frac{(\eta_{0}^{-})^{2}}{4},\kappa_{\eta_{0}^{-}}^{-}-a,\beta^{-}(\hat{r}^{-}),\beta^{-}(\eta_{0}^{-})\right\\}>0,$
(4.33)
$\mathsf{C}^{\pm}:=\frac{1}{2}(C^{\pm})^{2}((C^{\pm})^{2}+(C^{\pm}+1)^{2})>0,$
(4.34) $\gamma^{-}:=\frac{1}{C^{-}+\mathsf{C}^{-}}>0$ (4.35)

and

$d_{0}:=\mathrm{dist}_{\mathscr{L}}(\mathscr{F}^{+}_{r_{0}^{+}/2},\mathscr{F}^{-}_{r_{0}^{-}/2})>0,$
(4.36)

where the constants $C^{-},\beta^{-}(\hat{r}^{-}),\beta^{-}(\eta_{0}^{-})$ are
those from (H5’) and $\kappa_{r}^{\pm}$ for $r>0$ are defined in (4.10). The
fact that $d_{0}>0$ follows from Lemma 4.2 and (H1’). We can finally state the
following assumption:

######  (H7’).

Assume that (H1’) and (H2’) hold. Moreover, assume that

$-a<\mathcal{E}_{\max}^{-}$ (4.37)

and

$\\{v\in\mathscr{H}:\mathcal{E}(v)<0\\}\subset\mathscr{F}^{-}_{r_{0}^{-}/2}.$
(4.38)

Assumption (H7’) is essentially the abstract version of (H6).

###  4.2 Statement of the abstract results

Let us define the space

$\displaystyle X:=\left\\{U\in
H^{1}_{\mathrm{loc}}({\mathbb{R}},\mathscr{L}):\exists T\geq 1,\right.$
$\displaystyle\forall t\geq T,\hskip
5.69054pt\mathrm{dist}_{\mathscr{L}}(U(t),\mathscr{F}^{+})\leq\frac{r_{0}^{+}}{2},$
(4.39) $\displaystyle\left.\forall t\leq-T,\hskip
5.69054pt\mathrm{dist}_{\mathscr{L}}(U(t),\mathscr{F}^{-})\leq\frac{r_{0}^{-}}{2}\right\\}.$
(4.40)

The statement of the main abstract result is as follows:

######  Theorem 5 (Main abstract result).

Assume that (H3’), (H4’), (H5’), (H6’) and (H7’) hold. Then, the following
holds:

  1. 1.

Existence. There exists $c^{\star}>0$ and
$\mathbf{U}\in\mathcal{A}({\mathbb{R}})\cap X$, $\mathcal{A}({\mathbb{R}})$ as
in (4.29) and $X$ as in (4.39), such that $(c^{\star},\mathbf{U})$ solves
(4.1) with conditions at infinity (4.2), (4.3) and $\mathbf{U}$ is a global
minimizer of $\mathbf{E}_{c}$ in $X$, that is, $\mathbf{E}_{c}(\mathbf{U})=0$.
Moreover, for all $t\in{\mathbb{R}}$,
$\mathbf{U}(t)=\mathfrak{P}(\mathbf{U}(t))$, where $\mathfrak{P}$ is as in
(H6’).

  2. 2.

Uniqueness of the speed. The speed $c^{\star}$ is unique in the following
sense: if $\overline{c^{\star}}>0$ is such that

$\inf_{U\in X}\mathbf{E}_{\overline{c^{\star}}}(U)=0$ (4.41)

and there exists $\overline{\mathbf{U}}\in\mathcal{A}({\mathbb{R}})\cap X$
such that $(\overline{c^{\star}},\overline{\mathbf{U}})$ solves (4.1) and
$\mathbf{E}_{\overline{c^{\star}}}(\overline{\mathbf{U}})<+\infty$, then
$\overline{c^{\star}}=c^{\star}$.

  3. 3.

Exponential convergence. There exists a constant $M^{+}>0$ such that for all
$t\in{\mathbb{R}}$ we have

$\lVert\mathbf{U}(t)-\mathbf{v}^{+}(\mathbf{U})\rVert_{\mathscr{L}}\leq
M^{+}e^{-ct},$ (4.42)

for some $\mathbf{v}^{+}(\mathbf{U})\in\mathscr{F}^{+}$.

######  Remark 4.1.

Given the definition of $X$ in (4.39), we have that for any $U\in X$ and
$\tau\in{\mathbb{R}}$ it holds $U(\cdot+\tau)\in X$ and for any $c>0$ it holds
$\mathbf{E}_{c}(U(\cdot+\tau))=e^{-c\tau}\mathbf{E}_{c}(U)$. Such a thing
implies

$\forall c>0,\hskip 5.69054pt\inf_{U\in
X}\mathbf{E}_{c}(U)\in\\{-\infty,0\\}.$ (4.43)

Moreover, we see that in case $c>0$ is such that $\inf_{U\in
X}\mathbf{E}_{c}(U)=0$ one can find plenty of examples of minimizing sequences
in $X$ which cannot ever reasonably produce a global minimizer. Indeed,
consider any function $\tilde{U}\in X$ such that $E_{c}(\tilde{U})>0$ and then
take the minimizing sequence $(\tilde{U}(\cdot+n))_{n\in{\mathbb{N}}}$.

######  Remark 4.2.

A more general statement can be given about the uniqueness of the speed, which
in particular works for eventual non-minimizing solutions. See Proposition
5.3.

Theorem 5 will be shown to contain Theorem 1 in section 6. Notice that, as
before, the conditions at infinity (4.2) are rather weak (and not really of
heteroclinic type), since we do not have convergence to an element of
$\mathscr{F}^{-}$ as $t\to-\infty$. It is however clear that the conditions at
infinity (4.2), (4.3) are enough to ensure that the solution given by Theorem
5 is not constant. In any case, we can impose an additional assumption in
order to obtain stronger conditions at $-\infty$ on the solution:

######  (H8’).

Hypothesis (H7’) is fulfilled and, additionally:

$-a<\frac{(d_{0}\gamma^{-})^{2}}{2},$ (4.44)

where $d_{0}$ and $\gamma^{-}$ were defined in (4.36) and (4.35) respectively.

Then we can show the following exponential convergence result

######  Theorem 6.

Assume that (H2’), (H3’), (H4’), (H5’), (H6’), (H7’) and (H8’) hold. Then, if
$(c^{\star},\mathbf{U})$ is the solution given by Theorem 5, it holds that
$\gamma^{-}>c^{\star}$ ($\gamma^{-}$ as in (4.35)) and there exists
$\mathfrak{M}^{-}>0$ such that for all $t\in{\mathbb{R}}$

$\lVert\mathbf{U}(t)-\mathbf{v}^{-}(\mathbf{U})\rVert_{\mathscr{L}}\leq
M^{-}e^{(\gamma^{-}-c^{\star})t}$ (4.45)

for some $\mathbf{v}^{-}(U)\in\mathscr{F}^{-}$.

Theorem 6 corresponds to Theorem 2. Finally, we will prove the following
result:

######  Theorem 7.

Assume that (H2’), (H3’), (H4’), (H5’), (H6’), (H7’) and (H8’) hold. Let
$(c^{\star},\mathbf{U})$ be the solution given by Theorem 5. Then, if
$\tilde{\mathbf{U}}\in\mathcal{A}({\mathbb{R}})\cap X$ is such that

$\mathbf{E}_{c^{\star}}(\tilde{\mathbf{U}})=0$ (4.46)

then we have that $(c^{\star},\tilde{\mathbf{U}})$ solves (4.1) and

$c^{\star}=\frac{-a}{\int_{\mathbb{R}}\lVert\tilde{\mathbf{U}}^{\prime}(t)\rVert_{\mathscr{L}}^{2}dt}.$
(4.47)

In particular, the quantity
$\int_{\mathbb{R}}\lVert\tilde{\mathbf{U}}^{\prime}(t)\rVert_{\mathscr{L}}^{2}dt$
is finite. Moreover, we have

$c^{\star}=\sup\\{c>0:\inf_{U\in
X}\mathbf{E}_{c}(U)=-\infty\\}=\inf\\{c>0:\inf_{U\in X}\mathbf{E}_{c}(U)=0\\}$
(4.48)

as well as the bound

$c^{\star}\leq\frac{\sqrt{-2a}}{d_{0}}<\min\left\\{\frac{\sqrt{2\mathcal{E}_{\max}^{-}}}{d_{0}},\gamma^{-}\right\\}$
(4.49)

with $\mathcal{E}_{\max}^{-}$ as in (4.33), $d_{0}$ as in (4.36) and
$\gamma^{-}$ as in (4.35). The second inequality follows from the bounds on
$-a$ given by (H7’) and (H8’).

Theorem 7 corresponds to Theorem 4.

##  5 Proof of the abstract results

###  5.1 Scheme of the proofs

As pointed out several times, the structure of the proofs of our abstract
results, Theorems 5, 6 and 7, is analogous to that in Alikakos and Katzourakis
[7], which has its roots in Alikakos and Fusco [5]. In fact, most of their
results also carry into the abstract setting with the suitable modifications.
In fact, the structure of our proofs should be rather compared with subsection
2.6 in the book by Alikakos, Fusco and Smyrnelis [6], which slightly modifies
and simplifies the argument in [7]. We will also rely on some arguments
provided in Smyrnelis [69], when an analogous abstract approach is taken for
the stationary problem. As usual, most of the intermediate results we prove
hold under smaller subsets of assumptions (with respect to the set of all
assumptions that we dropped in the previous section). Therefore, for the sake
of clarity and generality, the necessary assumptions (and only these) that we
use to prove a result are specified in its statement.

Despite the previous facts, and as pointed before, several important
difficulties not present in [7] arise when one tries to tackle the same
problem in the abstract setting we introduced in the previous section. One of
those extra difficulties is due to the fact that, in our setting, we need deal
with two different norms in the configuration space of the curves,
$\mathscr{L}$ and $\mathscr{H}$ (to be thought as $L^{2}$ and $H^{1}$
respectively, for simplification) and that the potential $\mathcal{E}$ is only
lower semicontinuous with respect to $\mathscr{L}$-convergence. An additional
difficulty comes from the fact that, due to the requirements of our original
problem, we are not looking at curves that join two isolated minimum points,
but rather two isolated minimum sets. This turns out to be an obstacle when
one tries to adapt argument in [7], even if one were to restrict to finite-
dimensional configuration spaces. However, this difficulty is successfully
dealt with using the precise knowledge about the projection mappings (namely
assumptions (H1’), (H4’) and (H5’)) is available. That is, one uses that, for
a suitable neighborhood of the minimum sets, the projection onto the sets
(both with the $\mathscr{L}$ and $\mathscr{H}$ norms) is well defined and
enjoys some type of continuity and differentiability properties. This idea, in
the Allen-Cahn systems setting, has to be be traced back to Schatzman [67].

We will now briefly sketch the scheme of the proof of Theorem 5. Recall that,
according to Remark 4.1, direct minimization of $\mathbf{E}_{c}$ in $X$ cannot
yield solutions to the problem, the reason being the action of the group of
translations. The spaces $X_{T}$, which were introduced in [7] (also in [5]
for the equal-depth case) and will be precisely presented in (5.3), are
defined in order to overcome this source of degeneracy, as they are no longer
invariant by the action of the group of translations. See the design in Figure
5.1. As a consequence, compactness is restored and the corresponding
minimization problem has a solution for all $c>0$ and $T\geq 1$. See Lemma 5.7
later on. In general, minimizers in $X_{T}$ solve the profile equation on a
(possibly proper) subset of ${\mathbb{R}}$ (see Lemma 5.8), meaning that they
are in general not solutions of (4.1). However, such constrained minimizers
are in fact solutions of (4.1) in the case they do not saturate the
constraints. Therefore, the goal will be to show the existence of the speed
$c^{\star}$ such that, for some $T\geq 1$, there exists a constrained
minimizer in $X_{T}$ which does not saturate the constraints. For that
purpose, a careful analysis of the behavior of the constrained minimizers is
needed. Indeed, one needs a uniform bound (independent on $T$ and continuous
on $c$) on the distance between the entry times, i. e. the times in which the
constrained minimizers enter $\mathscr{F}^{\pm}_{r_{0}^{\pm}/2}$. In the
balanced case this follows from the fact that the energy density is bounded
below by a positive constant outside
$\mathscr{F}^{-}_{r_{0}^{-}/2}\cup\mathscr{F}^{+}_{r_{0}^{+}/2}$ (see for
instance Smyrnelis [69]). However this is no longer true for our unbalanced
problem, which makes it more involved: If one does not have the positivity of
the energy density, the constraint solutions can oscillate between the regions
$\mathscr{F}^{\pm}_{r_{0}^{\pm}/2}$ (producing energy compensations) in larger
and larger intervals as $T\to\infty$, so that no $T$-independent bound can be
found. This is the main new difficulty with respect to the balanced setting,
as one needs new ideas in order to obtain a uniform bound on the distance
between the entry times. Our assumption (H7’) provides this control because
the energy density of the constrained minimizers is bounded below by a
positive constant in the interval given by the two entry times mentioned
before, meaning that we can argue as in the balanced case. The precise result
is Corollary 5.1. This is the main step in which our proof differs with that
in [7].

The natural question is what happens if we remove (H7’). A natural approach is
to replace (H7’) by an assumption more closely related to the one used in [7]
and [6]. This would lead to introduce a convexity assumption on the level sets
of $\mathcal{E}$, as well as some sort of strict monotonicity on well-chosen
segments. While this assumption can be worked out in the abstract setting and
it is applicable for the finite-dimensional situation considered in [7] (as we
show in the companion paper [59]), we believe it to be too restrictive to be
applied to our original problem.

In any case, after the uniform bound on the entry times of constrained
minimizers is obtained, one needs to find the speed $c^{\star}$ as, until this
point, the speed $c>0$ has been only considered as a parameter of the problem
without any special role. Our arguments adapts without major difficulty from
[6] and it goes as follows: One introduces a set which classifies the speeds
according to the value of the infimum of the corresponding energy on $X$
(which, due to the weight and the invariance by translations, is either
$-\infty$ or $0$). Such a set is $\mathcal{C}$, defined in (5.169).
Subsequently, one shows (Lemma 5.11) that $\mathcal{C}$ is open, bounded, non-
empty and that its positive limit points give rise to entire minimizing
solutions of the equations (since for those points one can find corresponding
constrained minimizers which do not saturate the constraints). The speed
$c^{\star}$ is then defined as the supremum of $\mathcal{C}$, which is in fact
the unique positive limit point of the set, as shown in Corollary 5.3. At this
point, the process of the proof of Theorem 5 is completed. Later on, we show
that the asymptotic behavior of the constrained solutions can be improved
under an additional assumption, namely an upper bound on the speed. This is
Proposition 5.4. Theorems 6 and 7 can be then proven.

Figure 5.1: One-dimensional representation of $X_{T}$. The blue line
represents a function $U$ belonging to $X_{T}$. The red lines contain the
points which are at $\mathscr{L}$-distance smaller than $r_{0}^{\pm}/2$ from
$\mathscr{F}^{\pm}$.

###  5.2 Preliminaries

Let $r_{0}^{-}$ and $r_{0}^{+}$ be the constants introduced in section 4 and
$\mathscr{F}_{r_{0}^{\pm}/2}^{\pm}$ be the corresponding closed balls as in
(4.4). Assume that (H1’) holds. For $T\geq 1$, we define the sets

$X_{T}^{-}:=\left\\{U\in
H^{1}_{\mathrm{loc}}({\mathbb{R}},\mathscr{L}):\forall t\leq-T,\hskip
2.84526ptU(t)\in\mathscr{F}^{-}_{r_{0}^{-}/2}\right\\},$ (5.1)
$X_{T}^{+}:=\left\\{U\in
H^{1}_{\mathrm{loc}}({\mathbb{R}},\mathscr{L}):\forall t\geq T,\hskip
2.84526ptU(t)\in\mathscr{F}^{+}_{r_{0}^{+}/2}\right\\}.$ (5.2)

Subsequently, we set

$X_{T}:=X_{T}^{-}\cap X_{T}^{+}.$ (5.3)

Recall the space $X$ introduced in (4.39). Notice that

$X=\bigcup_{T\geq 1}X_{T}.$ (5.4)

We have the following preliminary properties on the spaces $X_{T}$:

######  Lemma 5.1.

Assume that (H1’) holds. Let $c>0$ and $T\geq 1$. For any $U\in X_{T}$, we
have that

$\forall t\geq T,\hskip 5.69054pt\mathcal{E}(U(t))\geq 0.$ (5.5)

Moreover, the quantity $\mathbf{E}_{c}(U)$ as introduced in (4.23) is well
defined in $(-\infty,+\infty]$.

###### Proof.

Let $U\in X_{T}$. Notice that for $t\geq T$, we have that
$U(t)\in\mathscr{F}^{+}_{r_{0}/2}$. Therefore, (5.5) follows directly from
(4.11) in Lemma 4.1.

Let now $\mathcal{E}^{+}(U)\geq 0$ and $\mathcal{E}^{-}(U)\geq 0$ be,
respectively, the non-negative and the non-positive part of $\mathcal{E}(U)$,
so that $\mathcal{E}(U)=\mathcal{E}^{+}(U)-\mathcal{E}^{-}(U)$. We have that
$\mathcal{E}^{-}(U)$ is null on $[T,+\infty)$. That is

$\int_{-\infty}^{+\infty}\mathcal{E}^{-}(U(t))e^{ct}dt=\int_{-\infty}^{T}\mathcal{E}^{-}(U(t))e^{ct}dt\leq-\frac{a}{c}e^{cT}<+\infty,$
(5.6)

where $a$ is the minimum value from (H1’). Therefore, the negative part of the
energy density $\mathbf{e}_{c}(U)$ (see (4.23)) belongs to
$L^{1}({\mathbb{R}})$, which establishes the result. ∎

Lemma 5.1 shows that for any $T\geq 1$ and $c>0$, $\mathbf{E}_{c}$ is well
defined as an extended functional on $X_{T}$, at least if sufficient
hypothesis are made. Moreover, it gives the following useful inequalities:

######  Lemma 5.2.

Assume that (H1’) holds. Let $c>0$ and $T\geq 1$. For any $U\in X_{T}$, we
have that

$\int_{{\mathbb{R}}}\frac{\lVert
U^{\prime}(t)\rVert_{\mathscr{L}}^{2}}{2}e^{ct}dt\leq\mathbf{E}_{c}(U)-\frac{a}{c}e^{cT}$
(5.7)

and

$\int_{{\mathbb{R}}}\lvert\mathcal{E}(U(t))\rvert
e^{ct}dt\leq\mathbf{E}_{c}(U)-\frac{a}{c}e^{cT}.$ (5.8)

Finally, we have that for all $t\in{\mathbb{R}}$,

$\int_{t}^{+\infty}\lVert
U^{\prime}(s)\rVert_{\mathscr{L}}ds\leq\left(\left(\mathbf{E}_{c}(U)-\frac{a}{c}e^{cT}\right)\frac{e^{-ct}}{c}\right)^{\frac{1}{2}}$
(5.9)

###### Proof.

Using (5.5) in Lemma 5.1, we get that

$\int_{{\mathbb{R}}}\frac{\lVert
U^{\prime}(t)\rVert_{\mathscr{L}}^{2}}{2}e^{ct}dt\leq\mathbf{E}_{c}(U)-\int_{-\infty}^{T}\mathcal{E}(U(t))e^{ct},$
(5.10)

which, by (H1’), implies that (5.7) holds. Inequality (5.8) is obtained in the
same fashion. Finally, we have that (5.9) follows by combining (5.7) with
Cauchy-Schwartz inequality. ∎

The previous results allow to prove the following convergence properties at
$+\infty$ for finite energy functions in $X_{T}$:

######  Lemma 5.3.

Assume that (H1’) and (H5’) hold. Let $c>0$ and $T\geq 1$. Take $U\in X_{T}$
such that $\mathbf{E}_{c}(U)<+\infty$. Then, we have that there exists a
subsequence $(t_{n})_{n\in{\mathbb{N}}}$ in ${\mathbb{R}}$ such that
$t_{n}\to+\infty$ as $n\to\infty$ and

$\lim_{n\to\infty}\mathcal{E}(U(t_{n}))e^{ct_{n}}=0.$ (5.11)

Moreover, there exists $\mathbf{v}^{+}(U)\in\mathscr{F}^{+}$ such that for all
$t\in{\mathbb{R}}$ it holds

$\lVert
U(t)-\mathbf{v}^{+}(U)\rVert_{\mathscr{L}}^{2}\leq\left(\frac{\mathbf{E}_{c}(U)-\frac{a}{c}e^{cT}}{c}\right)e^{-ct}.$
(5.12)

That is, $U$ tends to $\mathbf{v}^{+}(U)$ at $+\infty$ with an exponential
rate of convergence and with respect to the $\mathscr{L}$-norm.

###### Proof.

We have by (5.8) in Lemma 5.2 that
$t\in{\mathbb{R}}\to\mathcal{E}(U(t))e^{ct}\in{\mathbb{R}}$ belongs to
$L^{1}({\mathbb{R}})$ because $\mathbf{E}_{c}(U)<+\infty$. Therefore,
combining with (5.5) in Lemma 4.1, we obtain (5.11).

Subsequently, notice that (5.9) in Lemma 5.2 and the fact that
$\mathbf{E}_{c}(U)<+\infty$ gives the existence of
$\mathbf{v}^{+}(U)\in\mathscr{F}^{+}$ such that $\lim_{t\to+\infty}\lVert
U(t)-\mathbf{v}^{+}(U)\rVert_{\mathscr{L}}=0$. Therefore, fix
$t\in{\mathbb{R}}$ and notice that for any $\tilde{t}>t$ we have

$\lVert U(\tilde{t})-U(t)\rVert_{\mathscr{L}}\leq\int_{t}^{\tilde{t}}\lVert
U^{\prime}(s)\rVert_{\mathscr{L}}ds\leq\int_{t}^{+\infty}\lVert
U^{\prime}(s)\rVert_{\mathscr{L}},$ (5.13)

which by (5.9) in Lemma 5.2 means that

$\lVert
U(\tilde{t})-U(t)\rVert_{\mathscr{L}}^{2}\leq\left(\frac{\mathbf{E}_{c}(U)-\frac{a}{c}e^{cT}}{c}\right)e^{-ct}.$
(5.14)

Therefore, passing to the limit $\tilde{t}\to+\infty$ we obtain (5.12), also
due to the fact that $U$ is continuous with respect to the $\mathscr{L}$-norm.

∎

######  Remark 5.1.

Notice that (5.12) in Lemma 5.3 does not imply convergence of $\mathcal{E}(U)$
towards 0 at $+\infty$, due to the fact that $\mathcal{E}$ is not continuous
with respect to the $\mathscr{L}$ norm.

######  Remark 5.2.

Regarding the behavior at $-\infty$, notice that we can only say that if $U\in
X_{T}$ is such that $\mathbf{E}_{c}(U)<+\infty$, then $\mathcal{E}(U)$ does
not go to $+\infty$ faster than $e^{ct}$ at the limit $t\to-\infty$. That is,
almost nothing can be said for generic finite energy solutions regarding their
behavior at $-\infty$.

###  5.3 The infima of $\mathbf{E}_{c}$ in $X_{T}$ are well defined

Once we have defined the spaces $X_{T}$, we show that the corresponding
infimum of $\mathbf{E}_{c}$ is well defined as a real number for all $c>0$.
Set

$\mathbf{m}_{c,T}:=\inf_{U\in X_{T}}\mathbf{E}_{c}(U)\in[-\infty,+\infty).$
(5.15)

We have the following:

######  Lemma 5.4.

Assume that (H1’) and (H2’) hold. Fix
$\hat{\mathbf{v}}^{\pm}\in\mathscr{F}^{\pm}$. Let $c>0$ and $T\geq 1$. For all
$T\geq 1$ the function

$\Psi(t):=\begin{cases}\hat{\mathbf{v}}^{-}&\mbox{ if }t\leq-1,\\\
\frac{1-t}{2}\hat{\mathbf{v}}^{-}+\frac{t+1}{2}\hat{\mathbf{v}}^{+}&\mbox{ if
}-1\leq t\leq 1,\\\ \hat{\mathbf{v}}^{+}&\mbox{ if }t\geq 1,\end{cases}$
(5.16)

belongs to $X_{T}$. Moreover, for all $c>0$

$\mathbf{E}_{c}(\Psi)<+\infty.$ (5.17)

Furthermore, we have

$-\infty<\mathbf{m}_{c,T}<+\infty.$ (5.18)

###### Proof.

It is clear that $\Psi\in X_{T}$. We now show that (5.17) holds. Notice first
that

$\int_{-\infty}^{1}\mathbf{e}_{c}(\Psi)=\int_{-\infty}^{1}ae^{ct}dt=\frac{a}{c}e^{c},$
(5.19)

where $a$ is the minimum value from (H1’). Subsequently, we have

$\int_{1}^{+\infty}\mathbf{e}_{c}(\Psi)=0$ (5.20)

and

$\displaystyle\int_{-1}^{1}\mathbf{e}_{c}(\Psi)$
$\displaystyle=\int_{-1}^{1}\left[\frac{\left\lVert\hat{\mathbf{v}}^{+}-\hat{\mathbf{v}}^{-}\right\rVert_{\mathscr{L}}^{2}}{8}+\mathcal{E}\left(\frac{1-t}{2}\hat{\mathbf{v}}^{-}+\frac{t+1}{2}\hat{\mathbf{v}}^{+}\right)\right]e^{ct}dt$
(5.21)
$\displaystyle\leq\left[\frac{\left\lVert\hat{\mathbf{v}}^{+}-\hat{\mathbf{v}}^{-}\right\rVert_{\mathscr{L}}^{2}}{4}+2\max_{t\in[-1,1]}\mathcal{E}\left(\frac{1-t}{2}\hat{\mathbf{v}}^{-}+\frac{t+1}{2}\hat{\mathbf{v}}^{+}\right)\right]\frac{e^{c}-e^{-c}}{c}$
(5.22)

and we have

$\max_{t\in[-1,1]}\mathcal{E}\left(\frac{1-t}{2}\hat{\mathbf{v}}^{-}+\frac{t+1}{2}\hat{\mathbf{v}}^{+}\right)<+\infty,$
(5.23)

by (H2’). Therefore, we have obtained $E_{c}(\Psi)<+\infty$, which readily
implies that $\mathbf{m}_{c,T}<+\infty$. In order to establish (5.18), we
still need to show that $\mathbf{m}_{c,T}>-\infty$. For that purpose, let
$U\in X_{T}$. By (5.5) in Lemma 5.1, we have

$\int_{T}^{+\infty}\mathbf{e}_{c}(U)\geq 0.$ (5.24)

We also have

$\int_{-\infty}^{T}\mathbf{e}_{c}(U)\geq\int_{-\infty}^{T}a\mathbf{e}^{ct}dt=\frac{a}{c}e^{cT}.$
(5.25)

That is

$\forall U\in X_{T},\hskip
5.69054pt\mathbf{E}_{c}(U)\geq\frac{a}{c}e^{cT}>-\infty,$ (5.26)

which means that $\mathbf{m}_{c,T}>-\infty$. ∎

The next goal will be to show that, under the proper assumptions, we have that
for any $c>0$ and $T\geq 1$, the infimum values defined in (5.15) are
attained. Such a fact is not hard to prove since the constraints that define
the spaces $X_{T}$ allow to restore compactness. It relies on some properties
that will be proven in the next subsection.

###  5.4 General continuity and semi-continuity results

We now provide some results which address continuity and semicontinuity
properties of the energies $\mathbf{E}_{c}$ in the spaces $X_{T}$. Such
properties will allow us to show that the infimum values defined in (5.15) are
attained under the proper assumptions. They will be also be useful in a more
advanced stage of the proof, when the constrains will be removed. For now, we
essentially adapt some results from [7] to our setting.

Our first result is essentially Lemma 26 in [7]:

######  Lemma 5.5.

Assume that (H1’) holds. Fix $T\geq 1$ and $U\in X_{T}$. Consider the set

$A_{T,U}:=\\{c>0:\mathbf{E}_{c}(U)<+\infty\\}.$ (5.27)

Then, if $c\in A_{T,U}$, then $(0,c]\subset A_{T,U}$. Moreover, the
correspondence

$c\in A_{T,U}\to\mathbf{E}_{c}(U)\in{\mathbb{R}}$ (5.28)

is continuous.

###### Proof.

Let $c\in A_{T,U}$. On the one hand, inequality (5.5) in Lemma 5.1 gives

$0\leq\int_{T}^{+\infty}\mathbf{e}_{c}(U(t))dt\leq\mathbf{E}_{c}(U)-\int_{-\infty}^{T}\mathcal{E}^{-}(U(t))e^{ct}dt\leq\mathbf{E}_{c}(U)-ae^{cT}<+\infty,$
(5.29)

which implies that a. e. in $(T,+\infty)$

$0\leq\mathbf{e}_{c}(U(\cdot))\in L^{1}((T,+\infty)).$ (5.30)

Therefore, if $c^{\prime}\leq c$ we have that a. e. in $(T,+\infty)$

$0\leq\mathbf{e}_{c^{\prime}}(U(\cdot))\leq\mathbf{e}_{c}(U(\cdot))\in
L^{1}(T,+\infty).$ (5.31)

On the other hand,

$\int_{-\infty}^{T}\lvert\mathbf{e}_{c}(U(t))\rvert
dt\leq\mathbf{E}_{c}(U)+2ae^{cT}<+\infty$ (5.32)

because $\mathbf{e}_{c}(U(\cdot))$ is non-negative a. e. in $[T,+\infty)$. The
previous inequality shows

$\lvert\mathbf{e}_{c}(U(\cdot))\rvert\in L^{1}((-\infty,T)),$ (5.33)

and we have that a. e. in $(-\infty,T)$

$\lvert\mathbf{e}_{c^{\prime}}(U(\cdot))\rvert\leq\lvert\mathbf{e}_{c}(U(\cdot))\rvert\in
L^{1}((-\infty,T)).$ (5.34)

Combining (5.31) and (5.34), we obtain that $\lvert e_{c}(U(\cdot))\rvert\in
L^{1}({\mathbb{R}})$, meaning that $c^{\prime}\in A_{T,U}$. Hence, we have
$(0,c]\subset A_{T,U}$ as we wanted to show.

Consider now a sequence $(c_{n})_{n\in{\mathbb{N}}}$ in $A_{T,U}$ such that
$c_{n}\to c_{\infty}\in A_{T,U}$. The sequence $(c_{n})_{n\in{\mathbb{N}}}$ is
convergent (so in particular it is bounded), meaning that in case it does not
attain its $\sup$ we must have $c_{\infty}=\sup_{n\in{\mathbb{N}}}c_{n}$.
Therefore, we can set

$\hat{c}:=\begin{cases}c_{\infty}&\mbox{ if
}c_{\infty}=\sup_{n\in{\mathbb{N}}}c_{n},\\\
\max_{n\in{\mathbb{N}}}c_{n}&\mbox{ otherwise},\end{cases}$ (5.35)

and we obviously have $\hat{c}\in A_{T,U}$. As a consequence, (5.31) and
(5.34) imply that a. e. in ${\mathbb{R}}$

$\forall n\in{\mathbb{N}},\hskip
5.69054pt\lvert\mathbf{e}_{c_{n}}(U(\cdot))\rvert\leq\lvert\mathbf{e}_{\hat{c}}(U(\cdot))\rvert\in
L^{1}({\mathbb{R}}).$ (5.36)

Since we also have
$\mathbf{e}_{c_{n}}(U(\cdot))\to\mathbf{e}_{c_{\infty}}(U(\cdot))$ pointwise
a. e. in ${\mathbb{R}}$, the Dominated Convergence Theorem gives the result. ∎

We now show a lower semicontinuity result, which in particular will imply the
existence of the constrained solutions:

######  Lemma 5.6.

Assume that (H1’), (H3’) and (H4’) hold. Let $T\geq 1$ be fixed. Let
$(U_{n}^{i})_{n\in{\mathbb{N}}}$ be a sequence in $X_{T}$ and
$(c_{n})_{n\in{\mathbb{N}}}$ a convergent sequence of positive real numbers
such that

$\sup_{n\in{\mathbb{N}}}\mathbf{E}_{c_{n}}(U_{n}^{i})<+\infty.$ (5.37)

Then, there exists a sequence $(U_{n})_{n\in{\mathbb{N}}}$ in $X_{T}$ and
$U_{\infty}\in X_{T}$ such that up to extracting a subsequence in
$(U_{n},c_{n})_{n\in{\mathbb{N}}}$ it holds

$\forall n\in{\mathbb{N}},\hskip
5.69054pt\mathbf{E}_{c_{n}}(U_{n})=\mathbf{E}_{c_{n}}(U_{n}^{i}),$ (5.38)
$\forall t\in{\mathbb{R}},\hskip 5.69054ptU_{n}(t)\rightharpoonup
U_{\infty}(t)\mbox{ weakly in }\mathscr{L}$ (5.39)
$U_{n}^{\prime}e^{c_{n}\mathrm{Id}/2}\rightharpoonup
U_{\infty}^{\prime}e^{c_{\infty}\mathrm{Id}/2}\mbox{ weakly in
}L^{2}({\mathbb{R}},\mathscr{L})$ (5.40)

and

$\mathbf{E}_{c_{\infty}}(U_{\infty})\leq\liminf_{n\to\infty}\mathbf{E}_{c_{n}}(U_{n}),$
(5.41)

where $c_{\infty}:=\lim_{n\to\infty}c_{n}$.

###### Proof.

Denote $M:=\sup_{n\in{\mathbb{N}}}E_{c_{n}}(U_{n}^{i})$, which is finite by
(5.37). We will now use (H4’). We assume that 2. holds, the argument when 1.
holds being similar and easier. Fix any $\mathbf{v}^{+}\in\mathscr{F}^{\pm}$
and for all $n\in{\mathbb{N}}$, set $v_{n}:=U_{n}^{i}(T)\in\mathscr{F}^{+}$.
Define

$U_{n}:t\in{\mathbb{R}}\to\hat{P}_{(v_{n},\mathbf{v}^{+})}(U^{i}_{n}(t)),$
(5.42)

where $\hat{P}_{\mathbf{v}^{+}}$ is the differentiable operator introduced in
(H4’). We apply the properties summarized in 2. of (H4’). Notice that for all
$n\in{\mathbb{N}}$ we have $U_{n}\in X_{T}$ due to (4.15). The energy equality
(5.38) follows from (4.16) and (4.17). Moreover, (4.14) implies that for all
$n\in{\mathbb{N}}$

$P^{+}(U_{n}(T))=P^{+}(\hat{P}_{(v_{n},\mathbf{v}^{+})}(U^{i}_{n}(T)))=P^{+}(\hat{P}_{(v_{n},\mathbf{v}^{+})}(v_{n}))=\mathbf{v}^{+},$
(5.43)

which in particular means

$\lVert U_{n}(T)-\mathbf{v}^{+}\rVert_{\mathscr{L}}\leq\frac{r_{0}^{+}}{2}.$
(5.44)

Notice now that $\mathcal{E}(U(\cdot))$ is non-negative in $[T,+\infty)$ as
$U\in X_{T}$ by (5.5) in Lemma 5.1, therefore

$\displaystyle\forall n\in{\mathbb{N}},\hskip
5.69054pt\frac{1}{2}\int_{\mathbb{R}}\lVert
U^{\prime}_{n}(t)\rVert_{\mathscr{L}}^{2}e^{c_{n}t}dt$ $\displaystyle\leq
M-\int_{\mathbb{R}}\mathcal{E}(U_{n}(t))e^{c_{n}t}dt$ (5.45)
$\displaystyle\leq
M-\int_{-\infty}^{T}\mathcal{E}(U_{n}(t))e^{c_{n}t}dt\leq\sup_{n\in{\mathbb{N}}}\left\\{M-\frac{a}{c_{n}}e^{c_{n}T}\right\\}<+\infty.$
(5.46)

That is, we have that
$(U_{n}^{\prime}e^{c_{n}\mathrm{Id}/2})_{n\in{\mathbb{N}}}$ is uniformly
bounded in $L^{2}({\mathbb{R}},\mathscr{L})$. Therefore, there exists
$\tilde{U}\in L^{2}({\mathbb{R}},\mathscr{L})$ such that

$U_{n}^{\prime}e^{c_{n}\mathrm{Id}/2}\rightharpoonup\tilde{U}\mbox{ weakly in
}L^{2}({\mathbb{R}},\mathscr{L})$ (5.47)

up to subsequences. Such a thing implies

$\int_{\mathbb{R}}\lVert\tilde{U}(t)\rVert_{\mathscr{L}}^{2}dt\leq\liminf_{n\to\infty}\int_{\mathbb{R}}\lVert
U^{\prime}_{n}(t)\rVert_{\mathscr{L}}^{2}e^{c_{n}t}dt.$ (5.48)

Now, notice that by (5.44) we have that $(U_{n}(T))_{n\in{\mathbb{N}}}$ is
bounded in $\mathscr{L}$. Therefore, up to an extraction there exists
$v_{\infty}\in\mathscr{L}$ such that

$U_{n}(T)\rightharpoonup v_{\infty}\mbox{ in }\mathscr{L}.$ (5.49)

As in [69], we point out that

$\forall t\in{\mathbb{R}},\forall n\in{\mathbb{N}},\hskip
5.69054ptU_{n}(t)=U_{n}(T)+\int_{T}^{t}U_{n}^{\prime}(s)ds.$ (5.50)

Now, notice that for all $t\in{\mathbb{R}}$ we have
$\mathbf{1}_{(0,t)}e^{-c_{n}\mathrm{Id}/2}\to\mathbf{1}_{(0,t)}e^{-c_{\infty}\mathrm{Id}/2}$
in $L^{\infty}({\mathbb{R}})$, where $\mathbf{1}$ states for the indicator
function of a set. Therefore, we obtain by (5.47) and (5.49)

$\forall t\in{\mathbb{R}},\hskip 5.69054ptU_{n}(t)\rightharpoonup
U_{\infty}(t):=v_{\infty}+\int_{T}^{t}\tilde{U}(s)e^{-c_{\infty}s/2}ds,$
(5.51)

which gives (5.39). Moreover, we have that $U_{\infty}\in
H^{1}_{\mathrm{loc}}({\mathbb{R}},\mathscr{L})$ and
$U_{\infty}^{\prime}=\tilde{U}e^{-c_{\infty}\mathrm{Id}/2}$, meaning by (5.47)
that (5.40) also holds.

Recall now that $\mathcal{E}$ is lower semicontinuous on $\mathscr{L}$ by
(H1’), so that (5.39) gives

$\forall t\in{\mathbb{R}},\hskip
5.69054pt\mathcal{E}(U_{\infty}(t))\leq\liminf_{n\to\infty}\mathcal{E}(U_{n}(t)).$
(5.52)

We need to show that $U_{\infty}\in X_{T}$ and to establish the inequality
(5.41).

  * •

We begin by showing that $U_{\infty}\in X_{T}$. We need to show that for all
$t\in[T,+\infty)$, it holds $U_{\infty}(t)\in\mathscr{F}^{+}_{r_{0}^{+}/2}$
and similarly for $(-\infty,-T]$. Fix $t\in[T,+\infty)$. We have that