# Heteroclinic traveling waves of 1D parabolic systems with degenerate stable
states

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

We study the existence of traveling waves for the parabolic system

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

where $W$ is a potential bounded below and possessing two minima at different
levels. We say that $\mathfrak{w}$ is a traveling wave solution of the
previous equation if there exist $c^{\star}>0$ and
$\mathfrak{u}\in\mathcal{C}^{2}({\mathbb{R}},{\mathbb{R}}^{k})$ such that
$\mathfrak{w}(t,x)=\mathfrak{u}(x-ct)$. For a class of potentials $W$,
heteroclinic traveling waves of the previous equation where shown to exist by
Alikakos and Katzourakis [4]. More precisely, assuming the existence of two
local minimizers of $W$ at different levels which, in addition, satisfy some
non-degeneracy assumptions, the authors in [4] show the existence of a speed
$c^{\star}>0$ and profile
$\mathfrak{u}\in\mathcal{C}^{2}({\mathbb{R}},{\mathbb{R}}^{k})$ such that
$\mathfrak{u}$ connects the two local minimizers at infinity. In this paper,
we show that the non-degeneracy assumption on the local minima can be dropped
and replaced by another one which allows for potentials possessing degenerate
minima. As we do in [22], our main result is in fact proven for curves which
take values in a general Hilbert space and the main result is deduced as a
particular case, in the spirit of the earlier works by Monteil and
Santambrogio [19] and Smyrnelis [28] devoted to the existence of stationary
heteroclinics.

##  1 Introduction

We are concerned with the following one-dimensional parabolic system of
reaction-diffusion type

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

where $w:[0,+\infty)\times{\mathbb{R}}\to{\mathbb{R}}^{k}$ and $W$ is an
unbalanced double-well potential with possibly degenerate minima, meaning that
we can find two minimum points of $W$ at different levels and we do not assume
that $D^{2}W$ is positive definite at such points. For potentials $W$ which
possess two non-degenerate minimum points $a^{-},a^{+}$ in ${\mathbb{R}}^{k}$
such that $W(a^{-})<0=W(a^{+})$ satisfying some additional assumptions
(namely, non-degeneracy and local radial monotonicity), Alikakos and
Katzourakis [4] (see also Lucia, Muratov and Novaga [17] and Risler [23])
showed the existence of traveling wave solutions for (1.1) with heteroclinic
behavior at infinity. More precisely, they showed that there exist
$c^{\star}>0$ and
$\mathfrak{u}\in\mathcal{C}^{2}({\mathbb{R}},{\mathbb{R}}^{k})$ such that

$-c^{\star}\mathfrak{u}^{\prime}-\mathfrak{u}^{\prime\prime}=-\nabla_{\mathbf{u}}W(\mathfrak{u})\mbox{
in }{\mathbb{R}}$ (1.2)

and

$\lim_{t\to\pm\infty}\mathfrak{u}(t)=a^{\pm}.$ (1.3)

Such a thing implies that
$\mathfrak{w}:[0,+\infty)\times{\mathbb{R}}\to{\mathbb{R}}^{k}$ defined as

$\mathfrak{w}(t,x)=\mathfrak{u}(x-ct)$ (1.4)

for $(t,x)\in[0,+\infty)\times{\mathbb{R}}$ is a solution to (1.1). More
precisely, $\mathfrak{w}$ is a traveling wave solution of (1.1) which
propagates with speed $c^{\star}$ and with profile $\mathfrak{u}$. Our main
contribution here is to show that such solutions exist when we consider some
class potentials $W$ with possibly degenerate and non-isolated minima. More
precisely, we assume that $W$ takes the minimum values on sets, instead of
isolated points, and that it satisfies suitable properties around such sets.
Our motivation comes from a closely related work by the author [22]. There,
traveling waves solutions for some classes of two dimensional parabolic Allen-
Cahn systems are obtained. The approach of the proof is to deduce the result
from a more general one, proved in an abstract setting on Hilbert spaces,
following ideas that worked for similar problems (see Monteil and Santambrogio
[19] and Smyrnelis [28]). More precisely, one assumes that the potential $W$
is defined in a Hilbert space $\mathscr{H}$ which has no restriction in its
dimension, so that in case $\mathscr{H}$ is infinite-dimensional and $W$ is
suitably chosen, one recovers a system which has more than one dimension in
space. In this paper, we show that the abstract approach used in our previous
work can be also used to recover results in the finite-dimensional setting.
For this we mean that $\mathscr{H}={\mathbb{R}}^{k}$, so that the resulting
equation is of the type (1.1), which is 1D on space. These results are not
included in [4]. The abstract result of [22] applies to our problem, but the
main result of this paper follows by a modified version of such abstract
result, in which a key assumption in [22] is replaced by another one playing
the same role. Such an assumption is the natural adaptation of the one used by
Alikakos and Katzourakis [4] to the case of degenerate minima, and we use it
following the ideas from [4] but with more involved arguments, which is not
surprising since the fact that the minima might be degenerate adds extra
difficulties. Nevertheless, at present we are only able to apply the abstract
result of this paper to the finite-dimensional problem, meaning that we do not
use the full strength of the abstract setting, which could be useful in other
situations. We also mention that the result we obtain is weaker than that by
Alikakos and Katzourakis [4] since we cannot prove convergence of the profile
to the global minimum at $-\infty$, only that it stays close to it for small
enough times. This might obey to merely technical limitations, but it could
also be due to the fact that we allow for degenerate minima. However, we show
that under an additional assumption (essentially, either an upper bound
$W(a^{+})$ or the set of global minimizers is a singleton, see (H4)) one
essentially recovers the full result from [4] for our setting.

The scheme of the proof of the abstract result of this paper is analogous to
that in [22], which is based on the variational approach of Alikakos and
Katzourakis [4], inspired by that in Alikakos and Fusco [2]. While the
existence of a variational structure in the context of (some) reaction-
diffusion problems is known since Fife and McLeod [9, 10], it has not been
widely used in the previous literature. Besides [4] and our [22], other
references which use such a variational structure for studying traveling waves
in reaction-diffusion problems are Bouhours and Nadin [7], Lucia, Muratov and
Novaga [16, 17], Muratov [20], Risler [23, 24, 25] and, more recently, Chen,
Chien and Huang [8].

In [4, 22], one considers a family of constrained minimization problems and
the main difficulty of the problem can be reduced to excluding a degenerate
oscillatory behavior for the minimizers of these problems, as the energy
density under consideration changes its sign due to the fact that the
connected minima are at different levels. In [4], this is done by imposing a
non-degeneracy assumption, as well as radial monotonicity properties, on the
local minima and then using the ODE system along with optimality of the
minimizers. For several reasons, an assumption of this type was not available
to us in [22], so that we needed to replace it by a different one which
allowed an analogous type of conclusion. Such an assumption consists
essentially on an upper bound on the difference between the energy of the
minima, which implies that the energy of the constrained minimizers to. The
abstract result of this paper further replaces the previous assumption by
another one which plays the same role (but which does not apply to the problem
considered in [21]) and still allows for potentials with degenerate minima.
More precisely, it allows us to apply to the constrained minimizers an
argument based on the use of the ODE system, in the spirit of [4] but more
involved in some aspects. The reader is referred to [22] for a detailed scheme
of proof as well as a discussion on related literature.

In order to conclude this paragraph, we recall that in the realm of scalar
reaction-diffusion problems one can obtain important results of existence and
qualitative properties of traveling waves by means of the maximum principle
and comparison results. There is a large literature on the subject, the more
classical papers are Fife and McLeod [9, 10],Aronson and Weinberger [5],
Berestycki and Nirenberg [6], all devoted to the Fisher KPP equation
(introduced by Fisher [11], Kolmogorov, Petrovsky and Piskunov [15]). In the
context of the Allen-Cahn equation (which falls in a different framework than
the Fisher-KPP), the stability of traveling waves was shown by Matano, Nara
and Taniguchi [18]. However, the maximum and comparison principles are no
longer available (in general) for vector-values problems (i. e., systems), so
that one needs different tools. There is also a huge amount of research in
this direction, we refer to the books by Smoller [27] as well as Volpert,
Volpert and Volpert [29]. However, the use of variational methods is not
covered in the previously cited books and it has not been extensively used in
the parabolic context, which is in contrast with the framework of dispersive
equations.

### Acknowledgments

I wish to thank my PhD advisor Fabrice Bethuel for his encouragement and 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 Statement of the main results
    1. 2.1 Main assumptions and result
    2. 2.2 Conditions at infinity
    3. 2.3 Min-max characterization of the speed
  3. 3 Extensions and open problems
    1. 3.1 Other classes of potentials
    2. 3.2 The problem of traveling waves for 2D parabolic Allen-Cahn systems
    3. 3.3 Traveling waves between homoclinics
  4. 4 The abstract setting
  5. 5 Proof of the abstract results
    1. 5.1 Preliminaries
    2. 5.2 Two (semi-)continuity results
    3. 5.3 Existence of an infimum for $\mathbf{E}_{c}$ in $Y_{T}$
    4. 5.4 The comparison result
    5. 5.5 Existence of an unconstrained solution
    6. 5.6 Uniqueness of the speed
    7. 5.7 Proof of Theorem 4 completed
    8. 5.8 Asymptotic behavior of the constrained solutions
    9. 5.9 Proof of Theorem 5 completed
    10. 5.10 Proof of Theorem 6 completed
    11. 5.11 On the proof of Theorem 7
  6. 6 Proofs of the main results
    1. 6.1 Proof of Theorem 1
    2. 6.2 Proof of Theorem 2
    3. 6.3 Proof of Theorem 3

###### Contents

  1. 1 Introduction
  2. 2 Statement of the main results
    1. 2.1 Main assumptions and result
    2. 2.2 Conditions at infinity
    3. 2.3 Min-max characterization of the speed
  3. 3 Extensions and open problems
    1. 3.1 Other classes of potentials
    2. 3.2 The problem of traveling waves for 2D parabolic Allen-Cahn systems
    3. 3.3 Traveling waves between homoclinics
  4. 4 The abstract setting
  5. 5 Proof of the abstract results
    1. 5.1 Preliminaries
    2. 5.2 Two (semi-)continuity results
    3. 5.3 Existence of an infimum for $\mathbf{E}_{c}$ in $Y_{T}$
    4. 5.4 The comparison result
    5. 5.5 Existence of an unconstrained solution
    6. 5.6 Uniqueness of the speed
    7. 5.7 Proof of Theorem 4 completed
    8. 5.8 Asymptotic behavior of the constrained solutions
    9. 5.9 Proof of Theorem 5 completed
    10. 5.10 Proof of Theorem 6 completed
    11. 5.11 On the proof of Theorem 7
  6. 6 Proofs of the main results
    1. 6.1 Proof of Theorem 1
    2. 6.2 Proof of Theorem 2
    3. 6.3 Proof of Theorem 3

##  2 Statement of the main results

###  2.1 Main assumptions and result

The main result of this paper is Theorem 1, which establishes the existence of
an heteroclinic traveling wave with the uniqueness properties on the speed
already found in [4] and exponential convergence at $+\infty$. The behavior at
$-\infty$ is however weaker than that proved in [4], which might be due to the
fact that we allow for degenerate minima. The necessary assumptions for
Theorem 1 are (H1), (H2) and (H3). In Theorem 2 we show that one can obtain
convergence of the profile at $-\infty$ (so that one recovers the full result
from [4] if one assumes either an upper bound on the difference of the energy
of the minima or that the set of global minimizers is a singleton, see (H4).
Finally, in Theorem 3 we show that the formula and min-max characterization of
the speed given in [4] also holds here, again under assumption (H4).

######  (H1).

$W\in\mathcal{C}^{3}({\mathbb{R}}^{k})$. There exist $\mathcal{A}^{-}$ and
$\mathcal{A}^{+}$ subsets of ${\mathbb{R}}^{k}$ and $\mathfrak{h}<0$ such that
for all $a^{-}\in\mathcal{A}^{-}$ we have $W(a^{-})=\mathfrak{h}<0$, for all
$a^{+}\in\mathcal{A}^{+}$ we have $W(a^{+})=0$ and for all
$u\in{\mathbb{R}}^{k}$ we have $W(\mathbf{u})\geq\mathfrak{h}$. There exist
$\rho_{0}^{\pm}>0$ such that for all $\mathbf{u}\in{\mathbb{R}}^{k}$ such that
$\mathrm{dist}(\mathbf{u},\mathcal{A}^{\pm})\leq\rho_{0}^{\pm}$ it holds

$\mathrm{dist}(\mathbf{u},\mathcal{A}^{\pm})^{2}\leq
C^{\pm}(W(\mathbf{u})-\min\\{\pm(-\mathfrak{h}),0\\})$ (2.1)

for some $C^{\pm}>0$. Moreover, there exists a unique
$a^{\pm}(\mathbf{u})\in\mathcal{A}^{\pm}$ such that

$\mathrm{dist}(\mathbf{u},\mathcal{A}^{\pm})=\lvert\mathbf{u}-a^{\pm}(\mathbf{u})\rvert$
(2.2)

and the mappings $p^{\pm}:\mathbf{u}\to a^{\pm}(\mathbf{u})$ defined on

$\mathcal{A}^{\pm}_{\rho^{\pm}}:=\\{\mathbf{u}\in{\mathbb{R}}^{k}:\mathrm{dist}(\mathbf{u},\mathcal{A}^{\pm})\leq
r_{0}^{\pm}\\}$ (2.3)

are $C^{2}$. Moreover, we have that there exist $c_{0}>0$ and $R_{0}>0$ such
that for all $\mathbf{u}\in{\mathbb{R}}^{k}$ such that
$\lvert\mathbf{u}\rvert\geq R_{0}$, we have

$\langle\nabla W(\mathbf{u}),\mathbf{u}\rangle\geq
c_{0}\lvert\mathbf{u}\rvert^{2}$ (2.4)

In particular, the existence of the projection mappings $p^{\pm}$ holds if the
sets $\mathcal{A}^{\pm}$ are convex and smooth. The next assumption writes as
follows:

######  (H2).

One of the two follows:

  1. 1.

$\mathcal{A}^{\pm}$ is bounded.

  2. 2.

For any
$(\mathbf{u},a^{\pm})\in\mathcal{A}_{\rho_{0}^{\pm}}^{\pm}\times\mathcal{A}^{\pm}$,
there exist maps
$p^{\pm}_{(\mathbf{u},a^{\pm})}:{\mathbb{R}}^{k}\to{\mathbb{R}}^{k}$ such that
$p^{\pm}_{(\mathbf{u},a^{\pm})}(\mathbf{u})=a^{\pm}$,
$W(p^{\pm}_{(\mathbf{u},a^{\pm})}(\mathbf{u}))=W(\mathbf{u})$ and
$\mathrm{dist}(p^{\pm}_{(\mathbf{u},a^{\pm})}(\mathbf{u}),\mathcal{A}^{\pm})=\mathrm{dist}(\mathbf{u},\mathcal{A}^{\pm})$.
Moreover, $p^{\pm}_{(\mathbf{u},a^{\pm})}$ is differentiable and $\lvert
D(p^{\pm}_{(\mathbf{u},a^{\pm})})(\mathbf{u}_{1},\mathbf{u}_{2})\rvert=\lvert\mathbf{u}_{2}\rvert$
for any $(\mathbf{u}_{1},\mathbf{u}_{2})\in\left({\mathbb{R}}^{k}\right)^{2}$.

In particular, if $W$ is invariant with respect to some group action inside
$\mathcal{A}_{\rho_{0}^{\pm}}^{\pm}$ and $\mathcal{A}^{\pm}$ is a single orbit
with respect to such an action, then 2. in (H2) is met. However, less rigid
structures are also allowed by 2. Finally, for each $h\in{\mathbb{R}}$ define
the level set

$W^{h}:=\\{\mathbf{u}\in{\mathbb{R}}^{k}:W(\mathbf{u})\leq h\\}$ (2.5)

and notice that by continuity and (H1) we can find $h_{0}>0$ such that

$W^{h_{0}}=W^{h_{0},-}\cup W^{h_{0},+}$ (2.6)

with $W^{h_{0},\pm}$ closed and disjoint and such that
$W^{h_{0},+}\subset\mathcal{A}_{\rho_{0}^{+}/2}^{+}$. In particular

$W^{h_{0},-}\cap\mathcal{A}_{\rho_{0}^{+}/2}^{+}=\emptyset.$ (2.7)

For each $h\leq h_{0}$, we can then write

$W^{h}=W^{h,-}\cup W^{h,+}$ (2.8)

with $W^{h,\pm}$ closed and disjoint and such that $W^{h,\pm}\subset
W^{h_{0},\pm}$. It is then clear that

$W^{0}=W^{0,-}\cup\mathcal{A}^{+},W^{\mathfrak{h}}=\mathcal{A}^{-}$ (2.9)

and for all $h<0$,

$W^{h}=W^{h,-}.$ (2.10)

Under these notations, our last assumption writes as follows:

######  (H3).

For any $h\leq h_{0}$, the set $W^{h,-}$ is convex. For any $h\leq h_{0}$,
$\mathbf{u}\in W^{h,-}$ and $a^{-}\in\mathcal{A}^{-}$ define the set

$I(h,\mathbf{u},a^{-}):=\\{\lambda\in{\mathbb{R}}:W(a^{-}+\lambda(\mathbf{u}-a^{-}))>h\mbox{
and }a^{-}+\lambda(\mathbf{u}-a^{-})\in W^{h_{0},-}\\}.$ (2.11)

Then, there exists $h_{-}\in(\mathfrak{h},0)$ such that

  1. 1.

$W^{h_{-}}\subset\mathcal{A}^{-}_{\rho_{0}^{-}/2}$.

  2. 2.

For some constant $\sigma>0$ it holds that for $\mathbf{u}\in W^{h_{0},-}$,
$a^{-}\in\mathcal{A}^{-}$ and $\theta\in
I((\mathfrak{h}+h_{-})/2,\mathbf{u},a^{-})$ we have

$\frac{d}{d\lambda}(W(a^{-}+\lambda(\mathbf{u}-a^{-})))(\theta)\geq\sigma.$
(2.12)

  3. 3.

For all $h\in(\mathfrak{h},h_{-}]$, there exists $\sigma(h)>0$ such that for
all $\mathbf{u}\in W^{h_{-}}$ such that $W(\mathbf{u})\geq h$, there exists
$\delta(\mathbf{u})>0$ such that for all
$\theta\in(1-\delta(\mathbf{u}),1+\delta(\mathbf{u}))$ we have

$\frac{d}{d\lambda}(W(p^{-}(\mathbf{u})+\lambda(\mathbf{u}-p^{-}(\mathbf{u})))(\theta)\geq\sigma(h)$
(2.13)

with $p(\mathbf{u})$ the projection given in (H1).

Essentially, assumption (H3) imposes some convexity on some suitable subsets
of the level sets of $W$, as well as some uniform monotinicity on segments.
Some explanatory designs are shown in Figures 2.1, 2.2, 2.3. One can see
assumption (H2) as an adaptation of the key assumption by Alikakos and
Katzourakis to the case of degenerate minima. Indeed, in case
$\mathcal{A}^{-}$ is reduced to a singleton, (H3) essentially reduces to the
hypothesis formulated in [4]. Finally, recall that, as in [4], multi-well
potentials can satisfy the assumptions (H1), (H2) and (H3) as long as they
possess a local minimum at a level higher than 0 and they are modified by an
additive constant. We now come back to the following equation for a pair
$(c,u)$

$-cu^{\prime}-u^{\prime\prime}=-\nabla_{\mathbf{u}}W(u)\mbox{ in
}{\mathbb{R}}$ (2.14)

and, assuming that (H3) holds, consider the conditions at infinity

$\exists T^{-}\in{\mathbb{R}}:\forall t\leq T^{-},\hskip 5.69054ptu(t)\in
W^{h_{-}}\mbox{ and }\exists
a^{+}(u)\in\mathcal{A}^{+}:\lim_{t\to+\infty}\lvert u(t)-a^{+}(u)\rvert=0.$
(2.15)

For $c>0$, we consider as in [4], the following weighted functional introduced
in Fife and McLeod [9, 10]

$E_{c}(v):=\int_{\mathbb{R}}e_{c}(v(t))dt:=\int_{\mathbb{R}}\left[\frac{\lvert
v^{\prime}(t)\rvert^{2}}{2}+W(v(t))\right]e^{ct}dt$ (2.16)

where $v$ belongs to the class

$\mathcal{S}:=\left\\{v\in
H^{1}_{\mathrm{loc}}({\mathbb{R}},{\mathbb{R}}^{k}):\exists T\geq 1:\forall
t\geq T,\hskip 5.69054ptv(t)\in\mathcal{A}^{+}_{\rho_{0}^{+}/2}\mbox{ and
}\forall t\leq-T,\hskip
5.69054ptv(t)\in\mathcal{A}^{-}_{\rho_{0}^{-}/2}\right\\}$ (2.17)

so that (2.14) is (at least formally) the Euler-Lagrange equation of $E_{c}$.
As in [4], we shall find the solution profile $\mathfrak{u}$ as a critical
point (in fact, a global minimizer) of $E_{c^{\star}}$ in $\mathcal{S}$ for a
suitable $c^{\star}$ which, in addition, satisfies some uniqueness properties.
In fact, assuming that (H3) holds we shall consider the class

$\overline{\mathcal{S}}:=\left\\{v\in
H^{1}_{\mathrm{loc}}({\mathbb{R}},{\mathbb{R}}^{k}):\exists T\geq 1:\forall
t\geq T,\hskip 5.69054ptv(t)\in\mathcal{A}^{+}_{\rho_{0}^{+}/2}\mbox{ and
}\forall t\leq-T,\hskip 5.69054ptv(t)\in W^{h_{-}}\right\\},$ (2.18)

which by 1. in (H3) satisfies $\overline{S}\subset\mathcal{S}$. As it was
pointed out in [4], for any $c>0$, $v\in\mathcal{S}$ and $\tau\in{\mathbb{R}}$
we have

$E_{c}(v(\cdot+\tau))=e^{-c\tau}E_{c}(v)$ (2.19)

which implies that for all $c>0$

$\inf_{v\in\mathcal{S}}E_{c}(v),\inf_{v\in\overline{\mathcal{S}}}E_{c}(v)\in\\{-\infty,0\\}$
(2.20)

and that in case the infimum above is 0, then for any $v\in\mathcal{S}$ with
$E_{c}(v)>0$ we have that $E_{c}(v(\cdot+n))\to 0$ as $n\to+\infty$. This
remark shows that one cannot expect to solve the minimization problem (2.20)
directly, so that an indirect approach is needed. We can now state the main
result of this paper:

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

Assume that (H1), (H2) and (H3) hold. Then, we have that

  1. 1.

Existence. There exist $c^{\star}>0$ and
$\mathfrak{u}\in\mathcal{C}^{2}_{\mathrm{loc}}({\mathbb{R}},{\mathbb{R}}^{k})\cap\overline{S}$
such that $(c^{\star},\mathfrak{u})$ fulfills (2.14) and (2.15) as well as the
variational characterization

$E_{c^{\star}}(\mathfrak{u})=0=\inf_{v\in\overline{\mathcal{S}}}E_{c^{\star}}(v).$
(2.21)

  2. 2.

Uniqueness of the speed. Assume that $\overline{c^{\star}}>0$ is such that

$\inf_{v\in\overline{\mathcal{S}}}E_{\overline{c^{\star}}}(v)=0$ (2.22)

and that $\overline{\mathfrak{u}}\in\mathcal{S}$ is such that
$(\overline{c^{\star}},\overline{\mathfrak{u}})$ solves (2.14) and
$E_{\overline{c^{\star}}}(\overline{\mathfrak{u}})<+\infty$. Then,
$\overline{c^{\star}}=c^{\star}$.

  3. 3.

Exponential convergence. There exists $b>c^{\star}/2$ such that

$\lim_{t\to+\infty}\lvert\mathfrak{u}(t)-a^{+}(\mathfrak{u})\rvert e^{ct}=0$
(2.23)

where $a^{+}(\mathfrak{u})$ is given by (2.15).

######  Remark 2.1.

Notice that the conditions at infinity (2.15) show that $\mathfrak{u}$ is not
constant, since we clearly have that
$\mathcal{A}^{-}_{\rho_{0}^{-}/2}\cap\mathcal{A}^{+}_{\rho^{+}_{0}/2}=\emptyset$.
Regarding the uniqueness of the speed, in particular it holds that $c^{\star}$
is unique among the class of global minimizers.

Figure 2.1: Representation of 1. in (H3). The curve represents the set
$\mathcal{A}^{-}$, while the inner convex shadowed region corresponds to the
level set $W^{h_{-}}$, which is contained in
$\mathcal{A}^{-}_{\rho_{0}^{-}/2}$ , the larger shadowed region. Finally, the
outer punctured line contains the convex region representing $W^{h_{0},-}$.
Figure 2.2: Representation of 2. in (H3). We have $\mathbf{u}$, which is
contained in $W^{h_{0},-}$ but not in $W^{(\mathfrak{h}+h_{-})/2}$ . The full
lines represent the segments $I((\mathfrak{h}+h_{-})/2,\mathbf{u},a_{1}^{-})$
and $I((\mathfrak{h}+h_{-})/2,\mathbf{u},a_{2}^{-})$ for $a_{1}^{-}$ and
$a_{2}^{-}$ in $\mathcal{A}^{-}$. The discontinuous lines complete the
previous segments into the segment starting in $a^{-}_{1}$ and $a^{-}_{2}$.
Figure 2.3: Representation of 3. in (H3). We take $\mathbf{u}\in W^{h_{-}}$
not contained in $\mathcal{A}^{-}$. The discontinuous red line is a segment
starting in $p^{-}(\mathbf{u})$ and containing $\mathbf{u}$. The blue line
represents the small portion of the segment centered in $\mathbf{u}$ contained
in the line going trough $p^{-}(\mathbf{u})$ and with length
$2\delta(\mathbf{u})$, $\delta(\mathbf{u})>0$. We ask uniform strict
monotonicity property (2.13) to hold in this segment.

Once the main result has been stated, we give some properties on the solution
$(c^{\star},\mathfrak{u})$, which are essentially the adaptation from [22] to
the current setting.

###  2.2 Conditions at infinity

We now explain how the condition at infinity (2.15) can be upgraded. In
particular, we would like conditions at infinity of the type

$\exists a^{\pm}(u)\in\mathcal{A}^{\pm}:\lim_{t\to\pm\infty}\lvert
u(t)-a^{\pm}(u)\rvert=0.$ (2.24)

Under the previous assumptions, we have no proof that $(c,\mathfrak{u})$
satisfies (2.24). However, we can establish such a behavior adding the
following assumption:

######  (H4).

We have one of the two following situations:

  1. 1.

$\mathcal{A}^{-}:=\\{a^{-}\\}$.

  2. 2.

We have that

$-\mathfrak{h}<\frac{\left(d\eta\right)^{2}}{2}$ (2.25)

where

$d:=\mathrm{dist}_{{\mathbb{R}}^{k}}(\mathcal{A}^{+}_{\rho_{0}^{+}/2},W^{h_{0},-})$
(2.26)

which is positive by the definition of $h_{0}$ and

$\eta:=\frac{1}{\frac{1}{2}(C^{-})^{2}((C^{-})^{2}+(C^{-}+1)^{2})+C^{-}}>0,$
(2.27)

where $C^{-}$ is the constant from (H1).

Essentially, (H4) requires that either $\mathcal{A}^{-}$ is a singleton (while
$\mathcal{A}^{+}$ does not need to be) or that the value $-\mathfrak{h}$ is
not too large. Then we have the following:

######  Theorem 2.

Assume that (H1), (H2), (H3) and (H4) hold. Let $(c^{\star},\mathfrak{u})$ be
a solution given by Theorem 1. Then, $\mathfrak{u}$ also satisfies (2.24).

###  2.3 Min-max characterization of the speed

We now give some results which characterize the speed $c^{\star}$. As before,
such results are very close to the ones obtained in [4] and [22]. More
specifically, in case (H4) holds, the speed $c^{\star}$ satisfies the
following additional properties, which include a min-max characterization as
in Heinze [13], Heinze, Papanicolau and Stevens [14]:

######  Theorem 3.

Assume that (H1), (H2), (H3) and (H4) hold. Let $(c^{\star},\mathfrak{u})$ be
a solution given by Theorem 1. Then, for any
$\tilde{\mathfrak{u}}\in\overline{\mathcal{S}}$ such that
$E_{c^{\star}}(\tilde{\mathfrak{u}})=0$ we have that
$(c^{\star},\tilde{\mathfrak{u}})$ solves (2.14) and

$c^{\star}=\frac{-\mathfrak{h}}{\int_{\mathbb{R}}\lvert\tilde{\mathfrak{u}}^{\prime}\rvert^{2}}.$
(2.28)

In particular, the quantity
$\int_{\mathbb{R}}\lvert\tilde{\mathfrak{u}}^{\prime}\rvert^{2}$ is well-
defined and constant among the class of globally minimizing profiles for
$E_{c^{\star}}$. Moreover, it holds

$c^{\star}=\sup\\{c>0:\inf_{v\in\overline{\mathcal{S}}}E_{c}(v)=-\infty\\}=\inf\\{c>0:\inf_{v\in\overline{\mathcal{S}}}E_{c}(v)=0\\}.$
(2.29)

######  Remark 2.2.

Combining the uniqueness part in Theorem 1 and Theorem 3, we obtain that for
any $c>c^{\star}$, the infimum

$\inf_{v\in\overline{\mathcal{S}}}E_{c}(v)=0$ (2.30)

is never attained.

##  3 Extensions and open problems

###  3.1 Other classes of potentials

Applying directly the abstract result from [22], it is possible to replace
(H3) by an assumption which consists on an upper bound on $\mathfrak{h}$.
Following the idea in [22], a particular family of potentials satisfying such
an assumption can be obtained by suitably perturbing a given multi-well
potential (or even a more general potential). More precisely, one can consider
a non-negative smooth potential $V$ vanishing in two sets $\mathcal{A}_{1}$
and $\mathcal{A}_{2}$ in ${\mathbb{R}}^{k}$ and with local properties as in
(H1) and (H2), with the obvious modifications. A suitable perturbation around
$\mathcal{A}_{2}$ gives a family of unbalanced potentials
$(W_{\delta})_{\delta\geq 0}$ such that $W_{0}=V$ and for
$\delta\in(0,\delta_{0}]$ (for some $\delta_{0}>0$), the abstract framework
from [22] applies to $W_{\delta}$ (up to an additive constant). In particular,
one has results as the ones of this paper for $W_{\delta}$, which was a type
of potential not allowed in [4].

###  3.2 The problem of traveling waves for 2D parabolic Allen-Cahn systems

Our initial motivation comes from the study of the parabolic Allen-Cahn system

$\partial_{t}w-\Delta w=-\nabla_{\mathbf{u}}V(w)\mbox{ in
}[0,+\infty)\times{\mathbb{R}}^{2},$ (3.1)

where $V$ is a standard, smooth, multi-well potential. In [22], we show by an
abstract approach that there exist traveling wave solutions
$w:(t,x_{1},x_{2})\in[0,+\infty)\times{\mathbb{R}}^{2}\to\mathfrak{U}(x_{1}-c^{\star}t,x_{2})\in{\mathbb{R}}^{k}$
of (3.1) such that

$\lim_{x_{1}\to\pm+\infty}\lVert\mathfrak{U}(x_{1},\cdot)-\mathfrak{q}^{\pm}\rVert_{H^{1}({\mathbb{R}},{\mathbb{R}}^{k})}=0$
(3.2)

where $\mathfrak{q}^{\pm}$ solve

$\left(\mathfrak{q}^{\pm}\right)^{\prime\prime}=\nabla_{\mathbf{u}}V(\mathfrak{q}^{\pm})\mbox{
in }{\mathbb{R}}$ (3.3)

and satisfy

$\lim_{t\to\pm\infty}\mathfrak{q}^{\pm}(t)=\sigma^{\pm}$ (3.4)

with
$\sigma^{\pm}\in\Sigma:=\\{\mathbf{u}\in{\mathbb{R}}^{k}:V(\mathbf{u})=0\\}$.
Moreover, $\mathfrak{q}^{-}$ and $\mathfrak{q}^{+}$ are (in some sense) local
minimizers of the energy

$E:q\in H^{1}_{\mathrm{loc}}({\mathbb{R}},{\mathbb{R}}^{k})\to
E(q):=\int_{\mathbb{R}}\left[\frac{\lvert
q^{\prime}(t)\rvert^{2}}{2}+V(q(t))\right]dt$ (3.5)

with different energy levels. That is

$E(\mathfrak{q}^{\pm})=\mathfrak{m}^{\pm}$ (3.6)

with $\mathfrak{m}^{+}>\mathfrak{m}^{-}$ and $\mathfrak{m}^{-}$ the global
minima. The main point is that $\mathfrak{m}^{+}>\mathfrak{m}^{+}$, as the
case $\mathfrak{m}^{+}=\mathfrak{m}^{-}$ had already been treated by several
authors and it is by now well understood. See Fusco [12], Monteil and
Santambrogio [19], Schatzman [26], Smyrnelis [28]. As already explained, in
[22] we have to assume that $0<\mathfrak{m}^{+}-\mathfrak{m}^{-}$ is bounded
above by a given constant. As in here, the approach of the proof is to deduce
the result from an abstract setting. In an attempt to extend our result to
other situations, we were able to replace the perturbation assumption by
another one in the abstract setting. The other assumption is the abstract
version of (H3) and the one we use here (see (H7’)). However, the inconvenient
is that such condition does not seem to apply when dealing with traveling
waves of the system (3.1). Even for simple examples of the potential $V$, we
cannot prove that such condition is satisfied, which does not allow us to
apply the abstract result. The reason is that properties such as convexity of
level sets and uniform strict monotonicity along segments are harder to verify
in infinite dimensions. However, if an example of potential $V$ such that
(H7’) is satisfied could be found, then we would have the existence of a pair
$(c^{\star},\mathfrak{U})$ satisfying the same properties than the solution
obtained in [22].

###  3.3 Traveling waves between homoclinics

One considers again the Allen-Cahn system

$\partial_{t}w-\Delta w=-\nabla_{\mathbf{u}}V(w)\mbox{ in
}[0,+\infty)\times{\mathbb{R}}^{2},$ (3.7)

with $V$ as in the previous paragraph. We use the same notation as above. This
time, we consider the conditions at infinity

$\lim_{x_{1}\to\pm+\infty}\lVert\mathfrak{U}(x_{1},\cdot)-\mathfrak{u}^{\pm}\rVert_{H^{1}({\mathbb{R}},{\mathbb{R}}^{k})}=0$
(3.8)

for the associated profile $\mathfrak{U}$. Here $\mathfrak{u}^{-}$ is the
constant homoclinic solution equal to $\sigma\in\Sigma$ and $\mathfrak{u}^{+}$
is such that $E(\mathfrak{u}^{+})>0$ and it solves

$\left(\mathfrak{u}^{+}\right)^{\prime\prime}=\nabla_{\mathbf{u}}V(\mathfrak{u}^{+})\mbox{
in }{\mathbb{R}}$ (3.9)

satisfying

$\lim_{t\to\pm\infty}\mathfrak{u}^{+}(t)=\sigma.$ (3.10)

That is, $\mathfrak{u}^{+}$ is a non-constant homoclinic emanating from
$\sigma$. If $\mathfrak{u}^{+}$ is locally minimizing and, moreover, it
satisfies the assumptions given by some of our abstract results111That is,
either the perturbative one that we use in [22] or the one that we give here.,
then the existence of a solution $(c^{\star},\mathfrak{U})$ is guaranteed. The
fact that $\mathfrak{u}^{+}\not=\mathfrak{u}^{+}$ implies that $\mathfrak{U}$
is not constant, and since $\mathfrak{u}^{-}$ and $\mathfrak{u}^{+}$ are
homoclinic, then $\mathfrak{U}$ can be thought as a heteroclinic between
homoclinics. Moreover, notice that such type of solution cannot exist in the
stationary case, as stationary waves join two different 1D solutions which are
global minimizers, which is not possible in the homoclinic case222Clearly, the
only globally minimizing heteroclinic is the constant one.. Even though for
the moment we have no explicit example of potential $V$ such that the
necessary assumptions are fulfilled, a previous result by the author [21]
suggest that homoclinic solutions might exist in some situations.

##  4 The abstract setting

As we advanced in the introduction, the main results of this paper are
obtained from exploiting the abstract setting introduced in [22]. The current
setting only differs to that in [22] by the modification of an assumption, but
the structure is analogous. As a consequence, we shall be more brief here in
the exposition. We begin by recalling the main notations. 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
$\langle\cdot,\cdot\rangle_{\mathscr{H}}$. We will take
$\mathcal{E}:\mathscr{L}\to(-\infty,+\infty]$ an unbalanced potential. We
impose a set of general assumptions on $\mathcal{E}$. Those assumptions are an
adaptation of those in [4] for the infinite-dimensional setting with non-
isolated minima. We will begin by fixing two sets $\mathscr{F}^{-}$ and
$\mathscr{F}^{+}$ in $\mathscr{L}$. For $r>0$, we define

$\mathscr{F}^{\pm}_{r}:=\left\\{v\in\mathscr{L}:\inf_{\mathbf{v}\in\mathscr{F}^{\pm}}\lVert
v-\mathbf{v}\rVert_{\mathscr{L}}\leq r\right\\},$ (4.1)

and

$\mathscr{F}^{\pm}_{\mathscr{H},r}:=\left\\{v\in\mathscr{H}:\inf_{\mathbf{v}\in\mathscr{F}^{\pm}}\lVert
v-\mathbf{v}\rVert_{\mathscr{H}}\leq r\right\\},$ (4.2)

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

$\forall v\in\mathscr{L},\forall\mathbf{v}^{-}\in\mathscr{F}^{-},\hskip
5.69054pt\mathcal{E}(v)\geq\mathcal{E}(\mathbf{v}^{-})=a$ (4.3)

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
(4.1)). There also exist $C^{\pm}>1$ such that

$\forall v\in\mathscr{F}^{\pm}_{r_{0}^{\pm}},\hskip
5.69054pt(C^{\pm})^{-1}\mathrm{dist}_{\mathscr{L}}(v,\mathscr{F}^{\pm})^{2}\leq\mathcal{E}(v)-\min\\{\pm(-a),0\\}.$
(4.4)

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

$\lVert
v-\mathbf{v}^{\pm}(v)\rVert_{\mathscr{L}}=\inf_{\mathbf{v}^{\pm}\in\mathscr{F}^{\pm}}\lVert
v-\mathbf{v}^{\pm}\rVert_{\mathscr{L}}.$ (4.5)

Moreover, the projection maps

$P^{\pm}:v\in\mathscr{F}^{\pm}_{r_{0}^{\pm}}\to\mathbf{v}^{\pm}(v)\in\mathscr{F}^{\pm}$
(4.6)

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

The abstract result consists essentially on establishing existence of a pair
$(c,\mathbf{U})$ in $(0,+\infty)\times Y$ ($Y$ is a function space to be
defined later, see (4.54)) which fulfills

$\mathbf{U}^{\prime\prime}-D_{\mathscr{L}}\mathcal{E}(\mathbf{U})=-c\mathbf{U}^{\prime}\mbox{
in }{\mathbb{R}}$ (4.7)

and satisfies the conditions at infinity

$\exists T^{-}\leq 0:\forall t\leq T^{-},\hskip
5.69054pt\mathbf{U}(t)\in\mathscr{F}^{-}_{r_{0}^{-}/2}\hskip 2.84526pt\mbox{
and }\hskip
2.84526pt\exists\mathbf{v}^{+}(\mathbf{U})\in\mathscr{F}^{+}:\lim_{t\to+\infty}\lVert\mathbf{U}(t)-\mathbf{v}^{+}(\mathbf{U})\rVert_{\mathscr{H}}=0.$
(4.8)

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. 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

$\kappa^{\pm}_{r}:=\inf\\{\mathcal{E}(v):\mathrm{dist}_{\mathscr{L}}(v,\mathscr{F}^{\pm})\in[r,r_{0}^{\pm}]\\}$
(4.9)

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

$\forall v\in\mathscr{F}^{+}_{r_{0}^{+}/2},\hskip 5.69054pt\mathcal{E}(v)\geq
0.$ (4.10)

###### Proof.

It follows directly from (4.4) in (H1’). ∎

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

######  (H2’).

We have that $\\{v\in\mathscr{L}:\mathcal{E}(v)<+\infty\\}$ is a linear
subspace of $\mathscr{L}$. Such a subspace will be denoted as $\mathscr{H}$.
Moreover, $\mathscr{H}$ can be endowed with an inner product which gives rise
to a norm $\lVert\cdot\rVert_{\mathscr{H}}$ which verifies
$\lVert\cdot\rVert_{\mathscr{L}}\leq\lVert\cdot\rVert_{\mathscr{H}}$.
Furthermore $\mathcal{E}$ is coercive in $\mathscr{H}$ in the following weak
sense: there exists a constant $C_{\textit{c}}$ such that

$\forall v\in\mathscr{H},\hskip 5.69054pt\lVert v\rVert_{\mathscr{H}}-\lVert
v\rVert_{\mathscr{L}}\leq C_{\textit{c}}(\mathcal{E}(v)-a)$ (4.11)

Moreover, $\mathcal{E}$ is a $C^{1}$ functional on
$(\mathscr{H},\lVert\cdot\rVert_{\mathscr{H}})$ with differential
$D\mathcal{E}:v\in\mathscr{H}\to D\mathcal{E}(v)\in\mathscr{H}$ and

$\forall R>0,\exists
C_{\mathcal{E}}(R)>0,\forall(v,w)\in\mathscr{H}^{2}:\lVert
v\rVert_{\mathscr{H}}\leq R,\hskip 5.69054pt\lvert
D\mathcal{E}(v)(w)\rvert\leq C_{\mathcal{E}}(R)\lVert w\rVert_{\mathscr{H}}.$
(4.12)

Finally, there exists an even smaller space $\mathscr{H}^{\prime}$ with an
inner product $\langle\cdot,\cdot\rangle_{\mathscr{H}^{\prime}}$ and
associated norm
$\lVert\cdot\rVert_{\mathscr{H}^{\prime}}\geq\lVert\cdot\rVert_{\mathscr{H}}$
such that we can find a continuous correspondence

$D_{\mathscr{L}}\mathcal{E}:v\in(\mathscr{H}^{\prime},\lVert\cdot\rVert_{\mathscr{H}^{\prime}})\to
D_{\mathscr{L}}\mathcal{E}(v)\in(\mathscr{L},\lVert\cdot\rVert_{\mathscr{L}})$
(4.13)

such that

$\forall v\in\mathscr{H}^{\prime},\forall w\in\mathscr{H},\hskip
5.69054ptD_{\mathscr{L}}\mathcal{E}(v)(w)=D\mathcal{E}(v)(w),$ (4.14)

where we have identified the derivative of $\mathcal{E}$ with an element of
$\mathscr{H}$ via Riesz’s Theorem.

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 (4.1) are closed in
$\mathscr{L}$.

Assumption (H3’) is necessary in order to establish the conditions at
infinity. Subsequently, we impose the following:

######  (H4’).

Assume that (H1’) holds. One of the two holds:

  1. 1.

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

  2. 2.

Fix
$(v,\mathbf{v}^{\pm})\in\mathscr{F}^{\pm}_{r_{0}^{\pm}}\times\mathscr{F}^{\pm}$.
There exist an associated map
$\hat{P}^{\pm}_{(v,\mathbf{v}^{\pm})}:\mathscr{L}\to\mathscr{L}$ such that

$P^{\pm}(\hat{P}^{\pm}_{(v,\mathbf{v}^{\pm})}(v))=\mathbf{v}^{\pm}$ (4.15)

and

$\mathrm{dist}_{\mathscr{L}}(\hat{P}^{\pm}_{(v,\mathbf{v}^{\pm})}(v),\mathscr{F}^{\pm})=\mathrm{dist}_{\mathscr{L}}(v,\mathscr{F}^{\pm}).$
(4.16)

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

$\forall(w_{1},w_{2})\in\mathscr{L}^{2},\hskip 5.69054pt\lVert
D(\hat{P}^{\pm}_{(v,\mathbf{v}^{\pm})})(w_{1},w_{2})\rVert_{\mathscr{L}}=\lVert
w_{2}\rVert_{\mathscr{L}}$ (4.17)
$\mathcal{E}(\hat{P}_{(v,\mathbf{v}^{\pm})}(v))=\mathcal{E}(v).$ (4.18)

Essentially, we impose that the projections $P^{\pm}$ from (H1’) are, in some
sense, invertible. 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
(4.2), there exists a unique
$\mathbf{v}_{\mathscr{H}}^{\pm}(v)\in\mathscr{F}_{r_{0}}^{\pm}$ such that

$\lVert
v-\mathbf{v}_{\mathscr{H}}^{\pm}(v)\rVert_{\mathscr{L}}=\inf_{\mathbf{v}^{\pm}\in\mathscr{F}^{\pm}}\lVert
v-\mathbf{v}^{\pm}\rVert_{\mathscr{L}}.$ (4.19)

Moreover, the projection maps

$P^{\pm}_{\mathscr{H}}:v\in\mathscr{F}^{\pm}_{\mathscr{H},r_{0}^{\pm}}\to\mathbf{v}_{\mathscr{H}}^{\pm}(v)\in\mathscr{F}^{\pm}$
(4.20)

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

$\forall v\in\mathscr{F}^{\pm}_{\mathscr{H},r_{0}^{\pm}},\hskip
5.69054pt\lVert P^{\pm}(v)-P^{\pm}_{\mathscr{H}}(v)\rVert_{\mathscr{H}}\leq
C^{\pm}\lVert v-P^{\pm}_{\mathscr{H}}(v)\rVert_{\mathscr{H}}.$ (4.21)

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

$\mathcal{E}(v)\leq\min\\{\pm(-a),0\\}+\beta^{\pm}(r^{\pm}),$ (4.22)

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

$\forall v\in\mathscr{F}_{\mathscr{H},r_{0}^{\pm}},\hskip
5.69054pt(C^{\pm})^{-2}\lVert
v-P^{\pm}_{\mathscr{H}}(v)\rVert_{\mathscr{H}}^{2}\leq\mathcal{E}(v)-\min\\{\pm(-a),0\\}\leq(C^{\pm})^{2}\lVert
v-P^{\pm}_{\mathscr{H}}(v)\rVert_{\mathscr{H}}^{2}.$ (4.23)

Assumption (H5’) is made in order to ensure the suitable local properties
around $\mathscr{F}^{\pm}$ in $\mathscr{H}$. 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

$\mathbf{E}_{c}(U):=\int_{\mathbb{R}}\mathbf{e}_{c}(U)(t)dt:=\int_{{\mathbb{R}}}\left[\frac{\lVert
U^{\prime}(t)\rVert_{\mathscr{L}}^{2}}{2}+\mathcal{E}(U(t))\right]e^{ct}dt.$
(4.24)

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

$\mathbf{E}_{c}(U;I)=\int_{I}\mathbf{e}_{c}(U)(t)dt.$ (4.25)

Notice that the integrals defined in (4.24) and (4.25) 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

$\max_{t\in I}\lVert\phi(t)\rVert_{\mathscr{H}}<C,$ (4.26)

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

$\forall v\in\mathscr{L},\hskip
5.69054pt\mathcal{E}(\mathfrak{P}(v))\leq\mathcal{E}(v)\mbox{ and
}\mathcal{E}(\mathfrak{P}(v))=\mathcal{E}(v)\Leftrightarrow\mathfrak{P}(v)=v,$
(4.27) $\forall(v_{1},v_{2})\in\mathscr{L}^{2},\hskip
5.69054pt\lVert\mathfrak{P}(v_{1})-\mathfrak{P}(v_{2})\rVert_{\mathscr{L}}\leq\lVert
v_{1}-v_{2}\rVert_{\mathscr{L}},$ (4.28)

and

$\mathfrak{P}|_{\mathscr{F}^{\pm}}=\mathrm{Id}|_{\mathscr{F}^{\pm}}.$ (4.29)

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

$\mathcal{A}(O):=\mathcal{C}^{2}_{\mathrm{loc}}(O,\mathscr{L})\cap\mathcal{C}^{1}_{\mathrm{loc}}(O,(\mathscr{H},\lVert\cdot\rVert_{\mathscr{H}}))\cap\mathcal{C}^{0}_{\mathrm{loc}}(O,(\mathscr{H}^{\prime},\lVert\cdot\rVert_{\mathscr{H}^{\prime}}))$
(4.30)

and $\mathbf{W}$ solves

$\mathbf{W}^{\prime\prime}-D_{\mathscr{L}}\mathcal{E}(\mathbf{W})=-c\mathbf{W}^{\prime}\mbox{
in }I,$ (4.31)

where $D_{\mathscr{L}}\mathcal{E}$ was introduced in (4.13). Moreover if
$I=(-\infty,t_{0})$ for some $t_{0}\in{\mathbb{R}}$ and there exists
$\mathbf{v}^{-}(\mathbf{W})\in\mathscr{F}^{-}$ such that

$\lim_{t\to-\infty}\lVert\mathbf{W}(t)-\mathbf{v}^{-}(\mathbf{W})\rVert_{\mathscr{L}}=0,$
(4.32)

and

$\mathcal{E}(\mathbf{W}(\cdot))-a\in L^{1}((-\infty,t_{0}])$ (4.33)

then

$\lim_{t\to-\infty}\lVert\mathbf{W}(t)-\mathbf{v}^{-}(\mathbf{W})\rVert_{\mathscr{H}}=0.$
(4.34)

Furthermore, if $I=(t_{0},+\infty)$ with $t_{0}\in{\mathbb{R}}$ and for some
$\mathbf{v}^{+}(\mathbf{W})\in\mathscr{F}^{+}$ we have

$\lim_{t\to+\infty}\lVert\mathbf{W}(t)-\mathbf{v}^{+}(\mathbf{W})\rVert_{\mathscr{H}}=0,$
(4.35)

as well as

$\mathcal{E}(\mathbf{W}(\cdot))\in L^{1}((-\infty,t_{0}])$ (4.36)

we then have

$\sup_{t\in[t_{0}+1,+\infty)}\lVert\mathbf{W}(t)\rVert_{\mathscr{H}^{\prime}}+\lVert\mathbf{W}^{\prime}(t)\rVert_{\mathscr{H}}+\lVert\mathbf{W}^{\prime\prime}(t)\rVert_{\mathscr{L}}<+\infty.$
(4.37)

Before stating the abstract result, we introduce the following constants:

$\eta_{0}^{-}:=\min\left\\{\sqrt{e^{-1}\frac{r_{0}^{-}}{4}\sqrt{2(\kappa^{-}_{r_{0}^{-}/4}-a)}},\frac{r_{0}^{-}}{4}\right\\}>0,$
(4.38) $\hat{r}^{-}:=\frac{r_{0}^{-}}{C^{-}+1}>0$ (4.39)
$\varepsilon_{0}^{-}:=\frac{1}{(C^{-})^{2}(C^{-}+1)}\min\left\\{\frac{(\eta_{0}^{-})^{2}}{4},\kappa_{\eta_{0}^{-}}^{-}-a,\beta^{-}(\hat{r}^{-}),\beta^{-}(\eta_{0}^{-})\right\\}>0,$
(4.40)
$\mathsf{C}^{\pm}:=\frac{1}{2}(C^{\pm})^{2}((C^{\pm})^{2}+(C^{\pm}+1)^{2})>0,$
(4.41) $\gamma^{-}:=\frac{1}{C^{-}+\mathsf{C}^{-}}>0$ (4.42)

and

$d_{0}:=\mathrm{dist}_{\mathscr{L}}(\mathscr{F}^{+}_{r_{0}^{+}/2},\mathscr{F}^{-}_{r_{0}^{-}/2})>0,$
(4.43)

where the constants $C^{-},\beta^{-}(\hat{r}^{-}),\beta^{-}(\eta_{0}^{-})$ are
those from (H5’) and $\kappa_{r}^{\pm}$ for $r>0$ are defined in (4.9). The
fact that $d_{0}>0$ follows from Lemma 4.2 and (H1’). We now replace the
perturbation assumption in [22] for a new one, the generalization of (H3). We
define for $\alpha\in{\mathbb{R}}$ the level sets

$\mathcal{E}^{\alpha}:=\\{v\in\mathscr{H}:\mathcal{E}(v)\leq\alpha\\}$ (4.44)

and notice that (H1’) and (H2’) imply that there exists $\alpha_{0}>0$ such
that

$\mathcal{E}^{\alpha_{0}}=\mathcal{E}^{\alpha_{0},-}\cup\mathcal{E}^{\alpha_{0},+}$
(4.45)

with $\mathcal{E}^{\alpha_{0},\pm}$ disjoint, $\mathscr{L}$-closed and such
that

$\mathcal{E}^{\alpha_{0},+}\subset\mathscr{F}^{+}_{r_{0}^{+}/2}.$ (4.46)

As a consequence, for each $\alpha\in[a,\alpha_{0}]$ we can write

$\mathcal{E}^{\alpha}=\mathcal{E}^{\alpha,-}\cup\mathcal{E}^{\alpha,+}$ (4.47)

with $\mathcal{E}^{\alpha,\pm}$ disjoint, $\mathscr{L}$-closed and such that
if $\alpha_{1}\leq\alpha_{2}$ in $[a,\alpha_{0}]$, then
$\mathcal{E}^{\alpha_{1},\pm}\subset\mathcal{E}^{\alpha_{2},\pm}$. In
particular, we have

$\forall\alpha\in[a,\alpha_{0}],\hskip
5.69054pt\mathcal{E}^{\alpha,-}\cap\mathscr{F}^{+}_{r_{0}^{+}/2}=\emptyset.$
(4.48)

Next, notice that $\mathcal{E}^{0,+}=\mathscr{F}^{+}$ and if $\alpha\in[a,0)$,
then $\mathcal{E}^{\alpha,+}=\emptyset$. Our next assumption writes then as
follows:

######  (H7’).

Assume that (H1’) and (H2’) hold. We suppose that for all
$\alpha\in[a,\alpha_{0}]$, we hav that $\mathcal{E}^{\alpha,-}$ is convex. For
all $\mathbf{v}^{-}\in\mathscr{F}^{-}$, $\alpha\in[a,\alpha_{0}]$ and
$v\in\mathcal{E}^{\alpha,-}$ define

$A(\alpha,\mathbf{v}^{-},v):=\\{\lambda\in{\mathbb{R}}:\mathcal{E}(\mathbf{v}^{-}+\lambda(v-\mathbf{v}^{-}))>\alpha\mbox{
and
}\mathbf{v}^{-}+\lambda(v-\mathbf{v}^{-})\in\mathcal{E}^{\alpha_{0},-}\\}.$
(4.49)

Then, there exists $\alpha_{-}\in(a,0)$ such that

  1. 1.

$\mathcal{E}^{\alpha_{-}}\subset\mathscr{F}^{-}_{r_{0}^{-}/2}$.

  2. 2.

There exists $\omega>0$ such that

$\forall
v\in\mathcal{E}^{\alpha_{0},-},\forall\mathbf{v}^{-}\in\mathscr{F}^{-},\forall\theta\in
A\left(\frac{\alpha_{-}+a}{2},\mathbf{v}^{-},v\right),\hskip
5.69054pt\frac{d}{d\lambda}(\mathcal{E}(\mathbf{v}^{-}+\lambda(v-\mathbf{v}^{-})))(\theta)\geq\omega.$
(4.50)

  3. 3.

We have that for all $\alpha\in(a,\alpha_{-}]$, there exists
$\omega(\alpha)>0$ such that, for all $v\in\mathcal{E}^{\alpha_{0},-}$ such
that $\mathcal{E}(v)\geq\alpha$, it holds

$\exists\delta(v)>0,\forall\theta\in(1-\delta(v),1+\delta(v)),\hskip
5.69054pt\frac{d}{d\lambda}(\mathcal{E}(P^{-}(v)+\lambda(v-P^{-}(v))))(\theta)\geq\omega(\alpha),$
(4.51)

where $P^{-}$ is the projection map defined in (H1’), which is well defined in
$\mathcal{E}^{\alpha_{-}}$ because
$\mathcal{E}^{\alpha_{-}}\subset\mathscr{F}_{r_{0}^{-}/2}^{-}$.

Finally, we have the following global bound property: There exists $R>0$ such
that for any $v\in\mathcal{E}^{\alpha_{0},-}$ there exists
$\mathbf{v}^{-}(v)\in\mathscr{F}^{-}$ such that $\lVert
v-\mathbf{v}^{-}(v)\rVert_{\mathscr{H}}\leq R$.

Let us next define the space

$\displaystyle X:=\left\\{U\in
H^{1}_{\mathrm{loc}}({\mathbb{R}},\mathscr{L}):\exists T\geq 1,\right.$
$\displaystyle\forall t\geq T,\hskip
5.69054pt\mathrm{dist}_{\mathscr{L}}(U(t),\mathscr{F}^{+})\leq\frac{r_{0}^{+}}{2},$
(4.52) $\displaystyle\left.\forall t\leq-T,\hskip
5.69054pt\mathrm{dist}_{\mathscr{L}}(U(t),\mathscr{F}^{-})\leq\frac{r_{0}^{-}}{2}\right\\},$
(4.53)

already introduced in [22]. If (H7’) holds, we define the class

$\displaystyle Y:=\left\\{U\in
H^{1}_{\mathrm{loc}}({\mathbb{R}},\mathscr{L}):\exists T\geq 1,\right.$
$\displaystyle\forall t\geq T,\hskip
5.69054pt\mathrm{dist}_{\mathscr{L}}(U(t),\mathscr{F}^{+})\leq\frac{r_{0}^{+}}{2},$
(4.54) $\displaystyle\left.\forall t\leq-T,\hskip
5.69054pt\mathcal{E}(U(t))\leq\alpha_{-}\right\\}$ (4.55)

so that $Y\subset X$ by (H7’). For working under assumption (H7’), the space
$Y$ becomes more suitable in certain aspects. We can now finally state the
abstract result

######  Theorem 4 (Main abstract result).

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

  1. 1.

Existence. There exists $c^{\star}>0$ and
$\mathbf{U}\in\mathcal{A}({\mathbb{R}})\cap Y$, $\mathcal{A}({\mathbb{R}})$ as
in (4.30), such that $(c^{\star},\mathbf{U})$ solves (4.7) with conditions at
infinity (4.8) and $\mathbf{U}$ is a global minimizer of $\mathbf{E}_{c}$ in
$Y$ (that is, $\mathbf{E}_{c}(\mathbf{U})=0$).

  2. 2.

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

$\inf_{U\in Y}\mathbf{E}_{\overline{c^{\star}}}(U)=0$ (4.56)

and there exists $\overline{\mathbf{U}}\in\mathcal{A}({\mathbb{R}})\cap X$
such that $(\overline{c^{\star}},\overline{\mathbf{U}})$ solves (4.7) and
$\mathbf{E}_{\overline{c^{\star}}}(\overline{\mathbf{U}})<+\infty$, then
$\overline{c^{\star}}=c^{\star}$.

  3. 3.

Exponential convergence. We have that for some $b>c^{\star}/2$ it holds

$\lVert\mathbf{U}(t)-\mathbf{v}^{+}(U)\rVert_{\mathscr{H}}e^{bt}=0$ (4.57)

where $\mathbf{v}^{+}(U)$ is given by (4.8).

If $\mathscr{F}^{-}$ is reduced to a simple point and (H7’) holds, we can
shown convergence with respect to the $\mathscr{L}$-norm as $t\to-\infty$:

######  Theorem 5.

Assume that (H3’), (H4’), (H5’), (H6’) and (H7’) hold. Assume moreover that
$\mathscr{F}^{-}=\\{\mathbf{v}^{-}_{i}\\}$. Then, if $(c^{\star},\mathbf{U})$
is the solution given by Theorem 4, we have

$\lim_{t\to-\infty}\lVert\mathbf{U}(t)-\mathbf{v}^{-}_{i}\rVert_{\mathscr{L}}=0.$
(4.58)

######  Remark 4.1.

Notice that (4.58) does not imply that $\mathcal{E}(\mathbf{U}(t))\to a$ as
$t\to-\infty$, since $\mathcal{E}$ is only supposed to be lower-semicontinuous
with respect to $\mathscr{L}$-convergence.

We can impose an additional assumption in order to obtain stronger conditions
at $-\infty$ on the solution:

######  (H8’).

Hypothesis (H7’) is fulfilled and, additionally:

$-a<\frac{(d_{\alpha_{0}}\gamma^{-})^{2}}{2}$ (4.59)

where $\gamma^{-}$ is as in (4.42) and

$d_{\alpha_{0}}:=\mathrm{dist}_{\mathscr{L}}(\mathcal{E}^{\alpha_{0},-},\mathscr{F}^{+}_{r_{0}^{+}/2})$
(4.60)

which is positive by (H7’) and Lemma 4.2.

If (H8’) holds, we can show the following:

######  Theorem 6.

Assume that (H3’), (H4’), (H5’), (H6’) and (H8’) hold. Finally, assume that
(H8’) holds. Then, if $(c^{\star},\mathbf{U})$ is the solution given by
Theorem 4, the function $\mathbf{U}$ satisfies in addition

$\lim_{t\to-\infty}\lVert\mathbf{U}(t)-\mathbf{v}^{-}(\mathbf{U})\rVert_{\mathscr{H}}=0$
(4.61)

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

Finally, we have the characterization of the speed:

######  Theorem 7.

Assume that (H3’), (H4’), (H5’), (H6’) and (H8’) hold. Finally, assume that
(H8’) holds. Let $(c^{\star},\mathbf{U})$ be the solution given by Theorem 4.
Then, if $\tilde{\mathbf{U}}\in\mathcal{A}({\mathbb{R}})\cap Y$ is such that

$\mathbf{E}_{c^{\star}}(\tilde{\mathbf{U}})=0$ (4.62)

then we have that $(c^{\star},\tilde{\mathbf{U}})$ solves (4.7) and

$c^{\star}=\frac{-a}{\int_{\mathbb{R}}\lVert\tilde{\mathbf{U}}^{\prime}(t)\rVert_{\mathscr{L}}^{2}dt}.$
(4.63)

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

$c^{\star}=\sup\\{c>0:\inf_{U\in
Y}\mathbf{E}_{c}(U)=-\infty\\}=\inf\\{c>0:\inf_{U\in
Y}\mathbf{E}_{c}(U)=0\\}.$ (4.64)

##  5 Proof of the abstract results

The strategy is essentially the same that the one used in [22], which was
adapted from [4]. See also the book by Alikakos, Fusco and Smyrnelis [3]. We
reproduce the main steps and provide the proof of the new results, which are
essentially those in subsection 5.4, for which we use (H7’) instead of the
bound assumption that we made in [22]. Therefore, each result here has its
equivalent one in [22] and in some cases the arguments adapt in a
straightforward way or the result can even be applied directly.

###  5.1 Preliminaries

Let $r_{0}^{-}$ and $r_{0}^{+}$ be the constants introduced before and
$\mathscr{F}_{r_{0}^{\pm}/2}^{\pm}$ be the corresponding closed balls as in
(4.1). 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)

where the constant . Subsequently, we set

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

In case (H7’) holds, we can define for $T\geq 1$

$Y_{T}^{-}:=\left\\{U\in
H^{1}_{\mathrm{loc}}({\mathbb{R}},\mathscr{L}):\forall t\leq-T,\hskip
2.84526pt\mathcal{E}(U(t))\leq\alpha_{-}\right\\},$ (5.4)

where $\alpha_{-}$ is the constant introduced in assumption (H7’).
Subsequently, we set

$Y_{T}:=Y_{T}^{-}\cap X_{T}^{+}.$ (5.5)

The spaces $Y_{T}$ will play here the role that the spaces $X_{T}$ played in
[22]. We have that

$X=\bigcup_{T\geq 1}X_{T}$ (5.6)

and

$Y=\bigcup_{T\geq 1}Y_{T}.$ (5.7)

The following property is immediate:

######  Lemma 5.1.

Assume that (H7’) holds. Then, we have that

$Y_{T}\subset X_{T}$ (5.8)

###### Proof.

From (H7’) it follows that whenever $v\in\mathscr{H}$ is such that
$\mathcal{E}(v)\leq\alpha_{-}$, it then holds that
$v\in\mathscr{F}^{-}_{r_{0}^{-}/2}$. The inclusion (5.8) then follows by the
definitions (5.3) and (5.5). ∎

Lemma 5.1 shows that whenever (H7’) holds, $Y_{T}$ is a subspace of $X_{T}$.
Therefore, properties that hold for $X_{T}$ (and that we proved in [22]) will
apply to $Y_{T}$. Next, we recall the following properties:

######  Lemma 5.2 ([22]).

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

$\forall t\geq T,\hskip 5.69054pt\mathcal{E}(U(t))\geq 0.$ (5.9)

Moreover, the quantity $\mathbf{E}_{c}(U)$ as introduced in (4.24) is well
defined in $(-\infty,+\infty]$.

######  Lemma 5.3 ([22]).

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,

$\lim_{t\to+\infty}\mathcal{E}(U(t))=0$ (5.10)

and

$\lim_{t\to+\infty}\lVert U(t)-\mathbf{v}^{+}(U)\rVert_{\mathscr{H}}=0$ (5.11)

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

Define now the infimum value

$\mathbf{b}_{c,T}:=\inf_{U\in Y_{T}}\mathbf{E}_{c}(U).$ (5.12)

It will be later shown that $\mathbf{b}_{c,T}$ is always attained. We first
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

$\Psi(t):=\begin{cases}\hat{\mathbf{v}}^{-}&\mbox{ if }t\leq-1,\\\
\frac{1-t}{2}\hat{\mathbf{v}}^{-}+\frac{t+1}{2}\hat{\mathbf{v}}^{+}&\mbox{ if
}-1\leq t\leq 1,\\\ \hat{\mathbf{v}}^{+}&\mbox{ if }t\geq 1,\end{cases}$
(5.13)

belongs to $X_{T}$. If (H7’) holds, we also have $\Psi\in Y_{T}$. Moreover,
for all $c>0$

$\mathbf{E}_{c}(\Psi)<+\infty.$ (5.14)

Furthermore, we have

$-\infty<\mathbf{b}_{c,T}<+\infty.$ (5.15)

The proof of Lemma 5.4 follows the same lines that the equivalent result in
[22], so we skip it.

###  5.2 Two (semi-)continuity results

We now recall and extend some results regarding continuity and semi-continuity
properties of $\mathbf{E}_{c}$. We first have:

######  Lemma 5.5 ([22]).

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

$A_{T,U}:=\\{c>0:\mathbf{E}_{c}(U)<+\infty\\}.$ (5.16)

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

$c\in A_{T,U}\to\mathbf{E}_{c}(U)\in{\mathbb{R}}$ (5.17)

is continuous.

Next, we extend a result from [22]:

######  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

$\sup_{n\in{\mathbb{N}}}\mathbf{E}_{c_{n}}(U_{n}^{i})<+\infty.$ (5.18)

Then, there exists a sequence $(U_{n})$ 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

$\forall n\in{\mathbb{N}},\hskip
5.69054pt\mathbf{E}_{c}(U_{n})=\mathbf{E}_{c}(U_{n}^{i}),$ (5.19) $\forall
t\in{\mathbb{R}},\hskip 5.69054ptU_{n}(t)\rightharpoonup U_{\infty}(t)\mbox{
weakly in }\mathscr{L}$ (5.20)
$U_{n}^{\prime}e^{c_{n}\mathrm{Id}/2}\rightharpoonup
U_{\infty}^{\prime}e^{c_{\infty}\mathrm{Id}/2}\mbox{ weakly in
}L^{2}({\mathbb{R}},\mathscr{L})$ (5.21)

and

$\mathbf{E}_{c_{\infty}}(U_{\infty})\leq\liminf_{n\to\infty}\mathbf{E}_{c_{n}}(U_{n}),$
(5.22)

where $c_{\infty}:=\lim_{n\to\infty}c_{n}$. Moreover, if (H7’) holds and the
sequence $(U_{n}^{i})_{n\in{\mathbb{N}}}$ is contained in $Y_{T}$, then
$U_{\infty}\in Y_{T}$.

###### Proof.

Except for the last part, the result is exactly the same as that in [22].
Notice that if $(U_{n}^{i})_{n\in{\mathbb{N}}}$ is contained in $Y_{T}$, then
by (5.20) and the lower-semicontinuity of $\mathcal{E}$ we have that for all
$t\leq-T$ it holds

$\mathcal{E}(U_{\infty}(t))\leq\liminf_{n\to\infty}\mathcal{E}(U_{n}(t))\leq\alpha_{-}$
(5.23)

which shows that $U_{\infty}\in Y_{T}$ and concludes the proof. ∎

###  5.3 Existence of an infimum for $\mathbf{E}_{c}$ in $Y_{T}$

We now show that for any $c>0$ and $T\geq 1$, the infimum value
$\mathbf{b}_{c,T}$ defined in (5.12) is attained. The result is as follows:

######  Lemma 5.7.

Assume that (H3’), (H4’) and (H7’) hold. Let $c>0$, $T\geq 1$ and
$\mathbf{b}_{c,T}$ be as in (5.12). Then, $\mathbf{b}_{c,T}$ is attained for
some $\mathbf{V}_{c,T}\in Y_{T}$.

###### Proof.

By (5.15) in Lemma 5.4, we have that there exists a minimizing sequence
$(U_{n})_{n\in{\mathbb{N}}}$ in $Y_{T}$. Therefore, Lemma 5.6 implies the
existence of $\mathbf{V}_{c,T}\in Y_{T}$. ∎

We next show that constrained minimizers are solutions of the equation (4.7)
on a (possibly proper) subset of ${\mathbb{R}}$.

######  Lemma 5.8.

Assume that (H6’) and (H7’) hold. Let $c>0$, $T\geq 1$ and $\mathbf{b}_{c,T}$
be as in (5.12). Let $\mathbf{V}_{c,T}\in Y_{T}$ be such that
$\mathbf{E}_{c}(\mathbf{V}_{c,T})=\mathbf{b}_{c,T}$. Then,
$\mathbf{V}_{c,T}\in\mathcal{A}((-T,T))$ and

$\mathbf{V}_{c,T}^{\prime\prime}-D_{\mathscr{L}}\mathcal{E}(\mathbf{V}_{c,T})=-c\mathbf{V}_{c,T}^{\prime}\mbox{
in }(-T,T).$ (5.24)

Moreover, if $t\geq T$ is such that

$\mathrm{dist}_{\mathscr{L}}(\mathbf{V}_{c,T}(t),\mathscr{F}^{+})<\frac{r_{0}^{+}}{2},$
(5.25)

then, there exists $\delta^{+}_{Y}(t)>0$ such that
$\mathbf{V}_{c,T}\in\mathcal{A}((t-\delta^{+}_{Y}(t),t+\delta^{+}_{Y}(t)))$
and

$\mathbf{V}_{c,T}^{\prime\prime}-D_{\mathscr{L}}\mathcal{E}(\mathbf{V}_{c,T})=-c\mathbf{V}_{c,T}^{\prime}\mbox{
in }(t-\delta^{+}_{Y}(t),t+\delta^{+}_{Y}(t)).$ (5.26)

Similarly, if $t\leq-T$ is such that there exist $\varepsilon^{-}_{Y}(t)>0$
and $\delta^{-}_{Y}(t)>0$ such that

$\forall s\in[t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)],\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(s))\leq\alpha_{-}-\varepsilon^{-}_{Y}(t)$
(5.27)

then,
$\mathbf{V}_{c,T}\in\mathcal{A}((t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)))$
and

$\mathbf{V}_{c,T}^{\prime\prime}-D_{\mathscr{L}}\mathcal{E}(\mathbf{V}_{c,T})=-c\mathbf{V}_{c,T}^{\prime}\mbox{
in }(t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)).$ (5.28)

###### Proof.

We begin by showing that

$\forall t\in{\mathbb{R}},\hskip
5.69054pt\mathbf{V}_{c,T}(t)=\mathfrak{P}(\mathbf{V}_{c,T}(t)).$ (5.29)

Arguing as in the proof of the equivalent result in [22], to prove (5.29) it
suffices to show that

$\mathbf{V}_{c,T}^{\mathfrak{P}}:t\in{\mathbb{R}}\to\mathbf{V}_{c,T}^{\mathfrak{P}}(t)$
(5.30)

belongs to $Y_{T}$. Indeed, the proof given in [22] implies that
$\mathbf{V}_{c,T}^{\mathfrak{P}}\in X_{T}$. Thus, the fact that
$\mathbf{V}_{c,T}\in\mathcal{A}((-T,T))$ and $\mathbf{V}_{c,T}$ solves (5.24)
is proven as in [22]. Next, notice that (4.27) implies that for all $t\leq-T$,
we have
$\mathcal{E}(\mathbf{V}_{c,T}^{\mathfrak{P}}(t))\leq\mathcal{E}(\mathbf{V}_{c,T}(t))\leq\alpha_{-}$.
Therefore, $\mathbf{V}_{c,T}^{\mathfrak{P}}\in Y_{T}$, which shows (5.29). Let
now $t\geq T$ such that (5.25) holds. Again, arguing as [22], we obtain that
there exists $\delta^{+}_{Y}(t)>0$ such that
$\mathbf{V}_{c,T}\in\mathcal{A}((t-\delta^{+}_{Y}(t),t+\delta^{+}_{Y}(t)))$
and (5.26) holds. To conclude the proof, let $t\leq-T$ be such that (5.27)
holds. Notice that $\mathcal{E}(\mathbf{V}_{c,T})$ is finite on $(-\infty,T]$,
which by assumption (H2’) implies that $\mathbf{V}_{c,T}$ takes values in
$\mathscr{H}$ on $(-\infty,T]$. Moreover $\mathbf{V}_{c,T}$ is
$\mathscr{L}$-continuous. Therefore, using (4.11) we get

$\forall s\in[t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)],\hskip
5.69054pt\lVert\mathbf{V}_{c,T}(s)\rVert_{\mathscr{H}}\leq R_{1},$ (5.31)

with

$0<R_{1}:=C_{c}(\alpha_{-}-a)+\max_{s\in[t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)]}\lVert\mathbf{V}_{c,T}(s)\rVert_{\mathscr{L}}<+\infty.$
(5.32)

Let
$\phi\in\mathcal{C}_{c}^{1}((t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)),(\mathscr{H},\lVert\cdot\rVert_{\mathscr{H}}))$
be such that

$\max_{s\in[t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)]}\lVert\phi(s)\rVert_{\mathscr{H}}\leq
R_{1},$ (5.33)

by (4.12), we have that if $v\in\mathscr{H}$ is such that $\lVert
v\rVert_{\mathscr{H}}\leq 3R_{1}$, it holds

$\forall s\in[t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)],\hskip 5.69054pt\lvert
D\mathcal{E}(v)(\phi(s))\rvert\leq
C_{\mathcal{E}}(3R_{1})\lVert\phi(s)\rVert_{\mathscr{H}}$ (5.34)

with $C_{\mathcal{E}}(3R_{1})$ independent on $v$. We have by (5.27) that

$\forall s\in[t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)],\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(s)+\phi(s))\leq\alpha_{-}-\varepsilon^{-}_{Y}(t)+D\mathcal{E}(h(s))(\phi(s)),$
(5.35)

where, for all $s\in[t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)]$,
$h(s)\in\mathscr{H}$ lies on the segment joining $\mathbf{V}_{c,T}(s)+\phi(s)$
and $\mathbf{V}_{c,T}(s)$, so that $\lVert h(s)\rVert_{\mathscr{H}}\leq
3R_{1}$ by (5.31) and (5.33). Therefore, we can plug (5.34) into (5.35) to
obtain

$\forall s\in[t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)],\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(s)+\phi(s))\leq\alpha_{-}-\varepsilon^{-}_{Y}(t)+C_{\mathcal{E}}(3R_{1})\lVert\phi(s)\rVert_{\mathscr{H}},$
(5.36)

meaning that if we choose any
$\phi\in\mathcal{C}_{c}^{1}((t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)),(\mathscr{H},\lVert\cdot\rVert_{\mathscr{H}}))$
such that

$\max_{s\in[t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)]}\lVert\phi(s)\rVert_{\mathscr{H}}\leq\min\left\\{R_{1},\frac{\varepsilon_{Y}^{-}(t)}{2C_{\mathcal{E}}(3R_{1})}\right\\}$
(5.37)

we get by (5.36) that

$\forall s\in[t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)],\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(s)+\phi(s))\leq\alpha_{-}-\frac{\varepsilon^{-}_{Y}(t)}{2}<\alpha_{-}$
(5.38)

so that $\mathbf{V}_{c,T}+\phi\in Y_{T}$. Therefore, we have that
$\mathbf{V}_{c,T}$ is a local minimizer of
$\mathbf{E}_{c}(\cdot;[t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)])$ in the sense
of Definition 4.1. Since, in addition, $\mathbf{V}_{c,T}$ satisfies (5.29), we
can apply (H6’) and we obtain that
$\mathbf{V}_{c,T}\in\mathcal{A}((t-\delta^{-}_{Y}(t),t+\delta^{-}_{Y}(t)))$
and (5.28) holds. As a consequence, the proof is finished. ∎

###  5.4 The comparison result

As in [22], we establish the necessary properties regarding the behavior of
the constrained minimizers. This is done by using the fact that the
constrained minimizers solve the ODE system along with comparison arguments.
Essentially, we adapt the proof of the analogous result from [4, 3] in the
framework of (H3), which makes it more involved but uses the same ideas. We
begin by recalling the definition of some useful constants. For $0<r\leq
r_{0}^{\pm}$, recall the definition of $\kappa_{r}^{\pm}$ introduced in (4.9),
Lemma 4.1. We define

$\eta^{+}_{0}:=\min\left\\{\sqrt{e^{-1}\frac{r_{0}^{+}}{4}\sqrt{2\kappa^{+}_{r_{0}^{+}/4}}},\frac{r_{0}^{+}}{4}\right\\}>0,$
(5.39) $\hat{r}^{+}:=\frac{r_{0}^{+}}{C^{+}+1}>0,$ (5.40)
$\varepsilon^{+}_{0}:=\frac{1}{\left(C^{+}\right)^{2}(C^{+}+1)}\min\left\\{\frac{(\eta_{0}^{+})^{2}}{4},\kappa^{+}_{\eta_{0}^{+}},\beta^{+}(\hat{r}^{+}),\beta^{+}(\eta_{0}^{+})\right\\}>0,$
(5.41)

where the constants $C^{\pm}$,
$\beta^{\pm}(\hat{r}^{\pm}),\beta(\eta_{0}^{\pm})$ were introduced in (H5’).
Recall that in (4.38), (4.39), (4.40) we introduced the analogous constants

$\eta_{0}^{-}:=\min\left\\{\sqrt{e^{-1}\frac{r_{0}^{-}}{4}\sqrt{2(\kappa^{-}_{r_{0}^{-}/4}-a)}},\frac{r_{0}^{-}}{4}\right\\}>0,$
(5.42) $\hat{r}^{-}:=\frac{r_{0}^{-}}{C^{-}+1}>0$ (5.43)

and

$\varepsilon_{0}^{-}:=\frac{1}{(C^{-})^{2}(C^{-}+1)}\min\left\\{\frac{(\eta_{0}^{-})^{2}}{4},\kappa_{\eta_{0}^{-}}^{-}-a,\beta^{-}(\hat{r}^{-}),\beta^{-}(\eta_{0}^{-})\right\\}>0.$
(5.44)

For any $U\in X_{T}$, define

$t^{-}(U,\varepsilon^{-}_{0}):=\sup\left\\{t\in{\mathbb{R}}:\mathcal{E}(U(t))\leq
a+\varepsilon^{-}_{0}\mbox{ and
}\mathrm{dist}_{\mathscr{L}}(U(t),\mathscr{F}^{-})\leq\frac{r_{0}^{-}}{2}\right\\}$
(5.45)

and

$t^{+}(U,\varepsilon^{+}_{0}):=\inf\left\\{t\in{\mathbb{R}}:\mathcal{E}(U(t))\leq\varepsilon^{+}_{0}\mbox{
and
}\mathrm{dist}_{\mathscr{L}}(U(t),\mathscr{F}^{+})\leq\frac{r_{0}^{+}}{2}\right\\}.$
(5.46)

We begin by proving two preliminary comparison results:

######  Lemma 5.9.

Assume that (H7’) holds. Let $t_{1}<t_{2}$ and $U\in
H^{1}([t_{1},t_{2}],\mathscr{L})$. Assume that

$U(t_{1}),U(t_{2})\in\mathcal{E}^{\alpha,-}\mbox{ with }\alpha\leq 0$ (5.47)

and

$\exists t_{3}\in(t_{1},t_{2}):\mathcal{E}(U(t_{3}))>\alpha.$ (5.48)

Then, there exists $\tilde{U}\in H^{1}([t_{1},t_{2}],\mathscr{L})$ such that

$\forall t\in(t_{1},t_{2}),\forall i\in\\{1,2\\},\hskip
5.69054ptU(t)\in\mathcal{E}^{\alpha,-},\hskip
2.84526pt\tilde{U}(t_{i})=U(t_{i})$ (5.49)

and

$\forall c>0,\hskip 5.69054pt\mathbf{E}_{c}(\tilde{U})<\mathbf{E}_{c}(U).$
(5.50)

The following is inspired by similar results from [4], although our version is
weaker:

######  Lemma 5.10.

Assume that (H7’) holds. Let $t_{1}<t_{2}$,
$\alpha\in\left(\frac{\alpha_{-}+a}{2},\alpha_{0}\right)$ and $U_{\alpha}$ be
the constant function

$U_{\alpha}:t\in[t_{1},t_{2}]\to v_{\alpha}\in\mathcal{E}^{\alpha,-},\hskip
5.69054pt\mathcal{E}(v_{\alpha})=\alpha.$ (5.51)

Then, there exists $\tilde{U}\in
H^{1}_{\mathrm{loc}}([t_{1},t_{2}],\mathscr{L})$ such that

$\forall t\in(t_{1},t_{2}),\forall i\in\\{1,2\\},\hskip
5.69054pt\mathcal{E}(\tilde{U}(t))<\alpha,\hskip
2.84526pt\tilde{U}(t_{i})=v_{\alpha}$ (5.52)

and

$\forall c>0,\hskip
5.69054pt\mathbf{E}_{c}(\tilde{U})<\mathbf{E}_{c}(U_{\alpha}).$ (5.53)

Before proving Lemmas 5.9 and 5.10, we recall the following property:

######  Lemma 5.11.

Let $C\subset\mathscr{L}$ be a closed convex set of $\mathscr{L}$ and $P_{C}$
the corresponding orthogonal projection. If $I\subset{\mathbb{R}}$ is a non-
empty interval and $U\in H^{1}_{\mathrm{loc}}(I,\mathscr{L})$, it holds that
$P_{C}(U(\cdot))\in H^{1}_{\mathrm{loc}}(I,\mathscr{L}))$ and

$\mbox{ for a. e. }t\in I,\hskip 5.69054pt\lVert
P_{C}(U(t))^{\prime}\rVert_{\mathscr{L}}\leq\lVert U(t)\rVert_{\mathscr{L}}.$
(5.54)

###### Proof.

The Hilbert projection Theorem readily implies that $P_{C}$ is 1-Lipschitz. As
a consequence, we have that $P_{C}(U(\cdot))\in
H^{1}_{\mathrm{loc}}(I,\mathscr{L}))$. Fix now $t\in I$, for any $s\in
I\setminus\\{t\\}$ we have

$\left\lVert\frac{P_{C}(U(s))-P_{C}(U(t))}{s-t}\right\rVert_{\mathscr{L}}\leq\left\lVert\frac{U(s)-U(t)}{s-t}\right\rVert_{\mathscr{L}},$
(5.55)

which gives (5.54) by applying Lebesgue’s differentiation Theorem. ∎

###### Proof of Lemma 5.9.

From (H7’), we have that $\mathcal{E}^{\alpha,-}$ is closed and convex in
$\mathscr{L}$. As a consequence, Lemma 5.11 applies to
$\mathcal{E}^{\alpha,-}$. Therefore, we can consider the corresponding
orthogonal projection $P^{\alpha,-}:\mathscr{L}\to\mathcal{E}^{\alpha,-}$.
Define now

$\forall t\in[t_{1},t_{2}],\hskip 5.69054pt\tilde{U}(t):=P^{\alpha,-}(U(t))$
(5.56)

which by Lemma 5.11 belongs to $H^{1}([t_{1},t_{2}],\mathscr{L})$. Moreover,
applying (5.47) we see that $\tilde{U}$ verifies (5.49). Now, notice that by
(H1’) implies that $t\in[t_{1},t_{2}]\to\mathcal{E}(U(t))$ is lower-
semicontinuous. Therefore, $\\{t\in[t_{1},t_{2}]:\mathcal{E}(U(t))>\alpha\\}$
is open, and by (5.48) we have that it is also non-empty. As a consequence,
there exists $I\subset[t_{1},t_{2}]$ non-empty an open such that

$\forall t\in I,\hskip 5.69054pt\mathcal{E}(U(t))>\alpha$ (5.57)

which implies that

$\int_{I}\mathcal{E}(\tilde{U}(t))e^{ct}dt<\int_{I}\mathcal{E}(U(t))e^{ct}dt$
(5.58)

and, since for all $v\in\mathscr{H}$ we have
$\mathcal{E}(P^{\alpha,-}(v))\leq\mathcal{E}(v)$ by definition of the
orthogonal projection and because $\alpha\leq 0$, we conclude

$\int_{t_{1}}^{t_{2}}\mathcal{E}(\tilde{U}(t))e^{ct}dt<\int_{t_{1}}^{t_{2}}\mathcal{E}(U(t))e^{ct}dt,$
(5.59)

which in combination with (5.54) in Lemma 5.11 gives (5.50) and concludes the
proof. ∎

###### Proof of Lemma 5.10.

Let $\varepsilon>0$, $\mathbf{v}^{-}\in\mathscr{F}^{-}$ and
$\phi\in\mathcal{C}^{1}_{c}([t_{1},t_{2}])$ such that $\phi>0$ in
$(t_{1},t_{2})$. Define the function

$\forall t\in[t_{1},t_{2}],\hskip
5.69054ptU_{\varepsilon}(t):=\mathbf{v}^{-}+(1-\varepsilon\phi(t))(v_{\alpha}-\mathbf{v}^{-})$
(5.60)

so that $U_{0}=U_{\alpha}$. It is clear that for all $\varepsilon>0$ we have
$U_{\varepsilon}\in H^{1}([t_{1},t_{2}],\mathscr{L})$. Moreover, we also have
that for $i\in\\{1,2\\}$, it holds $U_{\varepsilon}(t_{i})=v_{\alpha}$. For
all $t\in[t_{1},t_{2}]$, consider
$A\left(\frac{\alpha_{-}+a}{2},\mathbf{v}^{-},U(t)\right)$ as in (4.49). Since
$\alpha\in\left(\frac{\alpha_{-}+a}{2},\alpha_{0}^{-}\right)$,
$v_{\alpha}\in\mathcal{E}^{\alpha,-}$ and
$\lVert\phi\rVert_{C^{1}([t_{1},t_{2}])}<+\infty$, there exists
$\varepsilon_{1}>0$ independent on $t$ such that

$(1-\varepsilon_{1}\phi(t),1+\varepsilon_{1}\phi(t))\subset
A\left(\frac{\alpha_{-}+a}{2},\mathbf{v}^{-},U(t)\right).$ (5.61)

Therefore, using (4.50) we have that

$\displaystyle\forall\varepsilon\in(0,\varepsilon_{1}),\forall
t\in[t_{1},t_{2}],\hskip
5.69054pt\mathcal{E}(U_{\varepsilon}(t))-\mathcal{E}(U_{0}(t))$
$\displaystyle=-\int_{1-\varepsilon\phi(t)}^{1}\frac{d}{d\lambda}(\mathcal{E}(\mathbf{v}^{-}+\lambda(v_{\alpha}-\mathbf{v}^{-})))(\theta)d\theta$
(5.62) $\displaystyle\leq-\varepsilon\phi(t)\omega,$ (5.63)

where $\omega>0$. In particular, we have that for all
$\varepsilon\in(0,\varepsilon_{1})$, $U_{\varepsilon}$ fulfills (5.52). Now,
notice that

$\forall\varepsilon>0,\mbox{ for a. e. }t\in[t_{1},t_{2}],\hskip
5.69054ptU_{\varepsilon}^{\prime}(t)=-\varepsilon\phi^{\prime}(t)(v_{\alpha}-\mathbf{v}^{-})$
(5.64)

which means that, using also (5.63), we obtain

$\displaystyle\forall c>0,$
$\displaystyle\forall\varepsilon\in(0,\varepsilon_{1}),$ (5.65)
$\displaystyle\mathbf{E}_{c}(U_{\varepsilon})-\mathbf{E}_{c}(U_{0})\leq(-\varepsilon\lVert\phi\rVert_{L^{\infty}([t_{1},t_{2}])}\omega+\frac{1}{2}\varepsilon^{2}\lVert\phi^{\prime}\rVert_{L^{\infty}([t_{1},t_{2}])}^{2}\lVert
v_{\alpha}-\mathbf{v}^{-}\rVert_{\mathscr{L}}^{2})\frac{e^{ct_{2}}-e^{ct_{1}}}{c}$
(5.66)

so that if $\varepsilon\in(0,\varepsilon_{2})$, where

$\varepsilon_{2}:=\frac{2\lVert\phi\rVert_{L^{\infty}([t_{1},t_{2}])}\omega}{\lVert\phi^{\prime}\rVert_{L^{\infty}([t_{1},t_{2}])}^{2}\lVert
v_{\alpha}-\mathbf{v}^{-}\rVert_{\mathscr{L}}^{2}}>0$ (5.67)

then (5.53) holds for $U_{\varepsilon}$. Hence, we have shown that if
$\varepsilon\in(0,\varepsilon_{3})$ with
$\varepsilon_{3}:=\min\\{\varepsilon_{1},\varepsilon_{2}\\}>0$, then
$U_{\varepsilon}$ fulfills (5.52) and (5.53), which completes the proof. ∎

We will now define for $U\in Y_{T}$

$t_{2}^{-}(U):=\inf\\{t\in{\mathbb{R}}:\mathcal{E}(U(t))=\alpha_{0},U(t)\in\mathcal{E}^{\alpha_{0},-}\\}$
(5.68)

and

$t_{1}^{-}(U):=\sup\\{t\leq t_{2}^{-}(U):\mathcal{E}(U(t))\leq\alpha_{-}\\}$
(5.69)

where the constants $a<\alpha_{-}<0<\alpha_{0}$ are those from (H7’). The
following preliminary property is straightforward:

######  Lemma 5.12.

Let $U\in Y_{T}$ and $t_{2}^{-}(U)$, $t_{1}^{-}(U)$ be as in (5.68) and (5.69)
respectively. Then, it holds

$-T\leq t_{1}^{-}(U)\leq t_{2}^{-}(U)$ (5.70)

###### Proof.

We have that $\mathcal{E}(U(-T))\leq\alpha_{-}$, which implies that $-T\leq
t_{1}^{-}(U)$. The inequality $t_{1}^{-}(U)\leq t_{2}^{-}(U)$ follows from the
definition. ∎

Lemmas 5.9 and 5.10 apply mainly to obtain the following information on
constrained minimizers in $Y_{T}$:

######  Proposition 5.1.

Assume that (H3’), (H4’), (H6’) and (H7’) hold. Let $c>0$ and $T\geq 1$. Let
$\mathbf{V}_{c,T}$ be a constrained minimizer of $\mathbf{E}_{c}$ in $Y_{T}$
given by Lemma 5.7. Let $t_{1}^{-}:=t_{1}^{-}(\mathbf{V}_{c,T})$ and
$t_{2}^{-}:=t_{2}^{-}(\mathbf{V}_{c,T})$, where $t_{1}^{-}(\mathbf{V}_{c,T})$
and $t_{2}^{-}(\mathbf{V}_{c,T})$ are as in (5.69), (5.68) respectively. Then,
it holds that $\mathcal{E}(\mathbf{V}_{c,T}(t_{2}^{-}))=\alpha_{0}$ and

$t_{2}^{-}-t_{1}^{-}\leq\mathsf{T}_{1}(c),$ (5.71)

where

$\mathsf{T}_{1}(c):=\frac{2Rc+2\sqrt{R^{2}c^{2}+2R\omega}}{\omega}.$ (5.72)

Moreover, we have that

$\forall t\leq t_{1}^{-},\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(t))\leq\alpha_{-}$ (5.73)

and there exist $\delta_{0}^{-}>0$ and $t_{0}^{-}\leq t_{1}^{-}$ such that

$\begin{cases}\mathcal{E}(\mathbf{V}_{c,T}(\cdot))\mbox{ is monotone in
}(-\infty,t_{0}^{-}+\delta_{0}^{-}]\mbox{ and }\\\ \forall t\leq
t_{0}^{-}+\delta_{0}^{-},\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(t))<\alpha_{-}.\end{cases}$ (5.74)

###### Proof.

Notice that due to the definition of $t_{2}^{-}$ in (5.68) for all $t\leq
t_{2}^{-}$ we have $\mathbf{V}_{c,T}(t)\in\mathcal{E}^{\alpha_{0},-}$.
Therefore, $t_{2}^{-}<T$. Notice now that if $t_{1}^{-}=t_{2}^{-}$, then
(5.72) holds trivially. Assume then that $t_{1}^{-}<t_{2}^{-}$. By Lemma 5.12,
we have that $(t_{1}^{-},t_{2}^{-})\subset(-T,T)$ with $t_{2}^{-}<T$.
Therefore, since we assume that (H6’) holds we have that
$\mathbf{V}_{c,T}\in\mathcal{A}((-T,T))$ fulfills (5.24) by Lemma 5.8. The
definition of $\mathcal{A}((-T,T))$ in (4.30) implies that
$t\in[t_{1}^{-},t_{2}^{-}]\to\mathcal{E}(\mathbf{V}_{c,T}(t))$ is continuous,
which means that $\mathcal{E}(\mathbf{V}_{c,T}(t_{2}^{-}))=\alpha_{0}$. Due to
the definitions of $t_{1}^{-}$ and $t_{2}^{-}$ in (5.69) and (5.68)
respectively, we have that

$\forall t\in(t_{1}^{-},t_{2}^{-}),\hskip
5.69054pt\mathcal{E}(\mathbf{V}(t))\in(\alpha_{-},\alpha_{0}),\hskip
2.84526pt\mathbf{V}(t)\in\mathcal{E}^{\alpha_{0},-}$ (5.75)

which means that we can use (4.50) to obtain

$\forall\mathbf{v}^{-}\in\mathscr{F}^{-},\forall
t\in(t_{1}^{-},t_{2}^{-}),\hskip
5.69054pt\frac{d}{d\lambda}(\mathcal{E}(\mathbf{v}^{-}+\lambda(\mathbf{V}_{c,T}(t)-\mathbf{v}^{-}))(1)\geq\omega>0,$
(5.76)

which means, using (4.14) in (H2’)

$\forall\mathbf{v}^{-}\in\mathscr{F}^{-},\forall
t\in(t_{1}^{-},t_{2}^{-}),\hskip 5.69054pt\langle
D_{\mathscr{L}}(\mathbf{V}_{c,T}),\mathbf{V}_{c,T}(t)-\mathbf{v}^{-}\rangle_{\mathscr{L}}\geq\omega.$
(5.77)

By (H7’), there exists $\mathbf{v}^{-}_{c,T}\in\mathscr{F}^{-}$ such that
$\lVert\mathbf{V}_{c,T}(t_{2}^{-})-\mathbf{v}^{-}_{c,T}\rVert_{\mathscr{L}}\leq
R$. Define now the function

$\rho:t\in[t_{1}^{-},t_{2}^{-}]\to\lVert\mathbf{V}_{c,T}(t)-\mathbf{v}^{-}_{c,T}\rVert_{\mathscr{L}}^{2}\in{\mathbb{R}}$
(5.78)

so that

$\rho(t_{2}^{-})\leq R.$ (5.79)

By (5.24) and (H6’), we have that $\rho\in\mathcal{C}^{2}((t_{1},t_{2}))$. We
have

$\forall t\in(t_{1}^{-},t_{2}^{-}),\hskip
5.69054pt\rho^{\prime}(t)=\langle\mathbf{V}_{c,T}^{\prime}(t),\mathbf{V}_{c,T}(t)-\mathbf{v}^{-}_{c,T}\rangle_{\mathscr{L}}$
(5.80)

and, subsequently, using (5.24)

$\forall t\in(t_{1}^{-},t_{2}^{-}),\hskip
5.69054pt\rho^{\prime\prime}(t)=\lVert\mathbf{V}_{c,T}\rVert_{\mathscr{L}}^{2}+\langle
D_{\mathscr{L}}(\mathbf{V}_{c,T}(t)),\mathbf{V}_{c,T}(t)-\mathbf{v}^{-}_{c,T}\rangle_{\mathscr{L}}-c\rho^{\prime}(t),$
(5.81)

which, by (5.77) means that

$\forall t\in(t_{1}^{-},t_{2}^{-}),\hskip
5.69054pt\rho^{\prime\prime}(t)+c\rho^{\prime}(t)\geq\omega.$ (5.82)

Notice that (5.82) implies that $\rho$ does not possess any local maximum in
$[t_{1}^{-},t_{2}^{-}]$, as we have $\omega>0$. That is, either

  1. 1.

The function $\rho$ is non-decreasing in $[t_{1}^{-},t_{2}^{-}]$.

  2. 2.

The function $\rho$ possesses a unique local minimum at
$t_{3}^{-}\in(t_{1}^{-},t_{2}^{-})$.

Assume that 1. holds. Then for each $t\in(t_{1}^{-},t_{2}^{-})$ we can
integrate (5.82) in $[t_{1}^{-},t]$ to obtain

$c(\rho(t)-\rho(t_{1}^{-}))+\rho^{\prime}(t)-\rho^{\prime}(t_{1}^{-})\geq\omega(t-t_{1}^{-})$
(5.83)

so that, since we assume that $\rho^{\prime}\geq 0$ in $(t_{1}^{-},t_{2}^{-})$

$c\rho(t)+\rho^{\prime}(t)\geq\omega(t-t_{1}^{-})$ (5.84)

which, integrating now in $[t_{1}^{-},t_{2}^{-}]$ gives

$c\int_{t_{1}^{-}}^{t_{2}^{-}}\rho(t)dt+\rho(t_{2}^{-})\geq\frac{\omega}{2}(t_{2}^{-}-t_{1}^{-})^{2}.$
(5.85)

By (5.79), and since $\rho$ does not possess any local maximum, the previous
becomes,

$Rc(t_{2}^{-}-t_{1}^{-})+R\geq\frac{\omega}{2}(t_{2}^{-}-t_{1}^{-})^{2}.$
(5.86)

The roots of the polynomial

$\frac{\omega}{2}x^{2}-Rcx-R$ (5.87)

are

$x_{1}=\frac{Rc+\sqrt{R^{2}c^{2}+2R\omega}}{\omega}>0\mbox{ and
}x_{2}=\frac{Rc-\sqrt{R^{2}c^{2}+2R\omega}}{\omega}<0,$ (5.88)

therefore, since (5.86) holds, we must have

$t_{2}^{-}-t_{1}^{-}\leq x_{1}.$ (5.89)

Assume now that 1. does not hold. Then 2. holds. Notice that since $t_{3}^{-}$
is a local minimum of $\rho$ in $(t_{1}^{-},t_{2}^{-})$ we have
$\rho^{\prime}(t_{3}^{-})=0$. We first integrate (5.82) in $[t_{3}^{-},t]$ for
$t\in(t_{3}^{-},t_{2}^{-})$ to get

$c(\rho(t)-\rho(t_{3}^{-}))+\rho^{\prime}(t)\geq\omega(t-t_{3}^{-})$ (5.90)

which means that

$\forall t\in(t_{3}^{-},t_{2}^{-}),\hskip
5.69054ptc\rho(t)+\rho^{\prime}(t)\geq\omega(t-t_{3}^{-}),$ (5.91)

which is exactly (5.84) with $t_{1}^{-}$ replaced by $t_{3}^{-}$. The same
reasoning as above gives

$t_{2}^{-}-t_{3}^{-}\leq x_{1}$ (5.92)

with $x_{1}$ as in (5.88). We now take $t\in(t_{1}^{-},t_{3}^{-})$ and
integrate (5.82) in $[t,t_{3}^{-}]$. We obtain

$-\rho^{\prime}(t)+c(\rho(t_{3}^{-})-\rho(t))\geq\omega(t_{3}^{-}-t)$ (5.93)

so that

$\forall t\in[t_{1}^{-},t_{3}^{-}],\hskip
5.69054pt-\rho^{\prime}(t)+c\rho(t_{3})\geq\omega(t_{3}^{-}-t)$ (5.94)

which, by integrating in $[t_{1}^{-},t_{3}^{-}]$ and using (5.79) along with
the fact that $\rho$ has no local maximum in $[t_{1}^{-},t_{2}^{-}]$, gives

$R+Rc(t_{3}^{-}-t_{1}^{-})\geq\frac{\omega}{2}(t_{3}^{-}-t_{1})^{2}$ (5.95)

which is exactly (5.86) but with $t_{2}^{-}$ replaced by $t_{3}^{-}$.
Therefore, we get

$t_{3}^{-}-t_{1}^{-}\leq x_{1}$ (5.96)

which in combination with (5.92) gives

$t_{2}^{-}-t_{1}^{-}\leq 2x_{1}.$ (5.97)

As a consequence, we have that either (5.89) or (5.97) holds. That is, we have
shown that (5.71), with $\mathsf{T}_{1}(c)$ as in (5.72), holds. We now show
that (5.73) holds. Otherwise, we would have $-T<t_{1}^{-}$ and some
$\tilde{t}\in(-T,t_{1}^{-})$ such that

$\mathcal{E}(\mathbf{V}_{c,T}(\tilde{t}))>\alpha_{-}$ (5.98)

and, since we have that $\mathcal{E}(\mathbf{V}_{c,T}(-T))$ and
$\mathcal{E}(\mathbf{V}_{c,T}(t_{1}^{-}))$ are smaller than $\alpha_{-}$, we
obtain a contradiction by Lemma 5.9. Therefore, (5.73) has been established.
We now show the existence $t_{0}^{-}\leq t_{1}^{-}$ and $\delta_{0}^{-}>0$
such that (5.74) holds. We begin by showing the existence of
$\tilde{t}_{0}^{-}\leq t_{1}^{-}$ such that

$\forall t\leq\tilde{t}_{0}^{-},\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(t))<\alpha_{-}.$ (5.99)

By contradiction, assume that there exists a sequence
$(t_{n})_{n\in{\mathbb{N}}}$ in $(-\infty,t_{1}^{-}]$ such that
$t_{n}\to-\infty$ as $n\to\infty$ and

$\forall n\in{\mathbb{N}},\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(t_{n}))=\alpha_{-}.$ (5.100)

Assume by contradiction that there exists a sequence
$(\tilde{t}_{n})_{n\in{\mathbb{N}}}$ in $(-\infty,t_{1}^{-}]$ such that
$\tilde{t}_{n}\to-\infty$ and

$\forall n\in{\mathbb{N}},\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(\tilde{t}_{n}))<\alpha_{-}$ (5.101)

which, in combinantion with (5.100), shows the existence of
$t_{a}<t_{b}<t_{c}\leq t_{1}^{-}$ and $\tilde{\alpha}_{-}<\alpha_{-}<0$ such
that

$\mathbf{V}_{c,T}(t_{a}),\mathbf{V}_{c,T}(t_{c})\in\mathcal{E}^{\tilde{\alpha}_{-}}=\mathcal{E}^{\tilde{\alpha}_{-},-}\mbox{
and }\mathcal{E}(\mathbf{V}_{c,T}(t_{b}))=\alpha_{-}>\tilde{\alpha}_{-}$
(5.102)

which, by Lemma 5.9, gives a contradiction with the minimality of
$\mathbf{V}_{c,T}$. As a consequence, we have that there exists $\tilde{t}\leq
t_{1}^{-}$ such that

$\forall t\leq\tilde{t},\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(t))=\alpha_{-}.$ (5.103)

By Lemma 5.10, we have that $\mathbf{V}_{c,T}$ cannot be constant in any non-
empty interval $I\subset(-\infty,\tilde{t}]$. Therefore, we have in particular
that

$\int_{-\infty}^{\tilde{t}}\frac{\lVert\mathbf{V}_{c,T}^{\prime}(t)\rVert_{\mathscr{L}}^{2}}{2}e^{ct}dt>0.$
(5.104)

Let $\tilde{U}$ be the function defined as

$\tilde{U}(t)=\begin{cases}\mathbf{V}_{c,T}\left(\frac{1}{2}t+\frac{1}{2}\tilde{t}\right)&\mbox{
if }t\leq\tilde{t}\\\ \mathbf{V}_{c,T}(t)&\mbox{ if
}t\geq\tilde{t}\end{cases}$ (5.105)

which is well defined and belongs to $Y_{T}$. Notice that for all
$t\leq\tilde{t}$, we have that
$\frac{1}{2}t+\frac{1}{2}\tilde{t}\leq\tilde{t}$, so that
$\mathcal{E}(\tilde{U}(t))=\alpha_{-}$ by (5.103). Therefore,

$\int_{-\infty}^{\tilde{t}}\mathcal{E}(\tilde{U}(t))e^{ct}dt=\int_{-\infty}^{\tilde{t}}\mathcal{E}(\mathbf{V}_{c,T}(t))e^{ct}dt,$
(5.106)

meaning that

$\int_{\mathbb{R}}\mathcal{E}(\tilde{U}(t))e^{ct}dt=\int_{\mathbb{R}}\mathcal{E}(\mathbf{V}_{c,T}(t))e^{ct}dt.$
(5.107)

Now, notice that for a. e. $t\leq\tilde{t}$ it holds
$U^{\prime}(t)=\frac{1}{2}\mathbf{V}_{c,T}\left(\frac{1}{2}t+\frac{1}{2}\tilde{t}\right)$.
Hence, we have that

$\displaystyle\int_{-\infty}^{\tilde{t}}\frac{\lVert\tilde{U}^{\prime}(t)\rVert_{\mathscr{L}}^{2}}{2}e^{ct}dt$
$\displaystyle=\frac{1}{4}\int_{-\infty}^{\tilde{t}}\frac{\lVert\mathbf{V}_{c,T}\left(\frac{1}{2}t+\frac{1}{2}\tilde{t}\right)\rVert_{\mathscr{L}}^{2}}{2}e^{ct}dt$
(5.108)
$\displaystyle\leq\frac{1}{4}\int_{-\infty}^{\tilde{t}}\frac{\lVert\mathbf{V}_{c,T}\left(\frac{1}{2}t+\frac{1}{2}\tilde{t}\right)\rVert_{\mathscr{L}}^{2}}{2}e^{c\left(\frac{1}{2}t+\frac{1}{2}\tilde{t}\right)}dt$
(5.109)
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\tilde{t}}\frac{\lVert\mathbf{V}_{c,T}^{\prime}(t)\rVert_{\mathscr{L}}^{2}}{2}e^{ct}dt,$
(5.110)

which, by (5.104), means that

$\int_{\mathbb{R}}\frac{\lVert\tilde{U}^{\prime}(t)\rVert_{\mathscr{L}}^{2}}{2}e^{ct}dt<\int_{\mathbb{R}}\frac{\lVert\mathbf{V}_{c,T}^{\prime}(t)\rVert_{\mathscr{L}}^{2}}{2}e^{ct}dt,$
(5.111)

so that, taking also into account (5.107), we have obtained

$\mathbf{E}_{c}(\tilde{U})<\mathbf{E}_{c}(\mathbf{V}_{c,T}),$ (5.112)

which is a contradiction, because $\mathbf{V}_{c,T}$ minimizes
$\mathbf{E}_{c}$ in $Y_{T}$. As a consequence, (5.100) cannot hold, which
means that there exists $\tilde{t}_{0}^{-}\leq t_{1}^{-}$ such that (5.99)
holds. We now establish the existence of $t_{0}^{-}\leq t_{1}^{-}$ and
$\delta_{0}^{-}>0$ such that (5.74) holds. Applying Lemma 5.9, we have that
$\mathcal{E}(\mathbf{V}_{c,T}(\cdot))$ does not possess any strict local
maximum, and at most one strict local minimum, in $(-\infty,t_{1}^{-}]$.
Therefore, there exist $t_{0}^{-}\leq\tilde{t}_{0}^{-}$ and $\delta_{0}^{-}>0$
such that $\mathcal{E}(\mathbf{V}_{c,T}(\cdot))$ is monotone in
$(-\infty,t_{0}^{-}+\delta_{0}^{-}]$ and
$t_{0}^{-}+\delta_{0}^{-}\leq\tilde{t}_{0}^{-}$. Therefore, (5.74) follows by
(5.99). ∎

For $U\in Y_{T}$, recall the definition of $t^{+}(U,\varepsilon_{0}^{+})$ in
(5.46). The goal now will be to provide a uniform bound on
$t^{+}(U,\varepsilon_{0}^{+})-t_{2}^{-}(U)$. The result is very close to the
second part of the comparison result in [22].

######  Proposition 5.2.

Assume that (H3’), (H4’), (H5’), (H6’) and (H7’) hold. Let $c>0$ and $T\geq
1$. Consider $\mathbf{V}_{c,T}\in Y_{T}$ a minimizer of $\mathbf{E}_{c}$ in
$Y_{T}$, which exists by Lemma 5.7. Let
$t_{2}^{-}:=t_{2}^{-}(\mathbf{V}_{c,T})$ as in (5.68) and
$t^{+}:=t^{+}(\mathbf{V}_{c,T},\varepsilon_{0}^{+})$ as in (5.46). Then, we
have that

$\forall t\geq t^{+},\hskip
5.69054pt\mathrm{dist}_{\mathscr{L}}(\mathbf{V}_{c,T}(t),\mathscr{F}^{+})<\frac{r_{0}^{+}}{2}$
(5.113)

Furthermore, we have that

$0<t^{+}-t_{2}^{-}\leq\mathsf{T}_{2}(c),$ (5.114)

where

$\mathsf{T}_{2}(c):=\frac{1}{c}\ln\left(\frac{-a}{\alpha_{{\star\star}}}+1\right)$
(5.115)

with $\alpha_{{\star\star}}>0$ a constant independent on $c$, $T$ and $U$.

###### Proof.

We claim that for all $t\geq t^{+}$, it holds that
$\mathcal{E}(\mathbf{V}_{c,T}(t))\geq 0$. Indeed, otherwise we could find
$\tilde{t}\geq t^{+}$ such that
$\mathcal{E}(\mathbf{V}_{c,T}(\tilde{t}))=\alpha<0$. Since we necessarily have
$t^{+}>-T$, and
$\mathbf{V}_{c,T}(-T)\in\mathcal{E}^{\max\\{\alpha_{-},\alpha\\}}$, by Lemma
5.9, we can find $\tilde{U}\in Y_{T}$ such that
$\mathbf{E}_{c}(\tilde{U})<\mathbf{E}_{c}(\mathbf{V}_{c,T})$, a contradiction.
Therefore, arguing as in the second part of the comparison result in [22] we
obtain that (5.113) must hold. In order to prove (5.114), we first show that
$t^{+}>t_{2}^{-}$. This follows from the fact that for all $t\leq t_{2}^{-}$
we must have $\mathbf{V}_{c,T}(t)\in\mathcal{E}^{\alpha_{0},-}$,
$\mathbf{V}_{c,T}(t^{+})\in\mathscr{F}^{+}_{r_{0}^{+}/2}$ and
$\mathscr{F}^{+}_{r_{0}^{+}/2}\cap\mathcal{E}^{\alpha_{0},-}=\emptyset$. See
(H7’). Let $\alpha_{{\star\star}}:=\min\\{\alpha_{0},\varepsilon_{0}^{+}\\}$.
We claim that

$\forall t\in[t_{2}^{-},t^{+}],\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(t))\geq\alpha_{{\star\star}}.$ (5.116)

Indeed, assume by contradiction that for some $t\in(t_{2}^{-},t^{+})$ we have
$\mathcal{E}(\mathbf{V}_{c,T}(t))=\alpha<\alpha_{{\star\star}}$. If
$\mathbf{V}_{c,T}(t)\in\mathscr{F}^{+}_{r_{0}^{+}/2}$, we have a contradiction
with the definition of $t^{+}$ since $\alpha<\varepsilon_{0}^{+}$. Assume then
$\mathbf{V}_{c,T}(t)\not\in\mathscr{F}^{+}_{r_{0}^{+}/2}$. Since
$\mathcal{E}^{\alpha,+}\subset\mathcal{E}^{\alpha_{0},+}\subset\mathscr{F}^{+}_{r_{0}^{+}/2}$
by (H7’), we have that $\mathbf{V}_{c,T}(t)\in\mathcal{E}^{\alpha,-}$. We have
that
$\mathbf{V}_{c,T}(-T)\in\mathcal{E}^{\alpha_{-}}\subset\mathcal{E}^{\alpha,-}$
and $\mathbf{V}_{c,T}(t_{2}^{-})=\alpha_{0}>\alpha$ by Proposition 5.1.
Therefore, Lemma 5.9 leads to a contradiction. As a consequence, we have shown
that (5.116) holds. Arguing as in the last part of the corresponding result in
[22], we obtain that (5.114), with $\mathsf{T}_{2}(c)$ as in (5.115) holds. ∎

Propositions 5.1 and 5.2 can be summarized as follows:

######  Corollary 5.1.

Assume that (H3’), (H4’), (H5’), (H6’) and (H7’) hold. Let $c>0$ and $T\geq
1$. Consider $\mathbf{V}_{c,T}\in Y_{T}$ a minimizer of $\mathbf{E}_{c}$ in
$Y_{T}$, which exists by Lemma 5.7. Let
$t_{1}^{-}:=t_{1}^{-}(\mathbf{V}_{c,T})$ as in (5.69) and
$t^{+}:=t^{+}(\mathbf{V}_{c,T},\varepsilon_{0}^{+})$ as in (5.46). Then it
holds that

$\forall t\geq t^{+},\hskip
5.69054pt\mathrm{dist}_{\mathscr{L}}(\mathbf{V}_{c,T}(t),\mathscr{F}^{+})<\frac{r_{0}^{+}}{2},$
(5.117) $\forall t\leq t_{1}^{-},\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(t))\leq\alpha_{-}$ (5.118)

and there exist $\delta_{0}^{-}>0$ and $t_{0}^{-}\leq t_{1}^{-}$ such that

$\begin{cases}\mathcal{E}(\mathbf{V}_{c,T}(\cdot))\mbox{ is monotone in
}(-\infty,t_{0}^{-}+\delta_{0}^{-}]\mbox{ and }\\\ \forall t\leq
t_{0}^{-}+\delta_{0}^{-},\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(t))<\alpha_{-}.\end{cases}$ (5.119)

Moreover, we have

$0<t^{+}-t_{1}^{-}\leq\mathsf{T}_{{\star\star}}(c)$ (5.120)

where

$\mathsf{T}_{{\star\star}}(c):=\mathsf{T}_{1}(c)+\mathsf{T}_{2}(c)$ (5.121)

with $\mathsf{T}_{1}(c)$ as in (5.72) and $\mathsf{T}_{2}(c)$ as in (5.115).
That is,

$c\in(0,+\infty)\to\mathsf{T}_{{\star\star}}(c)\in(0,+\infty)$ (5.122)

is continuous and independent on $T$. Finally, if
$t_{2}^{-}:=t_{2}^{-}(\mathbf{V}_{c,T})$ as in (5.68), we have
$t_{2}^{-}\in(t_{1}^{-},t^{+})$,
$\mathbf{V}_{c,T}(t_{2}^{-})\in\mathcal{E}^{\alpha_{0},-}$ and

$\forall t\geq t_{2}^{-},\hskip 5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(t))\geq
0.$ (5.123)

Finally, we have the following property on constrained solutions:

######  Corollary 5.2.

Assume that (H3’), (H4’), (H5’), (H6’) and (H7’) hold. Let $c>0$ and $T\geq
1$. Consider $\mathbf{V}_{c,T}\in Y_{T}$ a minimizer of $\mathbf{E}_{c}$ in
$Y_{T}$, which exists by Lemma 5.7. Let
$t_{1}^{-}:=t_{1}^{-}(\mathbf{V}_{c,T})$ as in (5.69) and
$t^{+}:=t^{+}(\mathbf{V}_{c,T},\varepsilon_{0}^{+})$ as in (5.46) and
$t_{0}^{-}\leq t_{1}^{-}$, also given by Corollary 5.1. Then, there exists
$\delta_{Y,c,T}>0$ such that the set

$S_{Y,c,T}:=(-\infty,t_{0}^{-}+\delta_{Y,c,T})\cup(-T,T)\cup(t^{+}-\delta_{Y,c,T},+\infty)$
(5.124)

is such that $\mathbf{V}_{c,T}\in\mathcal{A}(S_{Y,c,T})$ (see (4.30)) and

$\mathbf{V}_{c,T}^{\prime\prime}-D_{\mathscr{L}}\mathcal{E}(\mathbf{V}_{c,T})=-c\mathbf{V}_{c,T}^{\prime}\mbox{
in }S_{Y,c,T}.$ (5.125)

###### Proof.

Using (5.119) in Corollary 5.1, we have that for each $\tilde{T}>0$ it holds

$\forall t\leq[t_{0}^{-}-\tilde{T},t_{0}^{-}+\delta_{0}^{-}],\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(t))\leq\tilde{\alpha}<\alpha_{-},$
(5.126)

so that (5.27) in Lemma 5.8 is satisfied for any closed interval contained in
$(-\infty,t_{0}^{-}]$. As a consequence, (5.28) follows in such interval. That
means that for $t_{0}^{-}\leq t_{1}^{-}$ and $\delta_{0}^{-}>0$ as in
Corollary 5.1, we have that
$\mathbf{V}_{c,T}\in\mathcal{A}((-\infty,t_{0}^{-}+\delta_{0}^{-}))$ and

$\mathbf{V}_{c,T}^{\prime\prime}-D_{\mathscr{L}}\mathcal{E}(\mathbf{V}_{c,T})=-c\mathbf{V}_{c,T}^{\prime\prime}\mbox{
in }(-\infty,t_{0}^{-}+\delta_{0}^{-}).$ (5.127)

Using again Lemma 5.8 in combination with (5.117) in Corollary 5.1, we obtain
for some $\delta_{0}^{+}>0$ that

$\mathbf{V}_{c,T}\in\mathcal{A}((-T,T)\cup(t^{+}-\delta_{0}^{+},+\infty))$
(5.128)

and

$\mathbf{V}_{c,T}^{\prime\prime}-D_{\mathscr{L}}\mathcal{E}(\mathbf{V}_{c,T})=-c\mathbf{V}_{c,T}^{\prime\prime}\mbox{
in }(-T,T)\cup(t^{+}-\delta_{0}^{+},+\infty).$ (5.129)

As a consequence, (5.125) follows by taking
$\delta_{Y,c,T}:=\min\\{\delta_{0}^{-},\delta_{0}^{+}\\}>0$. ∎

###  5.5 Existence of an unconstrained solution

We can now establish the existence of an unconstrained solution for the proper
speed. Let

$\mathcal{D}:=\\{c>0:\exists T\geq 1\mbox{ and }U\in Y_{T}\mbox{ such that
}\mathbf{E}_{c}(U)<0\\}.$ (5.130)

We have:

######  Lemma 5.13.

Assume that (H3’), (H4’), (H5’) and (H7’) hold. Let $\mathcal{D}$ be the set
defined in (5.130). Then, $\mathcal{D}$ is open and non-empty and bounded such
that

$\sup\mathcal{D}\leq\frac{\sqrt{-2a}}{d_{\alpha_{0}}},$ (5.131)

where $d_{\alpha_{0}}>0$ is as in (4.60)

###### Proof.

Let $\Psi$ be the function defined in (5.13). Since (H7’) holds we have by
Lemma 5.4 that $\Psi\in Y_{T}$. Therefore, the argument from [22] applies here
to show that $\mathcal{D}\not=\emptyset$. Subsequently, let $c\in\mathcal{D}$.
By definition, there exists $T\geq 1$ such that
$\mathbf{E}_{c}(\mathbf{V}_{c,T})<0$, where $\mathbf{V}_{c,T}\in Y_{T}$ is a
minimizer of $\mathbf{E}_{c}$ in $Y_{T}$, which exists by Lemma 5.7. Let
$\mathbf{v}_{c,T}^{+}$ be such that
$\lVert\mathbf{V}_{c,T}(t)-\mathbf{v}_{c,T}^{+}\rVert_{\mathscr{H}}\to 0$ as
$t\to+\infty$, which occurs due to Lemma 5.3. Define for $t\geq T$

$\mathbf{V}_{c,T}^{t}(s):=\begin{cases}\mathbf{V}_{c,T}(s)&\mbox{ if }s\leq
t,\\\ (1+t-s)\mathbf{V}_{c,T}(t)+(s-t)\mathbf{v}_{c,T}^{+}&\mbox{ if }t\leq
s\leq t+1,\\\ \mathbf{v}_{c,T}^{+}&\mbox{ if }t+1\leq s.\end{cases}$ (5.132)

which belongs to $Y_{T}$. Arguing as in [22], we can show that for $t\geq T$
large enough and some $\delta>0$ we have for all
$\tilde{c}\in(c-\delta,c+\delta)$ that
$\mathbf{E}_{\tilde{c}}(\mathbf{V}_{c,T}^{t}(s))<0$. This shows that
$\mathcal{D}$ is open. In order to establish the bound (5.131), we use
Corollary 5.1. Let $c\in\mathcal{D}$, there exists then $T\geq 1$ such that
$\mathbf{E}_{c}(\mathbf{V}_{c,T})<0$, with $\mathbf{V}_{c,T}$ a minimizer of
$\mathbf{E}_{c}$ in $Y_{T}$. Let $t_{2}^{-}:=t_{2}^{-}(\mathbf{V}_{c,T})$ as
in (5.68) and $t^{+}:=t^{+}(\mathbf{V}_{c,T},\varepsilon_{0}^{+})$ as in
(5.46). Corollary 5.1 implies that
$\mathbf{V}_{c,T}(t_{2}^{-})\in\mathcal{E}^{\alpha_{0}-}$ and
$\mathbf{V}_{c,T}(t^{+})\in\mathscr{F}^{+}_{r_{0}^{+}/2}$. Therefore, the
definition of $d_{\alpha_{0}}$ in (4.60) implies

$\left(d_{\alpha_{0}}\right)^{2}\leq\lVert\mathbf{V}_{c,T}(t^{+})-\mathbf{V}_{c,T}(t_{2}^{-})\rVert_{\mathscr{L}}^{2}$
(5.133)

so that

$\left(d_{\alpha_{0}}\right)^{2}\leq
2\int_{{\mathbb{R}}}\frac{\lVert\mathbf{V}_{c,T}^{\prime}(t)\rVert_{\mathscr{L}}^{2}}{2}e^{ct}dt\left(\frac{e^{-ct_{2}^{-}}-e^{-ct^{+}}}{c}\right).$
(5.134)

Using (5.123) in Corollary 5.1, the inequality above becomes

$\left(d_{\alpha_{0}}\right)^{2}\leq
2\left(\mathbf{E}_{c}(\mathbf{V}_{c,T})-\frac{e^{ct_{2}^{-}}}{c}\right)\left(\frac{e^{-ct_{2}^{-}}-e^{-ct^{+}}}{c}\right)$
(5.135)

which, using that $\mathbf{E}_{c}(\mathbf{V}_{c,T})<0$ and $t^{+}>t_{2}^{-}$,
gives (5.131) and concludes the proof. ∎

We can now establish the existence of an unconstrained solution:

######  Proposition 5.3.

Assume that (H3’), (H4’), (H5’), (H6’) and (H7’) hold. Let
$\overline{c}\in\partial(\mathcal{D})\cap(0,+\infty)$, where
$\partial(\mathcal{D})$ stands for the boundary of the set $\mathcal{D}$
defined in (5.130). Then, there exists $\overline{T}\geq 1$ such that
$\mathbf{b}_{\overline{c},\overline{T}}=0$
($\mathbf{b}_{\overline{c},\overline{T}}$ as in (5.12)) and
$\overline{\mathbf{V}}\in Y_{\overline{T}}$ an associated minimizer of
$\mathbf{E}_{\overline{c}}$ which does not saturate the constraints, i. e.,

$\forall t\geq\overline{T},\hskip
5.69054pt\mathrm{dist}_{\mathscr{L}}(\overline{\mathbf{V}}(t),\mathscr{F}^{+})<\frac{r_{0}^{+}}{2}$
(5.136)

and

$\forall t\leq-\overline{T},\hskip
5.69054pt\mathcal{E}(\overline{\mathbf{V}}(t))<\alpha_{-}.$ (5.137)

In particular, the pair $(\overline{c},\overline{\mathbf{V}})$ solves (4.7).

######  Remark 5.1.

Notice that Lemma 5.13 implies that (under the necessary assumptions) the set
$\mathcal{D}$ is bounded, meaning that the set
$\partial(\mathcal{D})\cap(0,+\infty)$ is not empty. Such a fact, in
combination with Proposition 5.3, shows the existence of the unconstrained
solution.

###### Proof of Proposition 5.3.

We essentially mimic the proof from [22] with the obvious modifications. Lemma
5.13 implies that
$\partial(\mathcal{D})\subset{\mathbb{R}}\setminus\mathcal{D}$, so that
$\overline{c}\not\in\mathcal{D}$. The definition of $\mathcal{D}$ in (5.130)
implies then that

$\forall T\geq 1,\hskip 5.69054pt\mathbf{b}_{\overline{c},T}\geq 0.$ (5.138)

Let $(c_{n})_{n\in{\mathbb{N}}}$ be a sequence in $\mathcal{D}$ such that
$c_{n}\to\overline{c}$ as $n\to\infty$. By definition, for each
$n\in{\mathbb{N}}$ there exists $T_{n}\geq 1$ such that
$\mathbf{E}_{c_{n}}(\mathbf{V}_{c_{n},T_{n}})<0$ where, for each
$n\in{\mathbb{N}}$, $\mathbf{V}_{c_{n},T_{n}}$ is a minimizer of
$\mathbf{E}_{c_{n}}$ in $Y_{T_{n}}$. For each $n\in{\mathbb{N}}$, set
$t_{1,n}^{-}:=t_{1}^{-}(\mathbf{V}_{c_{n},T_{n}})$ as in (5.69) and
$t_{n}^{+}:=t^{+}(\mathbf{V}_{c_{n},T_{n}},\varepsilon_{0}^{+})$ as in (5.46).
Using (5.120) in Corollary 5.1 we have

$\forall n\in{\mathbb{N}},\hskip
5.69054pt0<t_{n}^{+}-t_{1,n}^{+}\leq\mathsf{T}_{{\star\star}}(c_{n})$ (5.139)

with

$c\in(0,+\infty)\to\mathsf{T}_{{\star\star}}(c)\in(0,+\infty)$ (5.140)

a continuous function. Since the sequence $(c_{n})_{n\in{\mathbb{N}}}$ is
bounded, we have that

$\mathsf{T}_{\star\star}:=\max\left\\{1,\sup_{n\in{\mathbb{N}}}\mathsf{T}_{\star\star}(c_{n})\right\\}<+\infty$
(5.141)

and

$\forall n\in{\mathbb{N}},\hskip
5.69054pt0<t^{+}_{n}-t^{-}_{1,n}\leq\mathsf{T}_{\star\star},$ (5.142)

so that we have a bound on $(t^{+}_{n}-t^{-}_{1,n})_{n\in{\mathbb{N}}}$.
Moreover, (5.117) and (5.118) in Corollary 5.1 imply

$\forall n\in{\mathbb{N}},\forall t\geq t^{+}_{1,n},\hskip
5.69054pt\mathrm{dist}_{\mathscr{L}}(\mathbf{V}_{c_{n},T_{n}}(t),\mathscr{F}^{+})<\frac{r_{0}^{+}}{2}$
(5.143)

and

$\forall n\in{\mathbb{N}},\forall t\leq t^{-}_{n},\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c_{n},T_{n}}(t))\leq\alpha_{-}.$ (5.144)

For each $n\in{\mathbb{N}}$, define the function
$\mathbf{V}_{c_{n},T_{n}}^{t_{n}^{+}}:=\mathbf{V}_{c_{n},T_{n}}(\cdot+t_{n}^{+})$.
Then, (5.142) implies that (5.143) and (5.144) write as

$\forall n\in{\mathbb{N}},\forall t\geq 0,\hskip
5.69054pt\mathrm{dist}_{\mathscr{L}}(\mathbf{V}_{c_{n},T_{n}}^{t_{n}^{+}}(t),\mathscr{F}^{-})<\frac{r_{0}^{-}}{2}.$
(5.145)

and

$\forall n\in{\mathbb{N}},\forall t\leq-\mathsf{T}_{\star\star},\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c_{n},T_{n}}^{t_{n}^{+}}(t))\leq\alpha_{-}$
(5.146)

so that for all $n\in{\mathbb{N}}$ we have
$\mathbf{V}_{c_{n},T_{n}}^{t_{n}^{+}}\in Y_{\mathsf{T}_{\star\star}}$.
Moreover, a computation shows

$\forall n\in{\mathbb{N}},\hskip
5.69054pt\mathbf{E}_{c_{n}}(\mathbf{V}_{c_{n},T_{n}}^{t_{n}^{+}})=e^{-c_{n}t_{n}^{+}}\mathbf{E}_{c_{n}}(\mathbf{V}_{c_{n},T_{n}})<0.$
(5.147)

Therefore, if we apply Lemma 5.6 with sequence of speeds
$(c_{n})_{n\in{\mathbb{N}}}$ and the sequence
$(\mathbf{V}_{c_{n},T_{n}}^{t_{n}^{+}})_{n\in{\mathbb{N}}}$ in
$Y_{\mathsf{T}_{\star\star}}$, we obtain $\overline{\mathbf{V}}\in
Y_{\mathsf{T}_{\star\star}}$ such that

$\mathbf{E}_{\overline{c}}(\overline{\mathbf{V}})\leq\liminf_{n\to\infty}\mathbf{E}_{c_{n}}(\mathbf{V}_{c_{n},T_{n}}^{t_{n}^{+}})\leq
0,$ (5.148)

which in combination with (5.138) implies that
$\mathbf{b}_{\overline{c},\mathsf{T}_{\star\star}}=0$. Therefore, we have
$\mathbf{E}_{\overline{c}}(\overline{\mathbf{V}})=0$, so that
$\overline{\mathbf{V}}$ is a minimizer of $\mathbf{E}_{\overline{c}}$ in
$Y_{\mathsf{T}_{\star\star}}$. Set
$t^{-}_{1,{\star\star}}:=t_{1}^{-}(\overline{\mathbf{V}})$ as in (5.69) and
$t^{+}_{{\star\star}}:=t^{+}(\overline{\mathbf{V}},\varepsilon_{0}^{\pm})$ as
in (5.46). Invoking (5.117) and (5.119) in Corollary 5.1, we obtain as before
that for some $t_{0}^{-}\leq t_{1}^{-}$ and $\delta_{0}^{-}>0$

$\forall t\geq t^{+}_{\star\star},\hskip
5.69054pt\mathrm{dist}_{\mathscr{L}}(\overline{\mathbf{V}}(t),\mathscr{F}^{+})<\frac{r_{0}^{+}}{2},$
(5.149) $\begin{cases}\mathcal{E}(\overline{\mathbf{V}}(\cdot))\mbox{ is
monotone in }(-\infty,t_{0}^{-}+\delta_{0}-]\mbox{ and }\\\ \forall t\leq
t_{0}^{-}+\delta_{0}^{-},\hskip
5.69054pt\mathcal{E}(\overline{\mathbf{V}}(t))<\alpha_{-}.\end{cases}$ (5.150)

If we set $\overline{T}=\max\\{1,t^{+}_{\star\star},-t_{0}^{-}\\}$, then
(5.149) and (5.150) imply that $\overline{\mathbf{V}}\in Y_{\overline{T}}$ and
that (5.136), (5.137) hold. Moreover, we have that
$\mathbf{E}_{\overline{c}}(\overline{\mathbf{V}})=0$, so that
$\overline{\mathbf{V}}$ is a minimizer of $\mathbf{E}_{\overline{c}}$ in
$Y_{\overline{T}}$ due to (5.138). We will now apply Corollary 5.2 with
$\overline{c}$, $\overline{T}$ as constants and $\overline{\mathbf{V}}$ as
constrained minimizer. By (5.149) and (5.150), we have that
$\overline{\mathbf{V}}\in\mathcal{A}(S_{Y,\overline{c},\overline{T}})$ and

$\overline{\mathbf{V}}^{\prime\prime}-D_{\mathscr{L}}\mathcal{E}(\overline{\mathbf{V}})=-\overline{c}\overline{\mathbf{V}}^{\prime}\mbox{
in }S_{Y,\overline{c},\overline{T}},$ (5.151)

with

$S_{Y,\overline{c},\overline{T}}:=(-\infty,t_{0}^{-}+\delta_{Y,\overline{c},\overline{T}})\cup(-\overline{T},\overline{T})\cup(t_{\star\star}^{+}-\delta_{Y,\overline{c},\overline{T}},+\infty),$
(5.152)

for some $\delta_{Y,\overline{c},\overline{T}}>0$. The choice of
$\overline{T}$ implies that $S_{Y,\overline{c},\overline{T}}={\mathbb{R}}$,
which concludes the proof. ∎

###  5.6 Uniqueness of the speed

We have the following:

######  Proposition 5.4 ([22]).

Assume that (H6’) holds. Let $X$ be the set defined in (4.52). Let
$(c_{1},c_{2})\in(0,+\infty)^{2}$ be such that there exist $\mathbf{U}_{1}$
and $\mathbf{U}_{2}$ in $X\cap\mathcal{A}({\mathbb{R}})$ such that
$(c_{1},\mathbf{U}_{1})$ and $(c_{2},\mathbf{U}_{2})$ solve (4.7) and for each
$i\in\\{1,2\\}$, $\mathbf{E}_{c_{i}}(\mathbf{U}_{i})<+\infty$. Assume moreover
that

$\forall i\in\\{1,2\\},\forall j\in\\{1,2\\}\setminus\\{i\\},\hskip
5.69054pt\mathbf{E}_{c_{i}}(\mathbf{U}_{j})\geq 0.$ (5.153)

Then, we have $c_{1}=c_{2}$.

This allows to show the following:

######  Corollary 5.3.

Assume that (H3’), (H4’), (H5’), (H6’) and (H7’) hold. Let

$c(\mathcal{D}):=\sup\mathcal{D}.$ (5.154)

Then, we have $\mathcal{D}=(0,c(\mathcal{D}))$.

###### Proof of Corollary 5.3.

The idea is the same than in [22]. As before, we need to show that

$\partial(\mathcal{D})\cap(0,+\infty)=\\{c(\mathcal{D})\\}.$ (5.155)

Let $\overline{c}\in\partial(\mathcal{D})\cap(0,+\infty)$, we need to show
that $\overline{c}=c(\mathcal{D})$ in order to finish the proof. Using
Proposition 5.3 we obtain $\mathbf{V}^{\mathcal{D}}$ and
$\overline{\mathbf{V}}$ in $Y\subset X$ such that
$(c(\mathcal{D}),\mathbf{V}(\mathcal{D}))$ and
$(\overline{c},\overline{\mathbf{V}})$ solve (4.7). Moreover,

$\inf_{U\in
Y}\mathbf{E}_{c(\mathcal{D})}(U)=\mathbf{E}_{c(\mathcal{D})}(\mathbf{V}^{\mathcal{D}})=0=\mathbf{E}_{\overline{c}}(\overline{\mathbf{V}})=\inf_{U\in
Y}\mathbf{E}_{\overline{c}}(U)$ (5.156)

so that we also have

$\mathbf{E}_{c(\mathcal{D})}(\overline{\mathbf{V}})\geq 0\mbox{ and
}\mathbf{E}_{\overline{c}}(\mathbf{V}^{\mathcal{D}})\geq 0.$ (5.157)

Therefore, the requirements of Proposition 5.4 are met, which means that
$\overline{c}=c(\mathcal{D})$. ∎

###  5.7 Proof of Theorem 4 completed

If (H7’) holds, using Proposition 5.3 and Corollary 5.3, we can argue as in
[22] to establish the existence of a solution solving (4.7) with condition
(4.8) and speed $c^{\star}=c(\mathcal{D})$. The statement regarding the
uniqueness of the speed $c^{\star}$ follows from Proposition 5.4. Finally, we
have that the exponential convergence (4.57) follows from Lemma 5.15 with
$b=(c^{\star}+\gamma^{+})/2>c^{\star}/2$. Thus, the proof of Theorem 4 is
completed.

∎

###  5.8 Asymptotic behavior of the constrained solutions

We essentially recall and extend the results given in [22]. Recall first the
following:

######  Lemma 5.14 ([22]).

Assume that (H6’) holds. Let $c>0$, $t_{1}<t_{2}$ and
$U\in\mathcal{A}((t_{1},t_{2}))$ such that

$U^{\prime\prime}-D_{\mathscr{L}}\mathcal{E}(U)=-cU^{\prime}\mbox{ in
}(t_{1},t_{2}).$ (5.158)

Then, we have the formula

$\forall t\in(t_{1},t_{2}),\hskip
5.69054pt\frac{d}{dt}\left(\mathcal{E}(U(t))-\frac{\lVert
U^{\prime}(t)\rVert_{\mathscr{L}}^{2}}{2}\right)=c\lVert
U^{\prime}(t)\rVert^{2}_{\mathscr{L}}.$ (5.159)

We also define the constant

$\gamma^{+}:=\frac{1}{e(C^{+}+\mathsf{C}^{+})}>0.$ (5.160)

We first have

######  Lemma 5.15.

Assume that (H3’), (H4’), (H5’), (H6’) and (H7’) holds. Let $c>0$ and $T\geq
1$. Let $\mathbf{V}_{c,T}\in Y_{T}$ be a constrained solution given by Lemma
5.7. Then, it holds that for some $\mathbf{v}_{c,T}^{+}\in\mathscr{F}^{+}$

$\lim_{t\to+\infty}\lVert\mathbf{V}_{c,T}(t)-\mathbf{v}_{c,T}^{+}\rVert_{\mathscr{H}}^{2}e^{\frac{c+\gamma^{+}}{2}t}=0$
(5.161)
$\lim_{t\to+\infty}\lVert\mathbf{V}_{c,T}^{\prime}(t)\rVert_{\mathscr{L}}^{2}e^{\frac{c+\gamma^{+}}{2}t}=0$
(5.162)

and

$\lim_{t\to+\infty}\mathcal{E}(\mathbf{V}_{c,T}(t))e^{\frac{c+\gamma^{+}}{2}t}=0,$
(5.163)

where $\gamma^{+}$ was defined in (5.160).

###### Proof.

It is clear that the arguments in [22] apply to a solution
$\mathbf{V}_{c,T}\in Y_{T}$ if (H7’) holds instead of the perturbation
assumption, using Corollaries 5.1 and 5.2 instead of the equivalent results in
[22]. ∎

Next, we have

######  Lemma 5.16.

Assume that (H3’), (H4’), (H5’), (H6’) and (H7’) hold. Let $c>0$ and $T\geq
1$. Assume moreover that $c<\gamma^{-}$, where $\gamma^{-}$ is defined in
(4.42). Let $\mathbf{V}_{c,T}$ be a constrained solution given by Lemma 5.7.
Then, it holds that for some $\mathbf{v}_{c,T}^{-}\in\mathscr{F}^{+}$

$\lim_{t\to-\infty}\lVert\mathbf{V}_{c,T}(t)-\mathbf{v}_{c,T}^{-}\rVert_{\mathscr{H}}^{2}=0$
(5.164)
$\lim_{t\to-\infty}\lVert\mathbf{V}_{c,T}^{\prime}(t)\rVert_{\mathscr{L}}^{2}=0$
(5.165)

and

$\lim_{t\to-\infty}\mathcal{E}(\mathbf{V}_{c,T}(t))=a.$ (5.166)

###### Proof.

Let $\mathbf{V}_{c,T}$ be a minimizer of $\mathbf{E}_{c}$ in $Y_{T}$ given by
Lemma 5.7. Let $t_{1}^{-}:=t_{1}^{-}(\mathbf{V}_{c,T})$ as in (5.69).
Corollary 5.1 implies that for all $t\leq t_{1}^{-}$, we have
$\mathcal{E}(\mathbf{V}_{c,T}(t))\leq\alpha_{-}$ and in particular
$\mathbf{V}_{c,T}(t)\in\mathscr{F}^{-}_{r_{0}^{-}/2}$ by (H7’). We have that
(5.119) in Corollary 5.1 as well as Corollary 5.2 give $t_{0}^{-}\leq
t_{1}^{-}$ such that

$\forall t\leq t_{0}^{-},\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(t))<\alpha_{-},$ (5.167)

and so that $\mathbf{V}_{c,T}$ solves

$\mathbf{V}_{c,T}^{\prime\prime}-D_{\mathscr{L}}\mathcal{E}(\mathbf{V}_{c,T})=-c\mathbf{V}_{c,T}^{\prime}\mbox{
in }(-\infty,t_{0}^{-}),$ (5.168)

meaning that Lemma 5.14 applies to $\mathbf{V}_{c,T}$ in
$(-\infty,t_{0}^{-})$. Next, define as above for $t\leq t_{0}^{-}$ and

$\tilde{V}^{-}_{t}(s):=\begin{cases}P^{-}(\mathbf{V}_{c,T}(t))&\mbox{ if
}s\leq t-1,\\\
(t-s)P^{-}(\mathbf{V}_{c,T}(t))+(s-t+1)\mathbf{V}_{c,T}(t)&\mbox{ if }t-1\leq
s\leq t,\\\ \mathbf{V}_{c,T}(s)&\mbox{ if }t\leq s.\end{cases}$ (5.169)

We claim that $\tilde{V}^{-}_{t}\in Y_{T}$. For that, notice first that if
$s\leq t-1$ then $\mathcal{E}(\tilde{V}^{-}_{t}(s))=a<\alpha_{-}$. We will
apply (4.51) in (H7’). Notice that, in particular, it implies that for all
$v\in\mathcal{E}^{\alpha_{-}}$, the correspondence

$\lambda\in[0,1]\to\mathcal{E}(P^{-}(v)+\lambda(v-P^{-}(v)))\in{\mathbb{R}}$
(5.170)

is non decreasing. Let $s\in[t-1,t]$. Notice that since
$\mathbf{V}_{c,T}(t)\in\mathcal{E}^{\alpha_{-}}$, we can apply the previous to
obtain

$\forall\lambda\in[0,1],\hskip
5.69054pt\mathcal{E}(P^{-}(\mathbf{V}_{c,T}(t))+\lambda(P^{-}(\mathbf{V}_{c,T}(t))-\mathbf{V}_{c,T}(t)))\leq\mathcal{E}(\mathbf{V}_{c,T}(t))<\alpha_{-}$
(5.171)

which by taking $\lambda=s-t+1\in[0,1]$ implies that
$\mathcal{E}(\tilde{V}^{-}_{t}(s))<\alpha_{-}$. Therefore, we have established
the claim. At this point, we can apply the same arguments as in [22] to
conclude the proof. ∎

We finally show the following result (which in fact generalizes Theorem 5) for
the case in which (H7’) holds and $\mathscr{F}^{-}$ is a singleton:

######  Lemma 5.17.

Assume that (H3’), (H4’), (H5’), (H6’) and (H7’) hold. Let $c>0$ and $T\geq
1$. Assume moreover that $\mathscr{F}^{-}=\\{\mathbf{v}^{-}\\}$. Let
$\mathbf{V}_{c,T}$ be a constrained solution given by Lemma 5.8. Then, it
holds

$\lim_{t\to-\infty}\lVert\mathbf{V}_{c,T}(t)-\mathbf{v}^{-}\rVert_{\mathscr{L}}=0.$
(5.172)

###### Proof.

We essentially argue similarly to [3] and [4], using the fact that
$\mathscr{F}^{-}=\\{\mathbf{v}^{-}\\}$. In such a case, (4.51) in assumption
(H7’) implies that

$\forall
v\in\mathcal{E}^{\alpha_{0},-},\exists\lambda(v)>1,\forall\theta\in(0,\lambda(v)),\hskip
5.69054pt\frac{d}{d\lambda}(\mathcal{E}(\mathbf{v}^{-}+\lambda(v-\mathbf{v}^{-}))\geq
0.$ (5.173)

Corollary 5.2 implies in particular the existence of
$t_{0}^{-}\in{\mathbb{R}}$ such that

$\forall t\leq t_{0}^{-},\hskip
5.69054pt\mathcal{E}(\mathbf{V}_{c,T}(t))<\alpha_{-},$ (5.174)
$\mathbf{V}\in\mathcal{A}((-\infty,t_{0}^{-}))$ (5.175)

and

$\mathbf{V}^{\prime\prime}_{c,T}-D_{\mathscr{L}}\mathcal{E}(\mathbf{V}_{c,T})=-c\mathbf{V}_{c,T}^{\prime}\mbox{
in }(-\infty,t_{0}^{-}).$ (5.176)

The combination of (5.173) and (5.174) gives that

$\forall t\leq
t_{0}^{-},\exists\lambda(t)>1,\forall\theta\in(0,\lambda(t)),\hskip
5.69054pt\frac{d}{d\lambda}(\mathcal{E}(\mathbf{v}^{-}+\lambda(\mathbf{V}_{c,T}(t)-\mathbf{v}^{-}))\geq
0$ (5.177)

which, using that $\mathbf{V}_{c,T}\in\mathcal{A}((-\infty,t_{0}))$ and (4.14)
in (H2’), implies

$\forall t<t_{0}^{-},\hskip 5.69054pt\langle
D_{\mathscr{L}}\mathcal{E}(\mathbf{V}_{c,T}(t)),\mathbf{V}_{c,T}(t)-\mathbf{v}^{-}\rangle_{\mathscr{L}}\geq
0.$ (5.178)

Define now the function

$\rho_{c,T}:t\in(-\infty,t_{0}^{-})\to\lVert\mathbf{V}_{c,T}(t)-\mathbf{v}^{-}\rVert_{\mathscr{L}}^{2}\in{\mathbb{R}}.$
(5.179)

Notice that (5.174) along with (H7’) implies that

$\forall t\leq t_{0}^{-},\hskip
5.69054pt\rho_{c,T}(t)\leq\frac{(r_{0}^{-})^{2}}{2}.$ (5.180)

The fact that $\mathbf{V}_{c,T}\in\mathcal{A}((-\infty,t_{0}^{-}))$ implies
that $\rho\in\mathcal{C}^{2}((-\infty,t_{0}^{-}))$. We have that

$\forall t<t_{0}^{-},\hskip
5.69054pt\rho_{c,T}^{\prime}(t)=2\langle\mathbf{V}_{c,T}(t)-\mathbf{v}^{-},\mathbf{V}_{c,T}^{\prime}(t)\rangle_{\mathscr{L}}$
(5.181)

and

$\forall t<t_{0}^{-},\hskip
5.69054pt\rho_{c,T}^{\prime\prime}(t)=2\lVert\mathbf{V}_{c,T}^{\prime}(t)\rVert_{\mathscr{L}}^{2}+2\langle\mathbf{V}_{c,T}(t)-\mathbf{v}^{-},\mathbf{V}_{c,T}^{\prime\prime}(t)\rangle_{\mathscr{L}}$
(5.182)

so that, using (5.176)

$\displaystyle\forall t<t_{0}^{-},\hskip
5.69054pt\rho_{c,T}^{\prime\prime}(t)$
$\displaystyle=2\lVert\mathbf{V}_{c,T}^{\prime}(t)\rVert_{\mathscr{L}}^{2}+2\langle
D_{\mathscr{L}}\mathcal{E}(\mathbf{V}_{c,T}(t)),\mathbf{V}_{c,T}(t)-\mathbf{v}^{-}\rangle_{\mathscr{L}}$
(5.183)
$\displaystyle-2c\langle\mathbf{V}_{c,T}(t)-\mathbf{v}^{-},\mathbf{V}_{c,T}^{\prime}(t)\rangle_{\mathscr{L}},$
(5.184)

which, by (5.178) and (5.181) gives the inequality

$\forall t<t_{0}^{-},\hskip
5.69054pt\rho_{c,T}^{\prime\prime}(t)+c\rho^{\prime}(t)\geq 0.$ (5.185)

Inequality (5.185) implies that $\rho_{c,T}$ does not possess any strict local
maximum. Therefore, we can find $\tilde{t}_{1}<t_{0}^{-}$ such that
$\rho_{c,T}$ is monotone in $(-\infty,\tilde{t}_{1})$. In combination with the
bound (5.180), monotony implies the existence of $l\in[0,(r_{0}^{-})^{2}/2]$
such that

$\lim_{t\to-\infty}\rho_{c,T}(t)=l.$ (5.186)

In order to conclude the proof, we need to show that $l=0$. Assume by
contradiction that $l\not=0$. Decreasing the value of $\tilde{t}_{1}$ if
necessary, we find $\tilde{l}>0$ such that for all $t\leq\tilde{t}_{1}$ it
holds $\mathcal{E}(\mathbf{V}_{c,T}(t))\geq\tilde{l}$. Therefore, using (4.51)
in (H7’), we have that

$\forall
t\leq\tilde{t}_{1},\exists\delta(t)>0,\forall\theta\in(1-\delta(t),1+\delta(t)),\hskip
5.69054pt\frac{d}{d\lambda}(\mathcal{E}(\mathbf{v}^{-}+(\mathbf{V}_{c,T}(t)-\mathbf{v}^{-})))\geq\omega(\tilde{l}),$
(5.187)

with $\omega(\tilde{l})>0$. In particular, it holds

$\forall t\leq\tilde{t}_{1},\hskip 5.69054pt\langle
D_{\mathscr{L}}\mathcal{E}(\mathbf{V}_{c,T}(t)),\mathbf{V}_{c,T}(t)-\mathbf{v}^{-}\rangle_{\mathscr{L}}\geq\omega(\tilde{l}),$
(5.188)

which by (5.181) and (5.182) gives

$\forall t\leq\tilde{t}_{1},\hskip
5.69054pt\rho_{c,T}^{\prime\prime}(t)+c\rho_{c,T}^{\prime}(t)\geq
2\omega(\tilde{l}).$ (5.189)

Suppose first that $\rho_{c,T}$ is non increasing in
$(-\infty,\tilde{t}_{1})$, meaning that for all $t\leq\tilde{t}_{1}$,
$\rho_{c,T}(\tilde{t}_{1})\leq\rho_{c,T}(t)$ and
$\rho_{c,T}^{\prime}(\tilde{t}_{1})\leq 0$. Moreover, integrating (5.189) in
$(s,\tilde{t}_{1})$ for $s<\tilde{t}_{1}$, we obtain

$\forall s<\tilde{t}_{1},\hskip
5.69054pt\rho_{c,T}^{\prime}(\tilde{t}_{1})-\rho_{c,T}^{\prime}(s)+c(\rho_{c,T}(\tilde{t}_{1})-\rho_{c,T}(s))\geq
2\omega(\tilde{l})(\tilde{t}_{1}-s).$ (5.190)

which becomes

$\forall s<\tilde{t}_{1},\hskip
5.69054pt2\omega(\tilde{l})(s-\tilde{t}_{1})\geq\rho_{c,T}^{\prime}(s)$
(5.191)

meaning that $\lim_{s\to-\infty}\rho_{c,T}^{\prime}(s)=-\infty$, which
contradicts (5.186). Therefore, $\rho_{c,T}$ must be non decreasing in
$(-\infty,\tilde{t}_{1}]$, which means that for all $t\leq\tilde{t}_{1}$,
$\rho_{c,T}^{\prime}(t)\geq 0$. We now fix $s<t<\tilde{t}_{1}$ and integrate
(5.189) in $[s,t]$. We obtain

$\rho_{c,T}^{\prime}(t)-\rho_{c,T}^{\prime}(s)+c(\rho_{c,T}(t)-\rho_{c,T}(s))\geq
2\omega(\tilde{l})(t-s),$ (5.192)

which, since $\rho_{c,T}$ is non decreasing and we have the bound (5.180),
gives

$\rho_{c,T}^{\prime}(t)\geq
2\omega(\tilde{l})(t-s)-c\frac{(r_{0}^{-})^{2}}{2}$ (5.193)

so that, by integrating again

$\forall s<\tilde{t}_{2},\hskip
5.69054pt\rho_{c,T}(\tilde{t}_{1})-\rho_{c,T}(s)\geq\omega(\tilde{t})(\tilde{t}_{1}-s)^{2}-c\frac{(r_{0}^{-})^{2}}{2}(\tilde{t}_{1}-s)$
(5.194)

which gives a contradiction by passing to the limit $s\to-\infty$, since the
right-hand side term tends in that case to $+\infty$. Therefore, we must have
that the limit $l$ defined in (5.186) equals 0, which concludes the proof. ∎

######  Remark 5.2.

The arguments in Lemma 5.17 do not seem to apply to the case in which
$\mathscr{F}^{-}$ is no longer a singleton. Indeed, in such an event, the
function $\rho_{c,T}$ would have to be defined as

$\rho_{c,T}:t\in(-\infty,t_{0}^{-})\to\lVert\mathbf{V}_{c,T}(t)-P^{-}(\mathbf{V}_{c,T}(t))\rVert_{\mathscr{L}}^{2}$
(5.195)

so that, when computing its derivatives, the (possibly nonzero) differential
of $DP^{-}$ appears and with it some terms which do not seem to have
necessarily the right sign. Recall also that the limit property given by Lemma
5.17 does not allow, by itself, to obtain information on
$\mathcal{E}(\mathbf{V}_{c,T}(\cdot))$ at $-\infty$, since $\mathcal{E}$ is
only lower semicontinuous with respect to $\mathscr{L}$-convergence.

###  5.9 Proof of Theorem 5 completed

It is a particular case of Lemma 5.17. Indeed, the solution
$(c^{\star},\mathbf{U})$ given by Theorem 4 is given by Proposition 5.3, so in
particular is a constrained solution in $Y_{T}$ for suitable $T\geq 1$ and
Lemma 5.17 applies.

∎

###  5.10 Proof of Theorem 6 completed

If (H7’) holds, then the same argument as in [22] using Proposition 5.3,
inequality (5.131) in Lemma 5.13 and (4.59) in (H8’) gives the result.

∎

###  5.11 On the proof of Theorem 7

The proof works exactly as in [22], so we skip it.

##  6 Proofs of the main results

The main results of this paper are obtained by the following, which links the
main setting with the abstract setting:

######  Lemma 6.1.

Assume that (H1), (H2) and (H3) hold. Let

$\mathscr{L}=\mathscr{H}=\mathscr{H}^{\prime}={\mathbb{R}}^{k}$ (6.1)

and $\mathcal{E}=W$. Then, (H1’), (H2’), (H3’), (H4’), (H5’), (H6’) and (H7’)
hold. Moreover, if (H4) holds, then (H8’) holds.

###### Proof.

The proof is straightforward. Indeed, assumption (H1’) follows from (H1).
Assumption (H2’) becomes tautological because
$\mathscr{L}=\mathscr{H}=\mathscr{H}^{\prime}={\mathbb{R}}^{k}$. Assumption
(H3’) is straightforward. Assumption (H4’) corresponds directly to (H2).
Assumption (H5’) is also tautological. Regarding (H6’), the existence of the
map $\mathfrak{P}$ follows from (2.4), see the proof of Lemma 2.3 in [22]. The
rest of the statement follows from elliptic regularity results or it is
tautological. Finally, we clearly have that (H8’) holds if (H4) holds. ∎

Notice that most of the strength of the abstract results will not be used as
we choose all the Hilbert spaces equal to ${\mathbb{R}}^{k}$. This also makes
all the

###  6.1 Proof of Theorem 1

We apply Theorem 4 with the choice of objects given by Lemma 6.1. It is clear
that (4.7) reads now as (2.14) and (4.8) reads as (2.15), which establishes
the existence part as we also have that $\mathcal{A}({\mathbb{R}})$ becomes a
subset of $\mathcal{C}^{2}_{\mathrm{loc}}({\mathbb{R}},{\mathbb{R}}^{k})$ by
its definition in (4.30). Due to the fact that $X=\mathcal{S}$,
$Y=\overline{\mathcal{S}}$ (with $X$ as in (4.52) and $Y$ as in (4.54)) the
uniqueness statement of Theorem 1 follows directly from that in Theorem 4.
Finally, it is clear that the exponential convergence in Theorem 4 writes as
that in Theorem 1, which concludes the proof. ∎

###  6.2 Proof of Theorem 2

We apply Theorems 5 and 6 with the choice of objects given by Lemma 6.1. Then,
we have that the case 1. in (H4) corresponds to the assumptions of Theorem 5
and case 2. in (H4) is the abstract assumption (H8’), required in Theorem 6.
Therefore, Theorem 2 follows. ∎

###  6.3 Proof of Theorem 3

We apply Theorem 4 with the choice of objects given by Lemma 6.1. It is then
clear that Theorem 3 holds, as recall that $Y=\overline{\mathcal{S}}$. ∎

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