# Asymptotic behavior of a diffused interface
volume-preserving mean curvature flow

Matteo Bonforte  Departamento de Matemáticas, Universidad Autónoma de Madrid
Campus de Cantoblanco, 28049, Madrid<EMAIL_ADDRESS>,  Francesco
Maggi  Department of Mathematics, The University of Texas at Austin, 2515
Speedway, Stop C1200, Austin TX 78712-1202, United States of America
<EMAIL_ADDRESS>and  Daniel Restrepo  Department of Mathematics, Johns
Hopkins University, Baltimore, MD, United States of America<EMAIL_ADDRESS>
###### Abstract.

We consider a diffused interface version of the volume-preserving mean
curvature flow in the Euclidean space, and prove, in every dimension and under
natural assumptions on the initial datum, exponential convergence towards
single “diffused balls”.

###### Contents

  1. 1 Introduction
  2. 2 Background material on the diffused isoperimetric problem
  3. 3 Estimates for the Lagrange multiplier functional
  4. 4 Existence, regularity and entropies of the flow (Proof of Theorem 1.2)
  5. 5 Subsequential bubbling resolution (Proof of Theorem 1.3)
  6. 6 Subsequential bubbling into diffused balls (Proof of Theorem 1.4)
  7. 7 Strict stability of the diffused isoperimetric problem (Proof of Theorem 1.5)
  8. 8 Exponential convergence to a single diffused bubble (Proof of Theorem 1.1)

##  1\. Introduction

###  1.1. Overview

In this paper we introduce a PDE reformulation of the classical volume-
preserving mean curvature flow (VPMCF) in $\mathbb{R}^{n}$ where the role of
the perimeter functional is played by the Allen–Cahn energy. From the physical
viewpoint this reformulation seems well justified, since it consists in
replacing the classical sharp interface model for surface tension based on
perimeter minimization with the equally interesting and important diffused
interface model based on the minimization of the Allen–Cahn energy. From the
mathematical viewpoint working in the diffused setting clears up the analysis
from the ambiguities brought in by the formation of singularities
characteristic of geometric flows, which are directly reflected into the
abundance of non-equivalent weak formulations of the VPMCF.

Our main result proves, in every dimension and under a variety of natural
assumptions on the initial datum, exponential convergence of the diffused
VPMCF towards single “diffused balls”. This result is indeed stronger than the
presently known analogous results for the classical VPMCF, see Remark 1.1
below.

The diffused interface model we adopt is based on the volume-constrained
minimization of the classical Allen–Cahn functional

$\mathcal{AC}_{\varepsilon}(u)=\varepsilon\,\int_{\mathbb{R}^{n}}|\nabla
u|^{2}+\frac{1}{\varepsilon}\,\int_{\mathbb{R}^{n}}W(u)\,,\qquad\varepsilon>0\,,$
(1.1)

defined on (dimensionless) density functions $u:\mathbb{R}^{n}\to[0,1]$, and
requiring the choice of (dimensionless) double-well potential
$W:[0,1]\to[0,\infty)$. We will always require that $W\in C^{2,1}[0,1]$ and
that $W$ satisfies the standard non-degeneracy assumptions

$\mbox{$W(0)=W(1)=0$}\,,\quad\mbox{$W>0$ on
$(0,1)$}\,,\quad\mbox{$W^{\prime\prime}(0),W^{\prime\prime}(1)>0$}\,,$ (1.2)

as well as the normalization

$\int_{0}^{1}\sqrt{W}=1\,.$ (1.3)

As usual, $\varepsilon$ has the dimensions of length, so that
$\mathcal{AC}_{\varepsilon}(u)$ has the dimensions of (codimension one) area.

Particular care must be put in the choice of the volume potential
$V:[0,1]\to[0,\infty)$ used to impose the volume constraint on $u$. Indeed,
while any choice of $V$ satisfying $V(1)>0$ and $V(0)=0$ will return the
correct volume constraint in the sharp interface limit $\varepsilon\to 0^{+}$
(and will thus be acceptable form the physical viewpoint), not every choice of
$V$ will result in a mathematical model that is either well-posed or feasible
of in-depth analysis.

A natural choice for $V$ is suggested by the classical isoperimetric lower
bound on $\mathcal{AC}_{\varepsilon}(u)$, and consists in taking

$\mbox{$V(r)=\Phi(r)^{n/(n-1)}$, where $\Phi(r)=\int_{0}^{r}\sqrt{W}$ for
$r\in[0,1]$}\,.$ (1.4)

Indeed, by a classical application of the Cauchy–Schwartz inequality and the
chain rule, we have that

$\varepsilon\,|\nabla u|^{2}+\frac{W(u)}{\varepsilon}\geq 2\,|\nabla
u|\,\sqrt{W(u)}=2\,|\nabla(\Phi\circ u)|\,,$

from which the isoperimetric lower bound

$\mathcal{AC}_{\varepsilon}(u)\geq 2\,|D[\Phi\circ u]|(\mathbb{R}^{n})\geq
2\,n\,\omega_{n}^{1/n}\,\mathcal{V}(u)^{(n-1)/n}\,,\qquad\mathcal{V}(u):=\int_{\mathbb{R}^{n}}V(u)\,,$
(1.5)

follows, so that111Here, $|Dv|$ denotes the total variation measure of $v\in
L^{1}_{\rm loc}(\mathbb{R}^{n})$, $\omega_{n}$ is the volume of the unit
radius ball in $\mathbb{R}^{n}$, $|E|$ and $P(E)$ denote the volume and
perimeter of $E\subset\mathbb{R}^{n}$, and $B^{(m)}$ stands for the ball of
volume $m$ with center at the origin in $\mathbb{R}^{n}$.

$n\,\omega_{n}^{1/n}\,m^{(n-1)/n}=P(B^{(m)})=\inf\\{P(E):|E|=m\\}\,,\qquad
m>0\,,$

is the optimal value of the (Euclidean) isoperimetric problem.

Thanks to (1.5), our choice (1.4) of $V$ is instrumental for obtaining a well-
posed diffused interface (Euclidean) isoperimetric problem,

$\Psi(\varepsilon,m)=\inf\Big{\\{}\mathcal{AC}_{\varepsilon}(u):\mathcal{V}(u)=m\,,u\in
L^{1}_{\rm loc}(\mathbb{R}^{n};[0,1])\Big{\\}}\,,\qquad\varepsilon\,,m>0\,.$
(1.6)

Indeed, by (1.5) and by the (sharp) isoperimetric inequality for functions of
bounded variation, we see that

$\Psi(\varepsilon,m)>2\,n\,\omega_{n}^{1/n}\,m^{(n-1)/n}\,,\qquad\forall\varepsilon,m>0\,,$
(1.7)

while a simple comparison argument shows that this inequality is saturated in
the limit as $\varepsilon\to 0^{+}$. It is important to keep in mind that
simpler choices of $V$, like $V(t)=t$ or $V(t)=t^{2}$, would have led222The
size of $V(t)$ for $t\to 0^{+}$ plays a crucial role here. Our choice of $V$
satisfies $V(t)={\rm O}(t^{2\,n/(n-1)})$ as $t\to 0^{+}$. This is not the only
property of $V$ that plays an important role in our analysis though, and the
close relation between $V$ and $W$ will be repeatedly used. to degenerate
minimization problems where every competitor has positive energy but where the
infimum of the energy is equal to zero.

Problem $\Psi(\varepsilon,m)$ has been studied in great detail in [MR24]. Some
of the results obtained therein will play an important role in the present
paper, and will therefore be summarized in Section 2. For the moment, with the
sole intent of formulating our main result as quickly as possible, we just
recall from [MR24] that, in the physical regime where
$\varepsilon\in(0,\varepsilon_{0}\,m^{1/n})$ for some universal constant333By
universal constant we mean a constant depending only on the dimension $n$ and
on the double-well potential $W$. By $C(a,b,\dots)$ we denote a constant
depending only on $n$, $W$, and the arguments $a$, $b$, etc..
$\varepsilon_{0}$, there is a unique minimizer $\zeta_{\varepsilon,m}$ in the
class of the radially symmetric, strictly decreasing, and everywhere positive
functions on $\mathbb{R}^{n}$ with maximum at the origin; and that $u$ is a
minimizer of $\Psi(\varepsilon,m)$ if and only if
$u=\tau_{x_{0}}[\zeta_{\varepsilon,m}]$ for some444We set
$\tau_{x_{0}}[v](x)=v(x-x_{0})$ for every $x,x_{0}\in\mathbb{R}^{n}$ and
$v:\mathbb{R}^{n}\to\mathbb{R}^{m}$. $x_{0}\in\mathbb{R}^{n}$. Each
$\zeta_{\varepsilon,m}$ solves the diffused constant mean curvature equation

$2\,\varepsilon^{2}\,\Delta\zeta_{\varepsilon,m}=W^{\prime}(\zeta_{\varepsilon,m})-\varepsilon\,\Lambda_{\varepsilon,m}\,V^{\prime}(\zeta_{\varepsilon,m})\qquad\mbox{on
$\mathbb{R}^{n}$}\,,$

where $\Lambda_{\varepsilon,m}\to 2\,(n-1)\,\omega_{n}^{1/n}\,m^{-1/n}$ as
$\varepsilon\to 0^{+}$ (and is thus positive in the physical regime
$\varepsilon\in(0,\varepsilon_{0}\,m^{1/n})$).

The diffused VPMCF is then defined as the $L^{2}$-gradient flow of
$\mathcal{AC}_{\varepsilon}$ with a Lagrange multiplier modification that
preserves $\mathcal{V}$ along the flow: that is, we consider the parabolic
initial value problem555Given $t\geq 0$ and
$u:\mathbb{R}^{n}\times[0,\infty)\to\mathbb{R}$, we set
$u(t):\mathbb{R}^{n}\to\mathbb{R}$ for the function defined by
$u(t)(x):=u(x,t)$ ($x\in\mathbb{R}^{n}$).

$\left\\{\begin{split}&\varepsilon^{2}\,\partial_{t}u=2\,\varepsilon^{2}\,\Delta
u-W^{\prime}(u)+\varepsilon\,\lambda_{\varepsilon}[u(t)]\,V^{\prime}(u)\,,\qquad\mbox{on
$\mathbb{R}^{n}\times(0,\infty)$}\,,\\\ &u(0)=u_{0}\,,\end{split}\right.$ (DF)

where we have introduced the Lagrange multiplier functional666Notice that
$\lambda_{\varepsilon}[v]$ is defined in $[0,\infty]$ on any
$v:\mathbb{R}^{n}\to[0,1]$ with $|\\{0<v<1\\}|>0$ – this condition guarantees
indeed that $\int_{\mathbb{R}^{n}}V^{\prime}(v)^{2}>0$.

$\lambda_{\varepsilon}[v]=\frac{\int_{\mathbb{R}^{n}}2\,\varepsilon^{2}\,|\nabla
v|^{2}\,V^{\prime\prime}(v)+W^{\prime}(v)\,V^{\prime}(v)}{\varepsilon\,\int_{\mathbb{R}^{n}}V^{\prime}(v)^{2}}\,,$
(1.8)

whose choice guarantees, on smooth solutions of the flow, that

$\frac{d}{dt}\,\mathcal{V}(u(t))=\int_{\mathbb{R}^{n}}V^{\prime}(u(t))\,\partial_{t}u(t)=0\,,\qquad\mbox{i.e.,
$\mathcal{V}(u(t))=\mathcal{V}(u_{0})$ for all $t>0$}\,.$

From the viewpoint of classical parabolic theory, (DF) presents some peculiar
features since it is a non-autonomous semilinear PDE, where the non-autonomy
is due to the Lagrange multiplier $\lambda_{\varepsilon}[u(t)]$, which, in
turn, is non-local in space (its determination requires knowledge of $u(t)$
over the whole $\mathbb{R}^{n}$). The following theorem is our main result
concerning the long-time behavior of (DF).

######  Theorem 1.1.

If $n\geq 2$ and $W\in C^{2,1}[0,1]$ satisfies (1.2) and (1.3), then there
exists a universal constant $\varepsilon_{0}>0$ with the following property.
If $\varepsilon\in(0,\varepsilon_{0})$, $u_{0}\in
W^{2,p}(\mathbb{R}^{n};[0,1])$ for all $p\geq 2$, $\mathcal{V}(u_{0})=1$, and

$\displaystyle\mbox{either
$\mathcal{AC}_{\varepsilon}(u_{0})<2\,\Psi(\varepsilon,1/2)$}\,,$ (1.9)
$\displaystyle\mbox{or ${\rm spt}\,u_{0}$ is compact}\,,$ (1.10)

then

$0\leq\mathcal{AC}_{\varepsilon}(u(t))-\Psi(\varepsilon,1)\leq
C(\varepsilon,u_{0})\,e^{-t/C(\varepsilon)}\,,\qquad\forall
t>\frac{1}{C(\varepsilon,u_{0})}\,,$ (1.11)

and there exists a unique $x_{0}\in\mathbb{R}^{n}$ (depending on $\varepsilon$
and $u_{0}$) such that for all $p>2$ and $t>1/C(\varepsilon,u_{0})$

$\big{\|}u(t)-\tau_{x_{0}}[\zeta_{\varepsilon,1}]\big{\|}_{(W^{2,p}\cap
W^{1,2})(\mathbb{R}^{n})}\leq
C(u_{0},\varepsilon,p)\,e^{-t/C(\varepsilon)}\,.$ (1.12)

######  Remark 1.1 (Asymptotic analysis of the VPMCF).

It is convenient to briefly review the state of the art concerning convergence
to equilibrium for the classical VPMCF (with the disclaimer that, due to
singularities formation, these various results may pertain to different weak
formulations of the VPMCF). First, convergence to a single ball has been
proved under a variety of suitable geometric restrictions on the initial datum
that can be shown to be preserved along the flow, and that exclude
singularities formation: these are uniform convexity [Hui87],
$C^{1}$-proximity to a sphere [ES98], star-shapedeness [KK20], or integral
pinching conditions [Li09]. In general, singularity formation may lead to
convergence to multiple balls, a phenomenon called bubbling (see [FJM22,
Theorem 1.4] for an example) and it is thus unclear for which class of initial
data one should expect convergence towards a single ball (of the same volume
as the initial datum) or towards multiple balls (all with a same volume equal
to a fraction of the initial one). Since perimeter decreases along the VPMCF,
a natural conjecture is that, if a unit volume initial datum has perimeter
strictly less than twice the perimeter of a ball of volume $1/2$, then
convergence to a single ball should follow (with exponential rates of
convergence). This conjecture has been proved for “flat flow” solutions of the
VPMCF, and in dimensions $n=2$ and $n=3$ respectively, in the recent papers
[JMPS23, JMOS24]. Moreover, again when $n=2,3$, in [JN23] it is proved that
flat flow solutions always resolve, as $t\to\infty$, as finite union of balls
with possibly moving centers. All these results are based on powerful
quantitative bubbling results for sets with $L^{2}$-small mean curvature
oscillation. The restriction to dimensions $n=2,3$ is strongly correlated, on
the one hand, to the fact that the $L^{2}$-oscillation of the mean curvature
is the dissipation of the VPMCF, and is thus the quantity to work with in this
setting; and that, on the other hand, the critical $L^{p}$-space for the
regularity theory of the mean curvature of a boundary in $\mathbb{R}^{n}$ is
$p=n-1$. For these reasons, the further extension of these methods to
dimensions $n\geq 4$ seems delicate.

######  Remark 1.2 (On assumption (1.9)).

Assumption (1.9) amounts to asking that the initial datum has strictly less
energy than twice that of two diffused balls of volume $1/2$. Hence, by
proving Theorem 1.1 under (1.9) we have proved the validity, in every
dimension, of the diffused analogue of the VPMCF-conjecture mentioned in
Remark 1.1.

######  Remark 1.3 (On assumption (1.10)).

Proving Theorem 1.1 under assumption (1.10) is somehow more striking than
doing so under assumption (1.9), since (1.10) allows for initial data with
arbitrarily large energy as well as for initial data that is arbitrarily close
to the characteristic functions of any bounded set with finite perimeter and
unit volume. Indeed, if $E\subset\mathbb{R}^{n}$ is a bounded set of finite
perimeter with $|E|=1$, then by a standard construction, we can find a family
$\\{v_{\varepsilon}\\}_{\varepsilon>0}$ of smooth and compactly supported
functions on $\mathbb{R}^{n}$ such that $v_{\varepsilon}\to 1_{E}$ in
$L^{1}(\mathbb{R}^{n})$ and $\mathcal{AC}_{\varepsilon}(v_{\varepsilon})\to
2\,P(E)$ as $\varepsilon\to 0^{+}$, with ${\rm
spt}\,v_{\varepsilon}\subset\subset\\{x:{\rm dist}(x,E)<1\\}$ for all
$\varepsilon>0$.

###  1.2. Open problems and metastability

Before presenting the proof of Theorem 1.1, and the various intermediate
results behind it, we briefly introduce some interesting problems related to
Theorem 1.1.

A first natural problem is addressing the existence of non-compactly supported
initial data such that the resulting flow does not converge to a single
diffused ball, but rather resolves into a superposition of time-drifting
diffused balls (compare with conclusion (1.21) in Theorem 1.4). Since the
physical or numerical relevance of non-compactly supported initial data is
unclear, this is probably a question of very theoretical flavor; still,
answering to it may be challenging.

In the $\varepsilon\to 0^{+}$-limit, and for suitably prepared initial data,
the diffused VPMCF should converge to a weak formulation of the VPMCF, and,
indeed, this kind of convergence has been established, in absence of volume-
preserving Lagrange multipliers, in [Ilm93, BOS06], and for the VPMCF but
under spherical symmetry assumptions on $\Omega$, in [BS97]. For this reason,
another natural problem related to Theorem 1.1 would be understanding whether
the $\varepsilon$-dependency of the decay rates (1.11) and (1.12) can be
dropped off or not: the corresponding $\varepsilon$-independent decay rates
could then be transferred to the VPMCF. It seems natural to conjecture that,
if $u_{0}$ is compactly supported and is such that
$\mathcal{AC}_{\varepsilon}(u_{0})<2\,\Psi(\varepsilon,1/2)$ (that is, if both
(1.9) and (1.10) hold), then $\varepsilon$-independent decay rates to a single
diffused ball hold true, thus providing a strategy to extend the results of
[JN23, JMPS23, JMOS24] to arbitrary dimensions.

We do not expect, however, that $\varepsilon$-independent decay rates to a
single diffused ball should hold for a generic compactly supported initial
datum $u_{0}$: such rates should hold, at best, only for
$t>T_{\varepsilon}=T_{\varepsilon}(u_{0})$ with $T_{\varepsilon}\to\infty$ as
$\varepsilon\to 0^{+}$. Indeed, a flow starting from a superposition of two
diffused balls truncated so to have compact support should spend a time
$T_{\varepsilon}\to\infty$ as $\varepsilon\to 0^{+}$ close to its “metastable”
initial datum, before eventually converging to a single diffused ball.

Evidence in support of such metastable scenario can be found in a series of
“slow-motion” results concerning the initial value problem

$\left\\{\begin{split}&\partial_{t}v=2\,\varepsilon^{2}\,\Delta
v-W^{\prime}(v)+\varepsilon\,\mu_{\varepsilon}[v(t)]\,\,,\qquad\mbox{on
$\Omega\times(0,\infty)$}\,,\\\ &\nabla_{\nu_{\Omega}}v(t)=0\,,\hskip
105.2751pt\qquad\mbox{on $\partial\Omega$}\,,\\\ &v(0)=u_{0}\,,\qquad\hskip
119.50148pt\mbox{on $\Omega$}\,,\end{split}\right.$ (1.13)

defined on a bounded open set $\Omega\subset\mathbb{R}^{n}$ with regular
boundary, and involving the Lagrange multiplier
$\mu_{\varepsilon}[v]=\int_{\Omega}W^{\prime}(v)/(\varepsilon\,|\Omega|)$,
where the choice777The boundedness of $\Omega$ allows one to work with the
simpler volume potential $V(t)=t$, which also brings some remarkable
simplifications in the form of the volume-preserving Lagrange multiplier.
Indeed, comparing the definition of $\lambda_{\varepsilon}$ with that of
$\mu_{\varepsilon}$ we see how the former choice, crucial in ensuring the
well-posedness of $\Psi(\varepsilon,m)$ as a minimization problem on
$\mathbb{R}^{n}$, leads to the presence of the possibly degenerate denominator
$\varepsilon\,\int_{\mathbb{R}^{n}}V^{\prime}(v)^{2}$ (in place of the
constant denominator $\varepsilon|\Omega|$), and of the additional term
$|\nabla v|^{2}\,V^{\prime\prime}(v)$($\approx|\nabla v|^{2}\,v^{2/(n-1)}$ for
$v$ small) at numerator. of $\mu_{\varepsilon}$ is such that
$\int_{\Omega}v(t)=\int_{\Omega}u_{0}$ for all $t\geq 0$. Problem (1.13) has
indeed been the object of study in several papers, as we are now going to
informally888Our review is informal in the sense that we will gloss over
specific assumptions made in the reviewed papers on the double well potential
$W$ and the initial data $u_{0}$. Moreover, the range of $u$ is often assumed
to be $\mathbb{R}$, while we have chosen to consider functions with range in
$[0,1]$. review. Before doing that, let us stress that in the slow-motion
literature it is customary to work with the parabolic operator
$\partial_{t}v-2\,\varepsilon^{2}\,\Delta v$ in place of the one used in (DF),
namely, $\varepsilon^{2}\,(\partial_{t}u-2\,\Delta u)$. In practice this is a
small difference, since one can pass from one setting to the other by just
rescaling solutions in time, according to the relation
$v(t)=u(\varepsilon^{2}\,t)$. The choice made in (DF) seems more natural,
since the resulting flow is the one that is expected to converge, in the
$\varepsilon\to 0^{+}$-limit, to the VPMCF. It is important to keep this
difference in mind when comparing Theorem 1.1 to results from the slow-motion
literature, which are typically formulated on (1.13).

In dimension $n=1$, when $\Omega$ is an interval, building up on the
pioneering work [BK90, BK91], in [Gra95] (see also [BHNN15]) the following
result is proved: if $u_{0}:\Omega\to\\{0,1\\}$ has $N$-many jumps and
$\\{u_{0}^{\varepsilon}\\}_{\varepsilon}$ is a family of initial data for
(1.13) such that $\|u_{0}^{\varepsilon}-u_{0}\|_{L^{1}(\Omega)}\to 0$ as
$\varepsilon\to 0^{+}$, then solutions
$\\{v^{\varepsilon}(t)\\}_{\varepsilon}$ of (1.13) are such that

$\lim_{\varepsilon\to
0^{+}}\sup_{t<C\,e^{C/\varepsilon}}\,\int_{\Omega}|v^{\varepsilon}(t)-u_{0}|=0\,.$
(1.14)

In terms of $u^{\varepsilon}(t)=v^{\varepsilon}(t/\varepsilon^{2})$, we thus
have
$\sup_{t<C\,\varepsilon^{2}\,e^{C/\varepsilon}}\,\int_{\Omega}|u^{\varepsilon}(t)-u_{0}|=0$
as $\varepsilon\to 0^{+}$, which is a non-trivial information since
$\varepsilon^{2}\,e^{C/\varepsilon}\to\infty$ as $\varepsilon\to 0^{+}$. When
$N\geq 2$ we can interpret this statement as a metastability result, since the
expectation is that $u^{\varepsilon}(t)$ will eventually converge to a single
transition layer.

Concerning dimensions $n\geq 2$, it is shown in [AF94, ABF98] that for initial
data resembling the characteristic function of a sufficiently small ball
contained in $\Omega$ and located near $\partial\Omega$, solutions
$\\{v^{\varepsilon}(t)\\}_{\varepsilon}$ to (1.13) will remain close to the
characteristic function of a ball contained in $\Omega$ on an interval of
times $(0,T_{\varepsilon})$ with $T_{\varepsilon}={\rm O}(e^{C/\varepsilon})$
as $\varepsilon\to 0^{+}$. This is another result supporting the metastability
scenario: indeed, the expected attractors of (1.13) as $t\to\infty$ should be
close, for $\varepsilon$ small, to minimizers of
$\inf\\{P(E;\Omega):E\subset\Omega\,,|E|=m\\}$; however, such minimizers, at
small volume $m$, look like half-balls centered at the point of highest mean
curvature of $\partial\Omega$ (see [Fal10, MM16]), and thus will never be
balls contained in $\Omega$.

Another slow-motion result in higher dimension has been obtained in [MR16,
LM19]. Its interest lies in the fact that it assumes the proximity of the
initial data to generic perimeter minimizing sets (thus, not necessarily to
balls/intervals); the drawback is that proximity of solutions
$v^{\varepsilon}(t)$ is shown to be preserved only on a time interval
$(0,T_{\varepsilon})$ with $T_{\varepsilon}=C/\varepsilon$ so that, in terms
of $u^{\varepsilon}(t)=v^{\varepsilon}(t/\varepsilon^{2})$, proximity
preservation is shown only on a time interval $(0,C\,\varepsilon)$, and no
information survives in the limit $\varepsilon\to 0^{+}$.

###  1.3. Analysis of the diffused VPMCF

We now provide a detailed breakdown of a series of results leading to Theorem
1.1.

The starting point of our analysis is to establish the existence of solutions
of the diffused VPMCF and their basic regularity properties. This is a
nontrivial task due to the presence of the Lagrange multiplier functional
$\lambda_{\varepsilon}$. In addition to being non-local and requiring
integrating on a non-compact set, this term brings two main technical
complications into our analysis: 1) the possible degeneration of the flow
because of the smallness of the $\int_{\mathbb{R}^{n}}V^{\prime}(u)^{2}$-term
at the denominator, and 2) the lack of regularity because of the $|\nabla
u|^{2}\,V^{\prime\prime}(u)$-term at numerator, since $V^{\prime\prime}(u)$ is
only Hölder continuous. These difficulties are addressed by first
approximating $V$ with regularized volume potentials $V_{\delta}$ such that
${\rm Lip}(V_{\delta}^{\prime\prime};[0,1])<\infty$, and by then discussing
the delicate limit as $\delta\to 0^{+}$. Boundedness, Lipschitz and Hölder
continuity estimates on $\lambda_{\varepsilon}$, and on its variant
$\lambda_{\varepsilon,\delta}$ associated with $V_{\delta}$, are discussed in
Section 3, and are obtained by borrowing the geometric viewpoint of
“nucleation lemmas” from the theory of isoperimetric clusters [Alm76]. With
these estimates at hand we can implement a fixed point argument to show the
existence of mild solutions to the modification of (DF) obtained by replacing
$V$ with $V_{\delta}$. We can then start bootstrapping regularity and
monotonicity properties for the $\delta$-approximating flows, up to the point
where enough information is obtained and the $\delta\to 0^{+}$ limit can be
addressed, thus establishing the following theorem (proved in Section 4).

######  Theorem 1.2 (Existence, regularity, and entropies of the flow).

If $\varepsilon>0$, $n\geq 2$, $W\in C^{2,1}[0,1]$ satisfies (1.2) and (1.3),
$u_{0}\in W^{2,p}(\mathbb{R}^{n};[0,1])$ for every $p\geq 2$, and
$\mathcal{V}(u_{0})=1$, then there exists a unique $u\in
C^{0}(\mathbb{R}^{n}\times[0,\infty))\cap C^{2}_{\rm
loc}(\mathbb{R}^{n}\times(0,\infty))$ that is a classical solution of (DF).
Moreover:

(i): for every $p\geq 2$ and $t_{0}>0$ we have

$\displaystyle\sup_{t>0}\big{\\{}\|\partial_{t}u(t)\|_{L^{p}(\mathbb{R}^{n})}\,,\|u(t)\|_{W^{2,p}(\mathbb{R}^{n})}\big{\\}}\leq
C(\varepsilon,u_{0},p)\,,$ (1.15)
$\displaystyle\sup_{t>t_{0}}\big{\\{}\|\partial_{tt}u(t)\|_{L^{p}(\mathbb{R}^{n})}\,,\|\partial_{t}u(t)\|_{W^{2,p}(\mathbb{R}^{n})}\,,\|u(t)\|_{W^{3,p}(\mathbb{R}^{n})}\big{\\}}\leq
C(\varepsilon,u_{0},p,t_{0})\,;$ (1.16)

(ii): $t\mapsto\lambda_{\varepsilon}[u(t)]$ is Lispchitz continuous on
$(t_{0},\infty)$ for every $t_{0}>0$, and is such that

$\sup_{t>0}\,\varepsilon\,|\lambda_{\varepsilon}[u(t)]|\leq
C\,\mathcal{AC}_{\varepsilon}(u_{0})^{2n+2}\,;$

(iii): $\mathcal{V}(u(t))=1$ for every $t\geq 0$;

(iv): $t\mapsto\mathcal{AC}_{\varepsilon}(u(t))$ is continuous and decreasing
on $[0,\infty)$ with

$\mathcal{AC}_{\varepsilon}(u(t_{2}))-\mathcal{AC}_{\varepsilon}(u(t_{1}))=-\varepsilon\,\int_{t_{1}}^{t_{2}}\,dt\int_{\mathbb{R}^{n}}(\partial_{t}u(t))^{2}\,,\qquad\forall
t_{2}\geq t_{1}>0\,;$ (1.17)

(v): $0<u<1$ on $\mathbb{R}^{n}\times(0,\infty)$;

(vi): the function $t\mapsto\int_{\mathbb{R}^{n}}(\partial_{t}u(t))^{2}$
belongs to $W^{1,1}(t_{0},\infty)$ for every $t_{0}>0$, with

$\frac{d}{dt}\,\int_{\mathbb{R}^{n}}(\partial_{t}u(t))^{2}=-\int_{\mathbb{R}^{n}}\Big{\\{}4\,|\nabla(\partial_{t}u)|^{2}+\frac{2}{\varepsilon}\,\Big{(}\frac{W^{\prime\prime}(u)}{\varepsilon}-\lambda_{\varepsilon}[u(t)]\,V^{\prime\prime}(u)\Big{)}\,(\partial_{t}u)^{2}\Big{\\}}(t)\,.$
(1.18)

In particular,

$\lim_{t\to\infty}\int_{\mathbb{R}^{n}}(\partial_{t}u(t))^{2}=0\,.$

It is well-know that the asymptotic behavior of a semilinear parabolic PDE can
be “subsequentially resolved” into a bubbling of mutually drifting stationary
states: a good exemplification of this kind of result, whose fundamental idea
is rooted into Lions’ concentration-compactness principle itself, is found,
for example, in [Fei97, Theorem 1.1]. Another powerful idea found in theory of
semilinear parabolic PDE is that when the initial datum is compactly
supported, then a sort of star-shapedness of the flow (see (5.45)) can be
established by means of the parabolic maximum principle, thus excluding
bubbling phenomena; for a general exemplification of this idea, see [Fei97,
Theorem 1.2]. In Section 5 we adapt these general methods to our specific
problem, which, again, does not follow in the range of application of the
general theory because of the non-autonomy of (DF) and because of the presence
of the non-local Lagrange multiplier $\lambda_{\varepsilon}$. Notice that we
are not considering (yet) the physical regime when $\varepsilon$ is small: in
particular, there is no way to attribute any geometric meaning to the
stationary states $\xi_{i}$ appearing in the statement, e.g. by relating them
to the minimizers $\zeta_{\varepsilon,m}$ of $\Psi(\varepsilon,m)$.

######  Theorem 1.3 (Subsequential bubbling, general $\varepsilon$).

Let $\varepsilon>0$, $n\geq 2$, $W\in C^{2,1}[0,1]$ satisfy (1.2) and (1.3),
$u_{0}\in W^{2,p}(\mathbb{R}^{n};[0,1])$ for every $p\geq 2$,
$\mathcal{V}(u_{0})=1$, and let $\\{u(t)\\}_{t\geq 0}$ be the diffused VPMCF
with $u(0)=u_{0}$.

Then, for every sequence $t_{j}\to\infty$ as $j\to\infty$, up to extracting a
subsequence, there are $M\in\mathbb{N}$, $\ell_{\varepsilon}>0$, sequences
$(x^{i}_{j})_{j}$ ($i=1,...,M$) satisfying $|x^{i}_{j}-x^{k}_{j}|\to\infty$ as
$j\to\infty$ ($i\neq k$), and radial solutions $\\{\xi_{i}\\}_{i=1}^{M}$ of

$2\,\varepsilon^{2}\,\Delta\xi_{i}=W^{\prime}(\xi_{i})-\varepsilon\,\ell_{\varepsilon}\,V^{\prime}(\xi_{i})\quad\mbox{on
$\mathbb{R}^{n}$}\,,$ (1.19)

such that

$\sum_{i=1}^{M}\mathcal{V}(\xi_{i})=1\,,\qquad\sum_{i=1}^{M}\,\mathcal{AC}_{\varepsilon}(\xi_{i})\leq\mathcal{AC}_{\varepsilon}(u_{0})\,,$
(1.20)

and, for all $p>2$,

$\lim_{j\to\infty}\Big{\|}u(t_{j})-\sum_{i=1}^{M}\tau_{x^{j}_{i}}[\xi_{i}]\Big{\|}_{(W^{2,p}\cap
W^{1,2})(\mathbb{R}^{n})}+\big{|}\lambda_{\varepsilon}[u(t_{j})]-\ell_{\varepsilon}\big{|}=0\,.$

Moreover, if $u_{0}$ has compact support in $\mathbb{R}^{n}$, then $M=1$ and
$x_{j}^{1}\to x_{*}$ as $j\to\infty$ for some $x_{*}\in\mathbb{R}^{n}$.

From this point on we work exclusively in the geometric regime when
$\varepsilon<\varepsilon_{0}$ for some small universal constant
$\varepsilon_{0}$ and relate the subsquential bubbling established in Theorem
1.3 to the diffused isoperimetric problem $\Psi(\varepsilon,m)$. A key tool in
achieving this result is the diffused Alexandrov theorem proved in [MR24,
Theorem 1.1-(iv)], whose statement is recalled in Section 2, and which
asserts, roughly speaking, that any solution of
$2\,\varepsilon^{2}\,\Delta\xi=W^{\prime}(\xi)-\varepsilon\,\ell\,V^{\prime}(\xi)$
on $\mathbb{R}^{n}$ with $\xi(x)\to 0$ as $|x|\to\infty$ and
$\varepsilon\,\ell\in(0,\nu_{0})$ for a sufficiently small universal constant
$\nu_{0}$, must satisfy, up to translations, $\xi=\zeta_{\varepsilon,m}$ and
$\ell=\Lambda_{\varepsilon,m}$, for some $m$ such that
$\varepsilon\in(0,\varepsilon_{0}\,m^{1/n})$.

######  Theorem 1.4 (Subsequential bubbling, small $\varepsilon$).

If $n\geq 2$, $W\in C^{2,1}[0,1]$ satisfies (1.2) and (1.3), $u_{0}\in
W^{2,p}(\mathbb{R}^{n};[0,1])$ for all $p\geq 2$, and $\mathcal{V}(u_{0})=1$,
then there is $\varepsilon_{0}^{*}=\varepsilon_{0}^{*}(n,W,u_{0})$ with the
following property.

If $\varepsilon\in(0,\varepsilon_{0}^{*})$ and $\\{u(t)\\}_{t\geq 0}$ is the
corresponding solution of (DF) with $u(0)=u_{0}$, then, for every
$t_{j}\to\infty$ as $j\to\infty$, up to extracting a subsequence in $j$, there
are $M\in\mathbb{N}$ and sequences $\\{(x_{j}^{i})_{j}\\}_{i=1}^{M}$ with
$|x_{j}^{i}-x_{j}^{k}|\to\infty$ as $j\to\infty$ ($i\neq k$), such that

$\lim_{j\to\infty}\Big{\|}u(t_{j})-\sum_{i=1}^{M}\tau_{x_{j}^{i}}\big{[}\zeta_{\varepsilon,1/M}\big{]}\Big{\|}_{(W^{2,p}\cap
W^{1,2})(\mathbb{R}^{n})}+\Big{|}\lambda_{\varepsilon}[u(t_{j})]-\Lambda_{\varepsilon,1/M}\Big{|}=0\,,$
(1.21)

for all $p>2$. Moreover, $M$ is uniquely characterized by the relation

$M\,\Psi\Big{(}\varepsilon,\frac{1}{M}\Big{)}=\lim_{t\to\infty}\mathcal{AC}_{\varepsilon}(u(t))\,.$
(1.22)

In particular, $M$ and the limit Lagrange multiplier
$\Lambda_{\varepsilon,1/M}$ depend only on $u_{0}$ and not on the specific
sequence $(t_{j})_{j}$ under consideration.

######  Remark 1.4.

For the quantification of $\varepsilon_{0}^{*}$ in terms of $u_{0}$, see (6.2)
below.

Under the assumption (1.9) or (1.10), Theorem 1.4 holds with $M=1$. Thus to go
from Theorem 1.4 to Theorem 1.1 we have to upgrade subsequential convergence
to full convergence as $t\to\infty$. The natural approach to this problem
consists in proving the differential inequality

$-\frac{d}{dt}\,\mathcal{I}_{\varepsilon}(t)\geq\frac{\mathcal{I}_{\varepsilon}(t)}{C(\varepsilon)}\,,\qquad\forall
t>1/C(\varepsilon,u_{0})\,,$

for the Fisher information/dissipation
$\mathcal{I}_{\varepsilon}(t)=\varepsilon\,\int_{\mathbb{R}^{n}}(\partial_{t}u(t))^{2}$
of the flow (compare with Theorem 1.2-(iv,vi)). This is the well-known Bakry-
Émery method, which requires establishing the strict stability of the
$\mathcal{V}$-constrained second variation

$\mathcal{Q}_{\varepsilon}[\zeta_{\varepsilon}](\varphi)=\int_{\mathbb{R}^{n}}2\,\varepsilon\,|\nabla\varphi|^{2}+\Big{\\{}\frac{W^{\prime\prime}(\zeta_{\varepsilon})}{\varepsilon}-\Lambda_{\varepsilon}\,V^{\prime\prime}(\zeta_{\varepsilon})\Big{\\}}\,\varphi^{2}\,,\qquad\varphi\in
W^{1,2}(\mathbb{R}^{n})\,,$ (1.23)

of the Allen–Cahn functional at the minimizer
$\zeta_{\varepsilon}=\zeta_{\varepsilon,1}$ of $\Psi(\varepsilon,1)$ (where we
are also setting $\Lambda_{\varepsilon}=\Lambda_{\varepsilon,1}$ for brevity).
This strict stability result is established in the following theorem, proved
in Section 7 (whereas the proof of Theorem 1.4 is finally discussed in Section
8).

######  Theorem 1.5 (Strict stability of $\mathcal{Q}_{\varepsilon}$ at
$\zeta_{\varepsilon}$).

If $n\geq 2$ and $W\in C^{2,1}[0,1]$ satisfies (1.2) and (1.3), then there
exists a positive universal constant $\varepsilon_{0}$ with the following
property.

For every $\varepsilon\in(0,\varepsilon_{0})$, if $\varphi\in
W^{1,2}(\mathbb{R}^{n})$ satisfies

$\displaystyle\int_{\mathbb{R}^{n}}\varphi\,V^{\prime}(\zeta_{\varepsilon})=0\,,\qquad\int_{\mathbb{R}^{n}}\varphi\,\nabla\zeta_{\varepsilon}=0\,,$
(1.24)

then

$\mathcal{Q}_{\varepsilon}[\zeta_{\varepsilon}](\varphi)\geq\frac{\varepsilon}{C}\,\int_{\mathbb{R}^{n}}\varphi^{2}\,;$
(1.25)

and, moreover, for a constant $C(\varepsilon)>0$,

$\mathcal{Q}_{\varepsilon}[\zeta_{\varepsilon}](\varphi)\geq\frac{1}{C(\varepsilon)}\,\int_{\mathbb{R}^{n}}|\nabla\varphi|^{2}+\varphi^{2}\,.$
(1.26)

It had already been proved in [MR24, Lemma 4.4] that if $\varphi$ is radial
(and satisfies (1.24)), then

$C\,\mathcal{Q}_{\varepsilon}[\zeta_{\varepsilon}](\varphi)\geq\int_{\mathbb{R}^{n}}\varepsilon\,|\nabla\varphi|^{2}+\frac{\varphi^{2}}{\varepsilon}\,,$
(1.27)

thus showing that the stability of
$\mathcal{Q}_{\varepsilon}[\zeta_{\varepsilon}]$ among radial variations is
much stronger than among generic variations (the costant $\varepsilon/C$ in
(1.25) is sharp as seen in the proof of the theorem itself). The proof of
Theorem 1.5 combines the decomposition of $\varphi$ as a Fourier series in
angular/radial variables with a geometric change of variables that allows to
relate (on special modes of such decomposition)
$\mathcal{Q}_{\varepsilon}[\zeta_{\varepsilon}]$ with the second variation of
the perimeter functional at a ball.

###  1.4. Organization of the paper

Section 2 contains a recap of our basic notation, main results from [MR24],
and useful properties of the various potentials involved in our analysis. The
other sections of the paper are organized as described in the previous
section.

Acknowledgements: FM has been supported by NSF Grant DMS-2247544. FM and DR
have been supported by NSF Grant DMS-2000034 and NSF FRG Grant DMS-1854344.

##  2\. Background material on the diffused isoperimetric problem

In this section we collect some background material that will be repeatedly
used in the sequel. In Section 2.1 we set some basic conventions and notation,
while in Section 2.2 we recall some key results contained in [MR24] and
concerning the diffused isoperimetric problem $\Psi(\varepsilon,m)$. Finally,
in Section 2.3 we collect some elementary inequalities concerning the various
potentials $W$, $\Phi$, and $V$ that will be often referred to in later
proofs, and in Section 2.4 we introduced some regularized volume potentials
$V_{\delta}$ which will be employed as a technical device in the proof of
Theorem 1.2.

###  2.1. Basic conventions

Throughout the paper $n\in\mathbb{N}$, $n\geq 2$, denotes the dimension of the
Euclidean space we are working in, and $W\in C^{2,1}[0,1]$ is a “double-well
potential” satisfying (1.2) and (1.3). By universal constant we mean a
positive real number depending on $W$ and the value of $n$ under
consideration. We denote by $C$ a generic universal constant whose value may
increase at each of its subsequent appearances. Universal constants that may
also depend on $\varepsilon>0$ are denoted by $C(\varepsilon)$, with the idea
that $C(\varepsilon)$ may diverge to $+\infty$ as $\varepsilon\to 0^{+}$.
Given $k\in\mathbb{N}$, we will write “$f(\varepsilon)={\rm
O}(\varepsilon^{k})$ as $\varepsilon\to 0^{+}$” if there exists a universal
constant $C$ such that $|f(\varepsilon)|\leq C\,\varepsilon^{k}$ for every
$\varepsilon\in(0,1/C)$; similar definitions are given for “${\rm O}(t)$ as
$t\to\infty$”, etc.

###  2.2. The diffused isoperimetric problem

Let us recall that for every $\varepsilon,m>0$ we set

$\Psi(\varepsilon,m)=\inf\Big{\\{}\mathcal{AC}_{\varepsilon}(u):\mathcal{V}(u)=m\,,u\in
W^{1,2}(\mathbb{R}^{n};[0,1])\Big{\\}}\,.$

Among the basic properties of $\Psi(\varepsilon,m)$ we have the scaling law

$\Psi(\varepsilon,m)=m^{(n-1)/n}\,\Psi\Big{(}\frac{\varepsilon}{m^{1/n}},1\Big{)}\,,\qquad\forall\varepsilon,m>0\,,$
(2.1)

and the isoperimetric lower bound and $\varepsilon\to 0^{+}$ limit

$\displaystyle\Psi(\varepsilon,m)\\!\\!$ $\displaystyle>$
$\displaystyle\\!\\!2\,c_{\rm
iso}(n)\,m^{(n-1)/n}\,,\qquad\forall\varepsilon,m>0\,,$ (2.2)
$\displaystyle\lim_{\varepsilon\to 0^{+}}\Psi(\varepsilon,m)\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!2\,c_{\rm
iso}(n)\,m^{(n-1)/n}\,,\qquad\forall m>0\,,$ (2.3)

where we have set $c_{\rm iso}(n)=n\,\omega_{n}^{1/n}$. A simple concentration
compactness argument (see [MNR23, Proof of Theorem A.1, steps one and two])
shows that $\Psi(\varepsilon,m)$ admits radially symmetric decreasing
minimizers for each $\varepsilon$ and $m$. Stronger properties are proved in
[MR24, Theorem 1.1, Theorem 6.1] under the geometric regime
$\varepsilon<\\!\\!<m^{1/n}$:

Theorem ${\bf\Psi}$ [MR24] If $n\geq 2$ and $W\in C^{2,1}[0,1]$ satisfies
(1.2) and (1.3), then there are positive universal constants $\varepsilon_{0}$
and $\nu_{0}$ with the following properties:

${\bf(i):}$ we have

$\displaystyle\mbox{$\Psi(\cdot,m)$ is strictly increasing on
$(0,\varepsilon_{0}\,m^{1/n})$}\,,$ (2.4)
$\displaystyle\mbox{$\Psi(\varepsilon,\cdot)$ is concave on $(0,\infty)$ and
strictly concave on $((\varepsilon/\varepsilon_{0})^{n},\infty)$}\,;$ (2.5)

moreover, if $0<\varepsilon<\varepsilon_{0}\,m^{1/n}$, then there is a
radially symmetric strictly decreasing minimizer $\zeta_{\varepsilon,m}$ of
$\Psi(\varepsilon,m)$ with maximum at the origin having the property that $u$
is a minimizer of $\Psi(\varepsilon,m)$ if and only if
$u=\tau_{x_{0}}[\zeta_{\varepsilon,m}]$ for some $x_{0}\in\mathbb{R}^{n}$;
and, for some $\Lambda_{\varepsilon,m}>0$, $\zeta_{\varepsilon,m}$ satisfies

$2\,\varepsilon^{2}\,\Delta\zeta_{\varepsilon,m}=W^{\prime}(\zeta_{\varepsilon,m})-\varepsilon\,\Lambda_{\varepsilon,m}\,V^{\prime}(\zeta_{\varepsilon,m})\qquad\mbox{on
$\mathbb{R}^{n}$}\,,$

where

$\lim_{\varepsilon\to
0^{+}}\,m^{1/n}\,\Lambda_{\varepsilon,m}=2\,(n-1)\,\omega_{n}^{1/n}\,.$ (2.6)

(ii): if $v\in C^{2}(\mathbb{R}^{n};[0,1])$ satisfies $v(x)\to 0^{+}$ as
$|x|\to\infty$ and solves

$2\,\varepsilon^{2}\,\Delta
v=W^{\prime}(v)-\varepsilon\,\lambda\,V^{\prime}(v)\qquad\mbox{on
$\mathbb{R}^{n}$}\,,$

for some positive $\lambda$ such that $\varepsilon\,\lambda<\nu_{0}$, then
there exist $x_{0}\in\mathbb{R}^{n}$ and $m>0$ such that

$v=\tau_{x_{0}}[\zeta_{\varepsilon,m}]\,,\qquad\varepsilon<\varepsilon_{0}\,m^{1/n}\,,\qquad\lambda=\Lambda_{\varepsilon,m}\,.$

###  2.3. Technical properties of $W$ and related potentials

In the linearization of (DF) we shall tacitly use the fact that, being $W\in
C^{2,1}[0,1]$, we have

$\Big{|}W(r)-W(s)-W^{\prime}(s)(r-s)-W^{\prime\prime}(s)\frac{(r-s)^{2}}{2}\Big{|}\leq
C\,|r-s|^{3}\,,\qquad\forall r,s\in[0,1]\,.$

We also observe that $1/C\leq W^{\prime\prime}\leq C$ on $[0,1]$, and that
there is a universal constant $\delta_{0}<1/2$ such that

$\begin{split}&\frac{1}{C}\leq\frac{W(r)}{r^{2}}\,,\frac{W^{\prime}(r)}{r}\leq
C\quad\hskip 42.67912pt\forall r\in(0,\delta_{0}]\,,\\\
&\frac{1}{C}\leq\frac{W(r)}{(1-r)^{2}}\,,\frac{-W^{\prime}(r)}{1-r}\leq
C\quad\hskip 11.38092pt\forall r\in[1-\delta_{0},1)\,.\end{split}$ (2.7)

Recalling that $\Phi(r)=\int_{0}^{r}\sqrt{W}$ and $V(r)=\Phi(r)^{n/(n-1)}$ for
$r\in[0,1]$, we use (2.7) to quantify the behaviors of $\Phi$ and $V$ near
$r=0$ and $r=1$. By (1.2), $\Phi\in C^{3}_{{\rm loc}}(0,1)$, with

$\Phi^{\prime}=\sqrt{W}\,,\quad\Phi^{\prime\prime}=\frac{W^{\prime}}{2\,\sqrt{W}}\,,\quad\Phi^{\prime\prime\prime}=\frac{W^{\prime\prime}}{2\,\sqrt{W}}-\frac{(W^{\prime})^{2}}{4\,W^{3/2}}\,,\qquad\mbox{on
$(0,1)$}\,.$

By (2.7) and (1.3) we thus see that $\Phi$ satisfies

$\begin{split}&\frac{1}{C}\leq\frac{\Phi(r)}{r^{2}}\,,\frac{\Phi^{\prime}(r)}{r}\,,\Phi^{\prime\prime}(r)\leq
C\,,\qquad\forall r\in(0,\delta_{0}]\,,\\\
&\frac{1}{C}\leq\frac{1-\Phi(r)}{(1-r)^{2}}\,,\frac{\Phi^{\prime}(r)}{1-r}\,,-\Phi^{\prime\prime}(r)\leq
C\,,\qquad\forall r\in[1-\delta_{0},1)\,.\end{split}$ (2.8)

By exploiting (2.8) and setting for brevity $a=W^{\prime\prime}(0)$, we see
that, as $r\to 0^{+}$,

$\displaystyle\Phi^{\prime\prime\prime}$ $\displaystyle=$
$\displaystyle\frac{2\,W^{\prime\prime}\,W-(W^{\prime})^{2}}{4\,W^{3/2}}=\frac{2\,(a+{\rm
O}(r))\,(a\,(r^{2}/2)+{\rm O}(r^{3}))-(a\,r+{\rm
O}(r^{2}))^{2}}{4\,(a\,(r^{2}/2)+{\rm O}(r^{3}))^{3/2}}$ $\displaystyle=$
$\displaystyle\frac{{\rm O}(r^{3})}{4\,a^{3/2}\,r^{3}+{\rm o}(r^{3})}\,,$

and by a similar computation for $r\to 1^{-}$, we find

$|\Phi^{\prime\prime\prime}|\leq C\quad\mbox{on
$(0,\delta_{0})\cup(1-\delta_{0},1)$}\,.$ (2.9)

By (2.8) and (2.9) we see that $\Phi\in C^{2,1}[0,1]$ with a universal
estimate on its $C^{2,1}[0,1]$-norm: in particular,

$\Big{|}\Phi(r)-\Phi(s)-\Phi^{\prime}(s)(r-s)-\Phi^{\prime\prime}(s)\frac{(r-s)^{2}}{2}\Big{|}\leq
C\,|r-s|^{3}\,,\qquad\forall r,s\in(0,1)\,.$ (2.10)

Since $V=\Phi^{1+\alpha}$ for $\alpha=1/(n-1)\in(0,1]$ (recall that $n\geq 2$)
and $\Phi(r)=0$ if and only if $r=0$, we easily see that $V\in C^{3}_{{\rm
loc}}(0,1)$, with

$\displaystyle V^{\prime}=(1+\alpha)\,\Phi^{\alpha}\,\Phi^{\prime}\,,\qquad
V^{\prime\prime}=(1+\alpha)\Big{\\{}\alpha\,\frac{(\Phi^{\prime})^{2}}{\Phi^{1-\alpha}}+\Phi^{\alpha}\,\Phi^{\prime\prime}\Big{\\}}\,,$
$\displaystyle|V^{\prime\prime\prime}|\leq
C(\alpha)\,\Big{\\{}\frac{(\Phi^{\prime})^{3}}{\Phi^{2-\alpha}}+\frac{\Phi^{\prime}\,|\Phi^{\prime\prime}|}{\Phi^{1-\alpha}}+\Phi^{\alpha}\,|\Phi^{\prime\prime\prime}|\Big{\\}}\,.$

By (2.10), and keeping track of the sign of $\Phi^{\prime\prime}$ and of the
fact that negative powers of $\Phi(r)$ are large only near $r=0$, but are
bounded near $r=1$, we find that

$\begin{split}&\frac{1}{C}\leq\frac{V(r)}{r^{2+2\alpha}}\,,\frac{V^{\prime}(r)}{r^{1+2\alpha}}\,,\frac{V^{\prime\prime}(r)}{r^{2\alpha}}\leq
C\,,\hskip
19.91684pt\quad|V^{\prime\prime\prime}(r)|\leq\frac{C}{r^{1-2\alpha}}\qquad\forall
r\in(0,\delta_{0}]\,,\\\
&\frac{1}{C}\leq\frac{1-V(r)}{(1-r)^{2}}\,,\frac{V^{\prime}(r)}{1-r}\leq
C\,,\quad|V^{\prime\prime}(r)|\,,|V^{\prime\prime\prime}(r)|\leq
C\,,\qquad\forall r\in[1-\delta_{0},1)\,.\end{split}$ (2.11)

In particular, $V^{\prime\prime}(r)\to\infty$ explodes as $r\to 0^{+}$.
However, $V\in C^{2,\gamma(n)}[0,1]$ (with
$\gamma(n)=\min\\{1,2/(n-1)\\}\in(0,1]$) and we have the second order Taylor
expansion

$\Big{|}V(r)-V(s)-V^{\prime}(s)\,(r-s)-V^{\prime\prime}(s)\frac{(r-s)^{2}}{2}\Big{|}\leq
C\,|r-s|^{2+\gamma(n)}\,,\qquad\forall r,s\in(0,1)\,.$

As much as the analogous expansion for $W$, this formula will be repeatedly
used in linearizing (DF).

###  2.4. Regularized volume potentials

As already discussed at the beginning of Section 1.3, the non-Lipschitzianity
of $V^{\prime\prime}$ near $0$ causes several technical problems, that call
for the introduction of regularized volume potentials
$V_{\delta}:[0,1]\to[0,1]$ ($\delta>0$) such that $V_{\delta}\in
C^{2,1}[0,1]$, with

$V_{\delta}(0)=0\,,\qquad
V_{\delta}^{\prime}(0)=V_{\delta}^{\prime}(1)=0\,,\qquad\lim_{\delta\to
0^{+}}\|V_{\delta}-V\|_{C^{2}[0,1]}=0\,.$

We will of course have

$\lim_{\delta\to 0^{+}}{\rm
Lip}(V_{\delta}^{\prime\prime};[0,1])=+\infty\,,\qquad\sup_{\delta>0}\,[V_{\delta}^{\prime\prime}]_{C^{0,\gamma(n)}}<\infty\,.$

Such potentials $V_{\delta}$ can be defined by first considering a family
$\\{\rho_{\delta}\\}_{\delta>0}$ of smooth mollifiers on $\mathbb{R}$ such
that ${\rm spt}\rho_{\delta}\subset\subset(-\delta^{2},\delta^{2})$, and then
by setting

$V_{\delta}=\rho_{\delta}\star\Big{(}L_{\delta}\circ\Phi\Big{)}^{n/(n-1)}\,,$
(2.12)

where

$L_{\delta}(r)=\left\\{\begin{split}&0\,,\qquad\hskip
28.45274ptr\in[0,\delta]\,,\\\ &\frac{r-\delta}{1-2\,\delta}\,,\qquad
r\in[\delta,1-\delta]\,,\\\ &1\,,\qquad\hskip
28.45274ptr\in[1-\delta,1]\,.\end{split}\right.$

By $W>0$ on $(0,1)$, (2.7) and (2.11), and up to further decreasing the value
of $\delta_{0}$ introduced above, we have

$V_{\delta}(r)\leq C\,r^{2}\leq C\,W(r)\,,\qquad V_{\delta}^{\prime}(r)\leq
C\,r\,,\qquad\forall r\in(0,1-\delta_{0})\,,$

for every $\delta\in\big{[}0,\delta_{0}]$, as well as (compare with (2.11))

$V_{\delta}(r)\geq\frac{1}{C}\,,\qquad V_{\delta}^{\prime}(r)\leq
C\,(1-r)\,\qquad\forall r\in(\delta_{0},1)\,.$ (2.13)

##  3\. Estimates for the Lagrange multiplier functional

This section is devoted to the analysis of the Lagrange multiplier functional
$\lambda_{\varepsilon}$, defined with values in $[0,\infty)$ on any given
function $v\in W^{1,2}(\mathbb{R}^{n};[0,1])$ with $|\\{0<v<1\\}|>0$ (and
assumption that guarantees $\int_{\mathbb{R}^{n}}V^{\prime}(u)^{2}>0$) by
setting

$\lambda_{\varepsilon}[v]=\frac{\int_{\mathbb{R}^{n}}2\,\varepsilon^{2}\,|\nabla
v|^{2}\,V^{\prime\prime}(v)+W^{\prime}(v)\,V^{\prime}(v)}{\varepsilon\,\int_{\mathbb{R}^{n}}V^{\prime}(v)^{2}}\,.$
(3.1)

In particular, we address the Lipschitz continuity properties of
$\lambda_{\varepsilon}$ in the Banach space $(X,\|\cdot\|)$ defined by

$X=(C^{0}\cap
W^{1,2})(\mathbb{R}^{n};[0,1])\,,\qquad\|u\|_{X}=\|u\|_{W^{1,2}(\mathbb{R}^{n})}+\|u\|_{C^{0}(\mathbb{R}^{n})}\,,$

that is the space we shall use to construct mild solutions of (DF). In fact,
we shall also need to consider the Lagrange multiplier functionals

$\lambda_{\varepsilon,\delta}[u]=\frac{\int_{\mathbb{R}^{n}}2\,\varepsilon^{2}\,|\nabla
u|^{2}\,V_{\delta}^{\prime\prime}(u)+W^{\prime}(u)\,V_{\delta}^{\prime}(u)}{\varepsilon\,\int_{\mathbb{R}^{n}}V_{\delta}^{\prime}(u)^{2}}\,,$
(3.2)

obtained by replacing $V$ with the regularized volume potentials $V_{\delta}$
introduced in the previous section. We shall also set $V_{0}=V$,
$\mathcal{V}_{0}=\mathcal{V}$, and use the notation

$\mathcal{AC}_{\varepsilon}(u;\Omega)=\int_{\Omega}\varepsilon\,|\nabla
u|^{2}+\frac{W(u)}{\varepsilon}\,,\qquad\mathcal{V}_{\delta}(u;\Omega)=\int_{\Omega}V_{\delta}(u)\,,$

for the localization to a Borel set $\Omega\subset\mathbb{R}^{n}$ of the
functionals $\mathcal{AC}_{\varepsilon}$ and $\mathcal{V}_{\delta}$.

######  Theorem 3.1.

If $n\geq 2$ and $W\in C^{2,1}[0,1]$ satisfies (1.2) and (1.3), then there
exist positive universal constants $\varepsilon_{0}$ and $\delta_{0}$ with the
following properties:

(i): if $u\in W^{1,2}(\mathbb{R}^{n};[0,1])$, $\varepsilon>0$, and
$\delta\in[0,\delta_{0}]$, then

$\int_{\mathbb{R}^{n}}u^{2}\leq
C\,\big{\\{}\varepsilon\,\mathcal{AC}_{\varepsilon}(u)+\mathcal{V}_{\delta}(u)\big{\\}}\,,$
(3.3)

and, with an $\varepsilon$-dependent universal constant $C(\varepsilon)$,

$\displaystyle C(\varepsilon)\,\int_{\mathbb{R}^{n}}|\nabla
u|^{2}\,\int_{\mathbb{R}^{n}}V_{\delta}^{\prime}(u)^{2}$ $\displaystyle\geq$
$\displaystyle\min\Big{\\{}1,\frac{\mathcal{V}_{\delta}(u)}{\mathcal{AC}_{\varepsilon}(u)}\Big{\\}}^{2\,n}\,,$
(3.4) $\displaystyle|\lambda_{\varepsilon,\delta}[u]|$ $\displaystyle\leq$
$\displaystyle
C(\varepsilon)\,\frac{\mathcal{AC}_{\varepsilon}(u)^{2n+1}}{\mathcal{V}_{\delta}(u)^{2n}}\,\int_{\mathbb{R}^{n}}|\nabla
u|^{2}\,.$ (3.5)

Furthermore, if $\delta=0$ and $\varepsilon\in(0,\varepsilon_{0})$, then (3.4)
and (3.5) hold with $C$ in place of $C(\varepsilon)$.

(ii): if $u,v\in X\setminus\\{0\\}$, $\varepsilon>0$, and
$\max\\{\|u\|_{X}\,,\|v\|_{X}\\}\leq C(\varepsilon)$, then

$\displaystyle|\lambda_{\varepsilon,\delta}[u]-\lambda_{\varepsilon,\delta}[v]|\\!\\!$
$\displaystyle\leq$
$\displaystyle\\!\\!C(\varepsilon,\delta)\,\frac{\|u-v\|_{X}}{\min\\{1,\mathcal{V}_{\delta}(u)^{2n}\\}\,\min\\{1,\mathcal{V}_{\delta}(v)^{2n}\\}}\,,\quad\forall\delta\in(0,\delta_{0}]\,,\,\,\,$
(3.6)
$\displaystyle|\lambda_{\varepsilon,\delta}[u]-\lambda_{\varepsilon,\delta}[v]|\\!\\!$
$\displaystyle\leq$
$\displaystyle\\!\\!C(\varepsilon)\,\frac{\|u-v\|_{X}^{\gamma(n)}}{\min\\{1,\mathcal{V}_{\delta}(u)^{2n}\\}\,\min\\{1,\mathcal{V}_{\delta}(v)^{2n}\\}}\,,\qquad\forall\delta\in[0,\delta_{0}]\,,$
(3.7)
$\displaystyle|\lambda_{\varepsilon,\delta}[u]-\lambda_{\varepsilon}[u]|\\!\\!$
$\displaystyle\leq$
$\displaystyle\\!\\!C(\varepsilon)\,\frac{\|V_{\delta}-V\|_{C^{2}[0,1]}}{\min\\{1,\mathcal{V}_{\delta}(u)^{2n}\\}\,\min\\{1,\mathcal{V}(u)^{2n}\\}}\,,\qquad\forall\delta\in[0,\delta_{0}]\,,$
(3.8)

where $\gamma(n)=\min\\{1,2/(n-1)\\}$; and if $u,v\in
W^{2,2}(\mathbb{R}^{n};[0,1])\setminus\\{0\\}$, then

$|\lambda_{\varepsilon,\delta}[u]-\lambda_{\varepsilon,\delta}[v]|\leq
C(\varepsilon)\frac{\max\\{\|u\|_{W^{2,2}},\|v\|_{W^{2,2}}\\}\,\|u-v\|_{W^{1,2}(\mathbb{R}^{n})}}{\min\\{1,\mathcal{V}_{\delta}(u)^{2n}\\}\,\min\\{1,\mathcal{V}_{\delta}(v)^{2n}\\}}\,,\qquad\forall\delta\in[0,\delta_{0}]\,.$
(3.9)

###### Proof.

Step one, diffused relative isoperimetry and nucleation: We prove two relative
isoperimetric inequalities in balls in the diffused setting and a consequent
nucleation type lemma modeled after [MPS22, Lemma 2.1]. This kind of result is
in turn inspired by a tool introduced by Almgren [Alm76] in the study of
isoperimetric clusters, see [Mag12, Lemma 29.10]. More precisely, we prove the
existence of universal constants $\eta_{0}$, $\sigma_{0}$ and $C$ with the
following properties:

(a): if $\delta\in[0,\delta_{0}]$, $\varepsilon,r>0$, and if $u\in
W^{1,2}(\mathbb{R}^{n};[0,1])$ satisfies

$\fint_{B_{r}}V_{\delta}(u)\leq\eta_{0}\,\Big{(}\frac{\varepsilon}{r}\Big{)}^{2\,n},$
(3.10)

then

$C\,\mathcal{AC}_{\varepsilon}(u;B_{r})\geq\frac{\varepsilon}{r}\,\mathcal{V}_{\delta}(u;B_{r})^{(n-1)/n}\,;$
(3.11)

(b): if $\varepsilon,r>0$ are such that $\varepsilon/r<\sigma_{0}$, and if
$u\in W^{1,2}(\mathbb{R}^{n};[0,1])$ satisfies

$\fint_{B_{r}}V(u)\leq\frac{1}{2}\,,$ (3.12)

then

$C\,\mathcal{AC}_{\varepsilon}(u;B_{r})\geq\mathcal{V}(u;B_{r})^{(n-1)/n}\,.$
(3.13)

(c): if $\varepsilon>0$ and

$\mathcal{B}=\Big{\\{}B_{\sqrt{n+1}\,R/2}(R\,z):z\in\mathbb{Z}^{n}\Big{\\}}\,,\qquad
R=\frac{\max\\{1,\varepsilon\\}}{\sigma_{0}\,(\sqrt{n+1}/2)}\,,$

then for every $u\in W^{1,2}(\mathbb{R}^{n};[0,1])$ we have

$\displaystyle
C\,\frac{\max\\{1,\varepsilon\\}}{\min\\{1,\varepsilon^{2\,n}\\}}\,\sup_{B\in\,\mathcal{B}}\fint_{B}V_{\delta}(u)$
$\displaystyle\geq$
$\displaystyle\min\Big{\\{}1,\Big{(}\frac{\mathcal{V}_{\delta}(u)}{\mathcal{AC}_{\varepsilon}(u)}\Big{)}^{n}\Big{\\}}\,,\qquad\forall\delta\in(0,\delta_{0}]\,,$
(3.14) $\displaystyle
C\,\max\\{1,\varepsilon\\}\,\sup_{B\in\,\mathcal{B}}\fint_{B}V(u)$
$\displaystyle\geq$
$\displaystyle\min\Big{\\{}1,\Big{(}\frac{\mathcal{V}(u)}{\mathcal{AC}_{\varepsilon}(u)}\Big{)}^{n}\Big{\\}}\,.$
(3.15)

We first derive conclusion (c) from conclusions (a) and (b): Let $u\in
W^{1,2}(\mathbb{R}^{n};[0,1])$. In proving (3.15) we can assume without loss
of generality that

$\sup_{B\in\mathcal{B}}\fint_{B}V(u)\leq\frac{1}{2}\,.$

In particular, since the choice of $R$ is such that
$\varepsilon/(\sqrt{n+1}R/2)<\sigma_{0}$ for every $\varepsilon>0$, we deduce
from conclusion (b) that (3.13) holds for every $B\in\mathcal{B}$. The
corresponding bounds can be used together with the fact that $\mathcal{B}$ is
a covering of $\mathbb{R}^{n}$ with finite overlapping (depending only on the
dimension $n$) to conclude that

$\displaystyle C(n)\,\mathcal{AC}_{\varepsilon}(u)$ $\displaystyle\geq$
$\displaystyle\sum_{B\in\mathcal{B}}\mathcal{AC}_{\varepsilon}(u;B)\geq\frac{1}{C}\,\sum_{B\in\mathcal{B}}\mathcal{V}(u;B)^{(n-1)/n}$
$\displaystyle\geq$
$\displaystyle\frac{1}{C}\,\frac{\sum_{B\in\mathcal{B}}\mathcal{V}(u;B)}{\sup_{B\in\mathcal{B}}\mathcal{V}(u;B)^{1/n}}\geq\frac{1}{C}\,\frac{\mathcal{V}(u)}{\sup_{B\in\mathcal{B}}\Big{(}\int_{B}V(u)\Big{)}^{1/n}}\,.$

Thanks to $|B|^{1/n}\leq C\,\max\\{1,\varepsilon\\}$ for every
$B\in\mathcal{B}$, (3) implies (3.15). To prove (3.14), let
$\delta\in(0,\delta_{0}]$, and let us assume without loss of generality that

$\sup_{B\in\mathcal{B}}\fint_{B}V_{\delta}(u)\leq\eta_{0}\,\Big{(}\frac{\varepsilon}{\sqrt{n+1}R/2}\Big{)}^{2\,n}\,,$
(3.17)

then we can apply (3.11) to each $B\in\mathcal{B}$, and conclude as in (3)
that

$C(n)\,\mathcal{AC}_{\varepsilon}(u)\geq\frac{\varepsilon}{C\,R}\,\frac{\mathcal{V}_{\delta}(u)}{\sup_{B\in\mathcal{B}}\mathcal{V}_{\delta}(u;B)^{1/n}}\geq\frac{\min\\{1,\varepsilon\\}}{C}\,\frac{\mathcal{V}_{\delta}(u)}{\sup_{B\in\mathcal{B}}\mathcal{V}_{\delta}(u;B)^{1/n}}\,.$

In particular, (3.14) follows by taking again into account that
$\varepsilon/R=C\,\max\\{1,\varepsilon\\}$.

We now prove conclusions (a) and (b): Up to a rescaling, we can take $r=1$ in
both conclusions. To prove conclusion (a), we notice that, by (3.10) and
(2.13),

$\eta_{0}\,\varepsilon^{2\,n}\geq\frac{1}{\omega_{n}}\,\int_{B_{1}\cap\\{u\geq
1/2\\}}V_{\delta}(u)\geq\frac{|B_{1}\cap\\{u\geq 1/2\\}|}{C}\,.$ (3.18)

Since $W(r)\geq r^{2}/C$ for $r\in[0,1/2]$, by (2.13), (3.18), and the Hölder
inequality, we get

$\displaystyle\int_{B_{1}}u^{2}$ $\displaystyle=$
$\displaystyle\int_{B_{1}\cap\\{u\leq 1/2\\}}u^{2}+\int_{B_{1}\cap\\{u\geq
1/2\\}}u^{2}$ (3.19) $\displaystyle\leq$ $\displaystyle
C\int_{B_{1}}W(u)+C\,\big{|}B_{1}\cap\\{u\geq
1/2\\}\big{|}^{1/n}\,\Big{(}\int_{B_{1}}u^{2n/(n-1)}\Big{)}^{(n-1)/n}$
$\displaystyle\leq$ $\displaystyle
C\int_{B_{1}}W(u)+C\,\eta_{0}^{1/n}\,\varepsilon^{2}\,\Big{(}\int_{B_{1}}u^{2n/(n-1)}\Big{)}^{(n-1)/n}\,.$

By combining (3.19) with the embedding of $L^{2n/(n-1)}(B_{1})$ into
$W^{1,2}(B_{1})$, we find that

$\displaystyle\int_{B_{1}}\varepsilon\,|\nabla
u|^{2}+\frac{W(u)}{\varepsilon}$ $\displaystyle\geq$
$\displaystyle\int_{B_{1}}\varepsilon\,|\nabla
u|^{2}+\frac{u^{2}}{C\varepsilon}-C\,\eta_{0}^{1/n}\,\varepsilon\,\Big{(}\int_{B_{1}}u^{2n/(n-1)}\Big{)}^{(n-1)/n}$
$\displaystyle\geq$
$\displaystyle\frac{\varepsilon}{C}\,\Big{(}\int_{B_{1}}u^{2n/(n-1)}\Big{)}^{(n-1)/n}-C\,\eta_{0}^{1/n}\,\varepsilon\,\Big{(}\int_{B_{1}}u^{2n/(n-1)}\Big{)}^{(n-1)/n}$
$\displaystyle\geq$
$\displaystyle\frac{\varepsilon}{C}\,\Big{(}\int_{B_{1}}u^{2n/(n-1)}\Big{)}^{(n-1)/n}\geq\frac{\varepsilon}{C}\,\Big{(}\int_{B_{1}}V_{\delta}(u)\Big{)}^{(n-1)/n}\,,$

provided $\eta_{0}$ is a sufficiently small universal constant, and where we
have used that $V_{\delta}(r)\leq C\,r^{2n/(n-1)}$ for every $r\in[0,1]$ and
$\delta\in[0,\delta_{0}]$. Having proved conclusion (a), we now prove
conclusion (b). Arguing by contradiction, we can assume the existence of
$\varepsilon_{k}\to 0^{+}$ and $\\{u_{k}\\}_{k}$ in $W^{1,2}(B_{1};[0,1])$
such that, for all $k\in\mathbb{N}$ and setting for brevity

$M_{k}=\int_{B_{1}}V(u_{k})=\int_{B_{1}}\Phi(u_{k})^{p}\,,\qquad
p=\frac{n}{n-1}\,,$

we have $M_{k}\leq\omega_{n}/2$ and

$\displaystyle\int_{B_{1}}\varepsilon_{k}|\nabla
u_{k}|^{2}+\frac{W(u_{k})}{\varepsilon_{k}}\leq\frac{M_{k}^{1/p}}{k}\,,$
(3.20)

for every $k\in\mathbb{N}$. Combining $|\nabla(\Phi\circ u)|=|\nabla
u|\,\sqrt{W(u)}$ with Young’s inequality as in (1.5), we deduce from (3.20)
and the $BV$-Poincaré inequality [AFP00, (3.41)] that

$\frac{1}{C}\Big{(}\int_{B_{1}}|\Phi(u_{k})-t_{k}|^{p}\Big{)}^{1/p}\leq\int_{B_{1}}|\nabla(\Phi\circ
u_{k})|\leq\frac{C}{k}\,M_{k}^{1/p}\,,$ (3.21)

where $t_{k}=\omega_{n}^{-1}\int_{B_{1}}\Phi(u_{k})$. In particular, there is
$c\in[0,1]$ such that, up to extracting a subsequence, $u_{k}\to c$ in
$L^{1}(B_{1})$ and a.e. in $B_{1}$ as $k\to\infty$. Since (3.20) implies
$\int_{B_{1}}W(u_{k})\to 0$ as $k\to\infty$, by Fatou’s lemma we find

$\int_{B_{1}}W(c)\leq\liminf_{k\to\infty}\int_{B_{1}}W(u_{k})=0\,.$

In particular, $c\in\\{0,1\\}$. Since $c=1$ would contradict
$M_{k}\leq\omega_{n}/2$ for every $k$, we conclude that $c=0$, and hence,
thanks also to $0\leq u_{k}\leq 1$, that $\Phi(u_{k})\to 0$ in $L^{p}(B_{1})$
(i.e., $M_{k}\to 0$) as $k\to\infty$. On noticing that $0\leq t_{k}\leq
C\,M_{k}^{1/p}$, we deduce by (3.21) that

$\displaystyle\|\Phi(u_{k})-(M_{k}/\omega_{n})^{1/p}\|_{L^{p}(B_{1})}\leq\|t_{k}-(M_{k}/\omega_{n})^{1/p}\|_{L^{p}(B_{1})}+\frac{C}{k}\,M_{k}^{1/p}\leq
C\,M_{k}^{1/p}\,,$ (3.22)

and, in particular, that

$\lim_{k\to\infty}\Big{|}B_{1}\cap\Big{\\{}\Phi(u_{k})\leq(M_{k}/2\omega_{n})^{1/p}\Big{\\}}\Big{|}=0\,.$

Since $\Phi(u_{k})\to 0$ in $L^{p}(B_{1})$ implies
$|B_{1}\cap\\{\Phi(u_{k})>1/2\\}|\to 0$ as $k\to\infty$, we conclude that

$\Big{|}B_{1}\cap\Big{\\{}\frac{1}{2}\geq\Phi(u_{k})\geq(M_{k}/2\omega_{n})^{1/p}\Big{\\}}\Big{|}\geq\frac{|B_{1}|}{2}\,,$
(3.23)

for $k$ large enough. In particular, thanks to $\Phi\leq C\,W$ on $[0,1/2]$
and to (3.20), for $k$ large enough we have

$\frac{M_{k}^{1/p}}{C\,\varepsilon_{k}}\leq\frac{1}{\varepsilon_{k}}\int_{B_{1}\cap\\{1/2\geq\Phi(u_{k})\geq(M_{k}/2\omega_{n})^{1/p}\\}}\Phi(u_{k})\leq\frac{1}{\varepsilon_{k}}\int_{B_{1}}W(u_{k})\leq\frac{M_{k}^{1/p}}{k}\,,$
(3.24)

that leads to a contradiction as $k\to\infty$.

Step two: We prove statement (a). To prove (3.3) it suffices to recall that
$W(r)\geq r^{2}/C$ for $r\in[0,1/2]$ and $V_{\delta}(r)\geq r^{2}/C$ for
$r\in[1/2,1]$, so that

$\int_{\mathbb{R}^{n}}u^{2}=\int_{\\{u\leq
1/2\\}}u^{2}+\int_{\\{u>1/2\\}}u^{2}\leq
C\,\int_{\mathbb{R}^{n}}W(u)+V_{\delta}(u)\,.$

To prove (3.5) we first notice that $|V_{\delta}^{\prime\prime}(r)|\leq C$ and
$|W^{\prime}(r)\,V^{\prime}_{\delta}(r)|\leq C\,W(r)$ for every $r\in[0,1]$
(and every $\delta\in[0,\delta_{0}])$, so that

$\Big{|}\int_{\mathbb{R}^{n}}2\varepsilon\,|\nabla
u|^{2}\,V_{\delta}^{\prime\prime}(u)+\frac{1}{\varepsilon}\,W^{\prime}(u)\,V_{\delta}^{\prime}(u)\Big{|}\leq
C\,\mathcal{AC}_{\varepsilon}(u)\,.$ (3.25)

Since $V_{\delta}(r)\leq C\,r^{2}$ for $r\in[0,1]$ and $u\in
L^{2}(\mathbb{R}^{n})$ we can find a sequence $R_{j}\to\infty$ such that
$\int_{\partial B_{R_{j}}}V_{\delta}(u)\to 0$ as $j\to\infty$, and thus apply
the divergence theorem to deduce that, for every $x_{0}\in\mathbb{R}^{n}$,

$\displaystyle(n-1)\,\int_{\mathbb{R}^{n}}\frac{V_{\delta}(u)}{|x-x_{0}|}\,dx$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{n}}V_{\delta}(u)\,{\rm
div}\,\Big{(}\frac{x-x_{0}}{|x-x_{0}|}\Big{)}\,dx$ $\displaystyle=$
$\displaystyle-\int_{\mathbb{R}^{n}}V^{\prime}_{\delta}(u)\,\nabla
u\cdot\frac{x-x_{0}}{|x-x_{0}|}\,dx\leq\|V^{\prime}_{\delta}(u)\|_{L^{2}(\mathbb{R}^{n})}\,\|\nabla
u\|_{L^{2}(\mathbb{R}^{n})}\,.$

Setting $R=2\,\max\\{1,\varepsilon\\}/\sigma_{0}\,\sqrt{n+1}$, we can apply
(3.14) to find $x_{0}\in\mathbb{R}^{n}$ such that

$C(\varepsilon)\,\fint_{B_{R}(x_{0})}V_{\delta}(u)\geq\min\Big{\\{}1,\frac{\mathcal{V}_{\delta}(u)}{\mathcal{AC}_{\varepsilon}(u)}\Big{\\}}^{n}$

which combined with (3) gives

$\|V^{\prime}_{\delta}(u)\|_{L^{2}(\mathbb{R}^{n})}\,\|\nabla
u\|_{L^{2}(\mathbb{R}^{n})}\geq\frac{(n-1)}{R}\,\int_{B_{R}(x_{0})}V_{\delta}(u)\geq\frac{1}{C(\varepsilon)}\,\min\Big{\\{}1,\frac{\mathcal{V}_{\delta}(u)}{\mathcal{AC}_{\varepsilon}(u)}\Big{\\}}^{n}\,,$
(3.27)

that is (3.4). In summary, by combining (3.25) and (3.27) we find

$\displaystyle|\lambda_{\varepsilon,\delta}[u]|$ $\displaystyle\leq$
$\displaystyle
C\,\frac{\mathcal{AC}_{\varepsilon}(u)}{\int_{\mathbb{R}^{n}}V_{\delta}^{\prime}(u)^{2}}\leq
C(\varepsilon)\,\frac{\mathcal{AC}_{\varepsilon}(u)}{\min\Big{\\{}1,\frac{\mathcal{V}_{\delta}(u)}{\mathcal{AC}_{\varepsilon}(u)}\Big{\\}}^{2n}}\,\int_{\mathbb{R}^{n}}|\nabla
u|^{2}$ (3.28)

that is (3.5). When $\varepsilon\in(0,\varepsilon_{0}]$ and $\delta=0$ we can
replace the constant $C(\varepsilon)$ in (3.5) with a plain universal constant
$C$ by exploiting the fact that (3.15) can be used in place of (3.14) (notice
indeed that $\max\\{1,\varepsilon\\}\leq C$ when
$\varepsilon\in(0,\varepsilon_{0}]$).

Step three: We prove statement (b). Setting for brevity,

$\mathcal{N}_{\varepsilon,\delta}[u]=\int_{\mathbb{R}^{n}}2\,\varepsilon\,|\nabla
u|^{2}\,V_{\delta}^{\prime\prime}(u)+\frac{W^{\prime}(u)\,V_{\delta}^{\prime}(u)}{\varepsilon}\,,\qquad\delta\in[0,\delta_{0}]\,,$

we notice that for every $\delta,\delta_{*}\in[0,\delta_{0}]$ and $u,v\in
X\setminus\\{0\\}$,

$\lambda_{\varepsilon,\delta}[u]-\lambda_{\varepsilon,\delta_{*}}[v]=\frac{\mathcal{N}_{\varepsilon,\delta}[u]-\mathcal{N}_{\varepsilon,\delta_{*}}[v]}{\int_{\mathbb{R}^{n}}V_{\delta}^{\prime}(u)^{2}}+\frac{\mathcal{N}_{\varepsilon,\delta_{*}}[v]\,\big{\\{}\int_{\mathbb{R}^{n}}V_{\delta_{*}}^{\prime}(v)^{2}-\int_{\mathbb{R}^{n}}V_{\delta}^{\prime}(u)^{2}\big{\\}}}{\int_{\mathbb{R}^{n}}V_{\delta}^{\prime}(u)^{2}\int_{\mathbb{R}^{n}}V_{\delta_{*}}^{\prime}(v)^{2}}\,.$
(3.29)

We first work on (3.29) with $\delta=\delta_{*}$, with the goal of proving
(3.6), (3.7), and (3.9). Since
$\mathcal{AC}_{\varepsilon}(u)\leq\|u\|_{W^{1,2}}^{2}/\varepsilon$, we deduce
from (3.4) that, if $\|u\|_{W^{1,2}}\leq C(\varepsilon)$, then

$\frac{1}{\int_{\mathbb{R}^{n}}V_{\delta}^{\prime}(u)^{2}}\leq\frac{C(\varepsilon)}{\min\\{1,\mathcal{V}_{\delta}(u)^{2n}\\}}\,.$
(3.30)

Recalling from (3.25) that $|\mathcal{N}_{\varepsilon,\delta}[u]|\leq
C\,\mathcal{AC}_{\varepsilon}(u)\leq C(\varepsilon)\,\|u\|_{W^{1,2}}$, and
using again first ${\rm Lip}(V_{\delta}^{\prime},[0,1])\leq C$, and then
$|V_{\delta}^{\prime}(r)|\leq C\,t$ for $r\in[0,1]$, we find that, if
$\|u\|_{W^{1,2}}\leq C(\varepsilon)$, then for every $v\in
W^{1,2}(\mathbb{R}^{n};[0,1])$

$\displaystyle\Big{|}\mathcal{N}_{\varepsilon,\delta}[v]\,\int_{\mathbb{R}^{n}}\big{(}V_{\delta}^{\prime}(v)^{2}-V_{\delta}^{\prime}(u)^{2}\big{)}\Big{|}\leq
C(\varepsilon)\,\int_{\mathbb{R}^{n}}|V_{\delta}^{\prime}(u)-V_{\delta}^{\prime}(v)|\,\big{(}|V_{\delta}^{\prime}(u)|+|V_{\delta}^{\prime}(v)|\big{)}$
$\displaystyle\leq
C(\varepsilon)\,\|u-v\|_{L^{2}}\,\Big{(}\int_{\mathbb{R}^{n}}|V_{\delta}^{\prime}(u)|^{2}+|V_{\delta}^{\prime}(v)|^{2}\Big{)}^{1/2}$
$\displaystyle\leq
C(\varepsilon)\,\max\\{\|u\|_{L^{2}},\|v\|_{L^{2}}\\}\,\|u-v\|_{L^{2}}\,.$
(3.31)

In summary, by (3.29), (3.30) and (3.31) for every $\varepsilon>0$ and $u,v\in
W^{1,2}(\mathbb{R}^{n};[0,1])\setminus\\{0\\}$ with
$\max\\{\|u\|_{W^{1,2}(\mathbb{R}^{n})},\|v\|_{W^{1,2}(\mathbb{R}^{n})}\\}\leq
C(\varepsilon)$ we have proved that

$\big{|}\lambda_{\varepsilon,\delta}[u]-\lambda_{\varepsilon,\delta}[v]\big{|}\leq
C(\varepsilon)\,\frac{|\mathcal{N}_{\varepsilon,\delta}[u]-\mathcal{N}_{\varepsilon,\delta}[v]|}{\min\\{1,\mathcal{V}_{\delta}(u)^{2n}\\}}+\frac{C(\varepsilon)\,\|u-v\|_{L^{2}}}{\min\\{1,\mathcal{V}_{\delta}(u)^{2n}\\}\,\min\\{1,\mathcal{V}_{\delta}(v)^{2n}\\}}\,.$
(3.32)

We first estimate that

$\displaystyle|\mathcal{N}_{\varepsilon,\delta}[u]-\mathcal{N}_{\varepsilon,\delta}[v]|$
$\displaystyle\leq$ $\displaystyle
2\,\varepsilon\,\int_{\mathbb{R}^{n}}|\nabla(u-v)|\,\big{(}|\nabla u|+|\nabla
v|)\,|V_{\delta}^{\prime\prime}(u)|$
$\displaystyle+2\,\varepsilon\,\int_{\mathbb{R}^{n}}|\nabla
v|^{2}\,|V_{\delta}^{\prime\prime}(u)-V_{\delta}^{\prime\prime}(v)|$
$\displaystyle+\frac{1}{\varepsilon}\,\int_{\mathbb{R}^{n}}|W^{\prime}(u)-W^{\prime}(v)|\,|V_{\delta}^{\prime}(u)|+\frac{1}{\varepsilon}\,\int_{\mathbb{R}^{n}}|W^{\prime}(v)|\,|V_{\delta}^{\prime}(u)-V_{\delta}^{\prime}(v)|\,.$

By recalling that ${\rm Lip}(W^{\prime};[0,1])\leq C$, ${\rm
Lip}(V_{\delta}^{\prime};[0,1])\leq C$, ${\rm
Lip}(V_{\delta}^{\prime\prime};[0,1])\leq C(\delta)$ (when $\delta>0$), and
$\max\\{|V_{\delta}^{\prime}(r)|,|W^{\prime}(r)|\\}\leq C\,r$ for $r\in[0,1]$
we find

$\displaystyle|\mathcal{N}_{\varepsilon,\delta}[u]-\mathcal{N}_{\varepsilon,\delta}[v]|\leq
C\,\max\big{\\{}\|\nabla u\|_{L^{2}},\|\nabla v\|_{L^{2}}\big{\\}}\,\|\nabla
u-\nabla v\|_{L^{2}}$ (3.34)
$\displaystyle+C(\varepsilon,\delta)\,\|u-v\|_{C^{0}}\,\int_{\mathbb{R}^{n}}|\nabla
v|^{2}+\frac{C}{\varepsilon}\,\|u-v\|_{L^{2}}\,\max\\{\|u\|_{L^{2}},\|v\|_{L^{2}}\\}$
$\displaystyle\leq$ $\displaystyle
C(\varepsilon)\,\max\big{\\{}\|u\|_{W^{1,2}},\|v\|_{W^{1,2}}\big{\\}}\,\|u-v\|_{W^{1,2}}+C(\delta)\,\|\nabla
v\|_{L^{2}}^{2}\,\|u-v\|_{C^{0}}\,;$

while, using $[V_{\delta}^{\prime\prime}]_{C^{0,\gamma(n)}[0,1]}\leq C$ (which
also holds when $\delta=0$) in place of ${\rm
Lip}(V_{\delta}^{\prime\prime};[0,1])\leq C(\delta)$, we find instead

$\displaystyle|\mathcal{N}_{\varepsilon,\delta}[u]-\mathcal{N}_{\varepsilon,\delta}[v]|$
(3.35) $\displaystyle\leq$ $\displaystyle
C(\varepsilon)\,\max\big{\\{}\|u\|_{W^{1,2}},\|v\|_{W^{1,2}}\big{\\}}\,\|u-v\|_{W^{1,2}}+C\,\|\nabla
v\|_{L^{2}}^{2}\,\|u-v\|_{C^{0}}^{\gamma(n)}\,;$

thanks to $\max\\{\|u\|_{X},\|v\|_{X}\\}\leq C(\varepsilon)$; with the
convention that $C(\varepsilon,\delta)=+\infty$ if $\delta=0$, we thus
conclude from (3), (3.34) and (3.35) that

$\displaystyle|\mathcal{N}_{\varepsilon,\delta}[u]-\mathcal{N}_{\varepsilon,\delta}[v]|\leq\min\Big{\\{}C(\varepsilon,\delta)\,\|u-v\|_{X},C(\varepsilon)\|u-v\|_{X}^{\gamma(n)}\Big{\\}}\,.$
(3.36)

By means of (3.32) and (3.36) we find that, if
$\max\\{\|u\|_{X},\|v\|_{X}\\}\leq C(\varepsilon)$, then (3.6) and (3.7) hold.
We now assume that $u,v\in W^{2,2}(\mathbb{R}^{n};[0,1])\setminus\\{0\\}$ and
prove (3.9). To this end let us first notice that an integration by parts
gives

$\displaystyle\int_{\mathbb{R}^{n}}|\nabla
u|^{2}\,V_{\delta}^{\prime\prime}(u)-\int_{\mathbb{R}^{n}}|\nabla
v|^{2}\,V_{\delta}^{\prime\prime}(v)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}(\nabla u-\nabla v)\cdot\nabla
v\,\,V_{\delta}^{\prime\prime}(v)+\int_{\mathbb{R}^{n}}\nabla u\cdot\nabla
u\,\,V_{\delta}^{\prime\prime}(u)-\nabla u\cdot\nabla
v\,\,V_{\delta}^{\prime\prime}(v)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}(\nabla u-\nabla v)\cdot\nabla
v\,V_{\delta}^{\prime\prime}(v)-\int_{\mathbb{R}^{n}}(\Delta
u)\,\big{(}V_{\delta}^{\prime}(u)-V_{\delta}^{\prime}(v)\big{)}\,,$

which can be used to replace (3) with

$\displaystyle|\mathcal{N}_{\varepsilon,\delta}[u]-\mathcal{N}_{\varepsilon,\delta}[v]|$
$\displaystyle\leq$ $\displaystyle
2\,\varepsilon\,\int_{\mathbb{R}^{n}}|\Delta
u|\,|V_{\delta}^{\prime}(u)-V_{\delta}^{\prime}(v)|+|\nabla(u-v)|\,|\nabla
v|\,|V_{\delta}^{\prime\prime}(v)|$
$\displaystyle+\frac{1}{\varepsilon}\,\int_{\mathbb{R}^{n}}|W^{\prime}(u)-W^{\prime}(v)|\,|V_{\delta}^{\prime}(u)|+\frac{1}{\varepsilon}\,\int_{\mathbb{R}^{n}}|W^{\prime}(v)|\,|V_{\delta}^{\prime}(u)-V_{\delta}^{\prime}(v)|\,.$

By ${\rm Lip}(W^{\prime};[0,1])\leq C$, ${\rm
Lip}(V_{\delta}^{\prime};[0,1])\leq C$ and
$\max\\{|V_{\delta}^{\prime}(r)|,|W^{\prime}(r)|\\}\leq C\,r$ for $r\in[0,1]$
we thus find

$|\mathcal{N}_{\varepsilon,\delta}[u]-\mathcal{N}_{\varepsilon,\delta}[v]|\leq
C(\varepsilon)\,\max\\{\|u\|_{W^{2,2}},\|v\|_{W^{1,2}}\\}\,\|u-v\|_{W^{1,2}}$

which, combined with (3.32) and with the assumption
$\max\\{\|u\|_{W^{2,2}},\|v\|_{W^{2,2}}\\}\leq C(\varepsilon)$, gives (3.9).

We now work on (3.29) with $u=v$ and $\delta_{*}=0$ to prove (3.8). To this
end, we first notice that

$\displaystyle\big{|}\mathcal{N}_{\varepsilon,\delta}[u]-\mathcal{N}_{\varepsilon}[u]\big{|}$
$\displaystyle\leq$ $\displaystyle 2\,\varepsilon\int_{\mathbb{R}^{n}}|\nabla
u|^{2}\,|V_{\delta}^{\prime\prime}(u)-V^{\prime\prime}(u)|+\frac{1}{\varepsilon}\int_{\mathbb{R}^{n}}|W^{\prime}(u)|\,|V_{\delta}^{\prime}(u)-V^{\prime}(u)|$
$\displaystyle\leq$ $\displaystyle
2\,\mathcal{AC}_{\varepsilon}(u)\,\|V_{\delta}-V\|_{C^{2}[0,1]}\,,$

while $V_{\delta}^{\prime}(0)=V^{\prime}(0)=0$ gives
$|V_{\delta}^{\prime}(u)-V^{\prime}(u)|\leq\|V_{\delta}^{\prime\prime}-V^{\prime\prime}\|_{C^{0}[0,1]}\,|u|$
on $\mathbb{R}^{n}$, and thus, arguing as in (3.31), that

$\displaystyle\Big{|}\int_{\mathbb{R}^{n}}V_{\delta}^{\prime}(u)^{2}-\int_{\mathbb{R}^{n}}V^{\prime}(u)^{2}\Big{|}\leq
C\,\|V_{\delta}-V\|_{C^{2}[0,1]}\,\|u\|_{L^{2}(\mathbb{R}^{n})}^{2}\,.$

Combining this last two estimates with (3.29), (3.30) and
$|\mathcal{N}_{\varepsilon,\delta}[u]|\leq C(\varepsilon)\,\|u\|_{X}$ we
immediately prove (3.8). ∎

##  4\. Existence, regularity and entropies of the flow (Proof of Theorem 1.2)

###### Proof of Theorem 1.2.

Step one, existence of mild solutions: For every $\varepsilon,\delta>0$ we
introduce the regularized flows

$\begin{cases}\varepsilon^{2}\,\partial_{t}u=2\,\varepsilon^{2}\,\Delta
u-W^{\prime}(u)+\varepsilon\,\lambda_{\varepsilon,\delta}[u(t)]\,V_{\delta}^{\prime}(u)\,,\qquad\mbox{on
$\mathbb{R}^{n}\times(0,\infty)$}\,,\\\ u(0)=u_{0}\,,\end{cases}$ (DFδ)

that are obtained by replacing $V$ with $V_{\delta}$ in (DF). If we set

$G(x,t)=\frac{e^{-|x|^{2}/8\,t}}{(8\,\pi\,t)^{n/2}}\,,\qquad(x,t)\in\mathbb{R}^{n}\times(0,\infty)\,,$
(4.1)

and $S_{t}v=v\star G(t)$ for $v:\mathbb{R}^{n}\to\mathbb{R}$ and $t>0$, then a
solution $u$ of

$\mbox{$u_{t}-2\,\Delta u=f$ on
$\mathbb{R}^{n}\times(0,\infty)$}\,,\qquad\mbox{$u(0)=u_{0}$ on
$\mathbb{R}^{n}$}\,,$ (4.2)

with data $f:\mathbb{R}^{n}\times(0,\infty)\to\mathbb{R}$ and
$u_{0}:\mathbb{R}^{n}\to\mathbb{R}$, is formally given by the Duhamel formula
(see, for example, [Eva98, Section 2.3.1]),

$\displaystyle u(x,t)$ $\displaystyle=$ $\displaystyle
S_{t}u_{0}(x)+\int_{0}^{t}\,S_{t-s}[f(s)](x)\,ds$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}G(x-y,t)\,u_{0}(y)\,dy+\int_{0}^{t}ds\int_{\mathbb{R}^{n}}G(x-y,t-s)\,f(y,s)\,dy\,.$

Solutions to the integral equation (4) are usually called mild solutions of
the parabolic PDE (4.2), and can be constructed by fixed points arguments. We
now set up the stage to prove the short-time existence of a unique mild
solution to (DFδ).

From now on we fix $u_{0}$ with

$u_{0}\in\bigcap_{p\geq
2}W^{2,p}(\mathbb{R}^{n};[0,1])\,,\qquad\mathcal{V}(u_{0})=1\,.$ (4.4)

By the properties of $V_{\delta}$ and $W$ it is easily seen that
$\mathcal{V}(u_{0})=1$ implies $\mathcal{V}_{\delta}(u_{0})\in(0,\infty)$ for
every $\delta>0$. Moreover, we also have

$\lim_{\delta\to 0^{+}}\mathcal{V}_{\delta}(u_{0})=\mathcal{V}(u_{0})=1\,.$
(4.5)

Given parameters $\tau>0$ and $\sigma\in(0,1)$ to be chosen in a moment, we
then introduce the vector space

$Y=\big{\\{}u\in
C^{0}\big{(}[0,\tau);B^{X}_{\sigma}(u_{0})\big{)}:u(0)=u_{0}\big{\\}}\,,$
(4.6)

where $B^{X}_{\sigma}(u_{0})$ denotes the ball in $X$ of radius $\sigma$ and
center $u_{0}$. If we pick $\sigma$ small enough depending on $u_{0}$, and
since $\mathcal{V}_{\delta}(u_{0})>0$ for every $\delta>0$, we find that (see
(4.10) below) $\mathcal{V}_{\delta}(u(t))>0$ for every $t\in[0,\tau)$: in
particular, $\lambda_{\varepsilon,\delta}[u(t)]$ is well defined for every
$t\in[0,\tau)$. Hence, for each $\varepsilon>0$, $\delta\in(0,\delta_{0}]$,
and $u\in Y$ we can define
$F_{\delta}[u]:\mathbb{R}^{n}\times[0,\tau)\to\mathbb{R}$ and
$T_{\delta}[u]:\mathbb{R}^{n}\times[0,\tau)\to\mathbb{R}$ by setting

$\displaystyle F_{\delta}[u]$ $\displaystyle=$
$\displaystyle-\frac{W^{\prime}(u)}{\varepsilon^{2}}+\lambda_{\varepsilon,\delta}[u(t)]\,\frac{V_{\delta}^{\prime}(u)}{\varepsilon}\,,$
(4.7) $\displaystyle T_{\delta}[u]$ $\displaystyle=$ $\displaystyle
S_{t}u_{0}+\int_{0}^{t}S_{t-s}[F_{\delta}[u](s)]\,ds\,.$ (4.8)

We claim that, if $\tau$ and $\sigma$ (introduced in the definition of $Y$)
are small enough with respect to $\varepsilon$, $\delta$, and $u_{0}$, then
$u\mapsto T_{\delta}[u]$ defines a contraction of the Banach space $Y$. By the
Banach fixed point theorem we will then deduce the existence of a unique $u\in
C^{0}([0,\tau);B^{X}_{\sigma}(u_{0}))$ such that $u(0)=u_{0}$ and

$u(t)=S_{t}u_{0}+\int_{0}^{t}S_{t-s}[F_{\delta}[u](s)]\,ds\,,$ (4.9)

for every $t\in(0,\tau)$. In particular, this claim will prove the existence
of a unique, short-time, mild solution $u$ of (DFδ).

To prove our claim, we begin by showing that if $\sigma$ is small enough in
terms of $u_{0}$ and $\delta$, then for every $u\in Y$ we have

$\mathcal{V}_{\delta}(u(t))\geq\frac{\mathcal{V}_{\delta}(u_{0})}{2}\,,\qquad\forall
t\in[0,\tau)\,.$ (4.10)

Indeed, by $|V_{\delta}^{\prime}(r)|\leq C\,|r|$ for every $r\in(0,1)$ and by
$\|u_{0}-u(t)\|_{X}<\sigma<1$ we find that

$\displaystyle\mathcal{V}_{\delta}(u(t))-\mathcal{V}_{\delta}(u_{0})$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}\,\int_{0}^{1}V_{\delta}^{\prime}\big{(}s\,u_{0}+(1-s)\,u(t)\big{)}\,(u_{0}-u(t))\,ds$
$\displaystyle\geq$
$\displaystyle-C\,\int_{\mathbb{R}^{n}}\,|u_{0}-u(t)|\,\big{(}|u_{0}|+|u(t)|\big{)}$
$\displaystyle\geq$
$\displaystyle-C\,\|u(t)-u_{0}\|_{L^{2}}\,\big{(}\|u_{0}\|_{L^{2}}+\|u(t)\|_{L^{2}}\big{)}$
$\displaystyle\geq$
$\displaystyle-C\,\sigma\,\big{(}2\,\|u_{0}\|_{L^{2}}+\sigma\big{)}\geq-\frac{\mathcal{V}_{\delta}(u_{0})}{2}\,,$

provided $\sigma$ is small enough in terms of $\delta$ and $u_{0}$ (recall
that $\mathcal{V}_{\delta}(u_{0})>0$). Having proved (4.10) we notice that,
combined with (3.5), (3.6), and $\mathcal{AC}_{\varepsilon}(u)\leq
C(\varepsilon)\,\|u\|_{W^{1,2}}$, it implies

$\displaystyle\big{|}\lambda_{\varepsilon,\delta}[u(t)]\big{|}$
$\displaystyle\leq$ $\displaystyle C(\varepsilon,\delta,u_{0})$ (4.11)
$\displaystyle\big{|}\lambda_{\varepsilon,\delta}[u(t)]-\lambda_{\varepsilon,\delta}[v(t)]\big{|}$
$\displaystyle\leq$ $\displaystyle
C(\varepsilon,\delta,u_{0})\,\|u(t)-v(t)\|_{X}\,,$ (4.12)

for every $u,v\in Y$ and every $t\in[0,\tau)$. Next we notice that if $u\in
Y$, then for every $t\in(0,\tau)$ we have

$|F_{\delta}[u(t)]|\leq
C(\varepsilon)\,\max\big{\\{}1,|\lambda_{\varepsilon,\delta}[u(t)]\big{\\}}\,|u(t)|\,,\qquad\mbox{on
$\mathbb{R}^{n}$}\,;$ (4.13)

moroever, since $|W^{\prime}(r)|\leq C\,|r|$ and $|V_{\delta}^{\prime}|\leq
C\,|r|$ for $r\in(0,1)$, $W^{\prime\prime}$ and $V_{\delta}^{\prime\prime}$
are bounded on $[0,1]$, and

$\nabla\big{(}F_{\delta}[u](t))=\Big{\\{}-\frac{W^{\prime\prime}(u)}{\varepsilon^{2}}+\lambda_{\varepsilon,\delta}[u(t)]\,\frac{V_{\delta}^{\prime\prime}(u)}{\varepsilon}\Big{\\}}\,\nabla
u(t)\,,$ (4.14)

then

$|\nabla F_{\delta}[u(t)]|\leq
C(\varepsilon)\,\max\big{\\{}1,|\lambda_{\varepsilon,\delta}[u(t)]\big{\\}}\,|\nabla
u(t)|\,,\qquad\mbox{on $\mathbb{R}^{n}$}\,.$ (4.15)

By combining (4.13) and (4.15) with (4.11), we see that if $u\in Y$, then
$F_{\delta}[u]\in C^{0}([0,\tau);X)$ and

$\displaystyle\|F_{\delta}[u](t)\|_{C^{0}}\\!\\!$ $\displaystyle\leq$
$\displaystyle\\!\\!C(\varepsilon)\,\max\big{\\{}1,|\lambda_{\varepsilon,\delta}[u(t)]\big{\\}}\,\|u(t)\|_{C^{0}}\,,$
(4.16) $\displaystyle\|F_{\delta}[u](t)\|_{L^{2}}\\!\\!$ $\displaystyle\leq$
$\displaystyle\\!\\!C(\varepsilon)\,\max\big{\\{}1,|\lambda_{\varepsilon,\delta}[u(t)]\big{\\}}\,\|u(t)\|_{L^{2}}\,,$
(4.17) $\displaystyle\|\nabla(F_{\delta}[u](t))\|_{L^{2}}\\!\\!$
$\displaystyle\leq$
$\displaystyle\\!\\!C(\varepsilon)\,\max\big{\\{}1,|\lambda_{\varepsilon,\delta}[u(t)]\big{\\}}\,\|\nabla
u(t)\|_{L^{2}},\qquad\forall t\in[0,\tau)\,.$ (4.18)

By combining these estimates with (4.11) we thus find

$\|F_{\delta}[u(t)]\|_{X}\leq
C(\varepsilon,\delta,u_{0})\,\|u\|_{X}\,,\qquad\forall t\in[0,\tau)\,.$ (4.19)

Since $\|S_{t}v\|_{C^{0}}\leq\|v\|_{C^{0}}$,
$\|S_{t}v\|_{L^{2}}\leq\|v\|_{L^{2}}$, and
$\|\nabla(S_{t}v)\|_{L^{2}}\leq\|\nabla v\|_{L^{2}}$ for every $v\in X$ and
every $t>0$, we deduce that if $u\in Y$, then $T_{\delta}[u]\in
C^{0}([0,\tau);X)$. Moreover, since $\|S_{t}u_{0}-u_{0}\|_{X}\to 0$ as $t\to
0^{+}$, if we pick $\tau$ small enough in terms of $\varepsilon$, $\delta$,
$u_{0}$ and $\sigma$ (where $\sigma$ has already been chosen small enough in
terms of $u_{0}$ and $\delta$), then we find

$\displaystyle\|T_{\delta}[u](t)-u_{0}\|_{X}$ $\displaystyle\leq$
$\displaystyle\|S_{t}u_{0}-u_{0}\|_{X}+t\,\sup_{0<s<t}\|F_{\delta}[u](t)\|_{X}$
$\displaystyle\leq$
$\displaystyle\|S_{t}u_{0}-u_{0}\|_{X}+C(\varepsilon,\delta,u_{0})\,t\,\|u(t)\|_{X}<\sigma\,,$

for every $t\in[0,\tau_{0})$, so that $T_{\delta}[u]\in Y$ for every $u\in Y$.
Having proved that $T_{\delta}$ is a self-map of $Y$, we now pick $u,v\in Y$
and notice that for every $t\in[0,\tau)$ we have

$\|T_{\delta}[u]-T_{\delta}[v]\|_{Y}=\sup_{0<t<\tau}\|T_{\delta}[u](t)-T_{\delta}[v](t)\|\leq\tau\,\sup_{0<t<\tau}\|F_{\delta}[u](t)-F_{\delta}[v](t)\|\,.$
(4.20)

Now for every $u,v\in Y$ and $t\in[0,\tau)$ we have, pointwise on
$\mathbb{R}^{n}$,

$\displaystyle|F_{\delta}[u](t)-F_{\delta}[v](t)|\leq\Big{\\{}\frac{{\rm
Lip}(W^{\prime})}{\varepsilon^{2}}+\big{|}\lambda_{\varepsilon,\delta}[u(t)]\big{|}\,\frac{{\rm
Lip}(V_{\delta}^{\prime})}{\varepsilon}\Big{\\}}\,|u(t)-v(t)|$
$\displaystyle\hskip
113.81102pt+\big{|}\lambda_{\varepsilon,\delta}[u(t)]-\lambda_{\varepsilon,\delta}[v(t)]\big{|}\,\frac{|V_{\delta}^{\prime}(v(t))|}{\varepsilon}$
(4.21) $\displaystyle\hskip 28.45274pt\leq
C(\varepsilon)\,\Big{\\{}\max\big{\\{}1,\big{|}\lambda_{\varepsilon,\delta}[u(t)]\big{|}\big{\\}}\,|u(t)-v(t)|+\big{|}\lambda_{\varepsilon,\delta}[u(t)]-\lambda_{\varepsilon,\delta}[v(t)]\big{|}\,|v(t)|\Big{\\}}\,,$

where we have used $|V_{\delta}^{\prime}(r)|\leq C\,|r|$ for $r\in[0,1]$;
then, by (4.11), (4.12), and (4.21), we find that

$\displaystyle\|F_{\delta}[u](t)-F_{\delta}[v](t)\|_{X}\leq
C(\varepsilon,\delta,u_{0})\,\|u(t)-v(t)\|_{X}\,,\qquad\forall
t\in[0,\tau)\,.$ (4.22)

Similarly, starting from (4.14), we find that, pointwise on $\mathbb{R}^{n}$,

$\displaystyle|\nabla(F_{\delta}[u](t))-\nabla(F_{\delta}[v](t))|\leq\Big{\\{}\frac{{\rm
Lip}(W^{\prime})}{\varepsilon^{2}}+\big{|}\lambda_{\varepsilon,\delta}[u(t)]\big{|}\,\frac{{\rm
Lip}(V_{\delta}^{\prime})}{\varepsilon}\Big{\\}}\,|\nabla u(t)-\nabla v(t)|$
$\displaystyle\hskip 113.81102pt+\Big{\\{}\frac{{\rm
Lip}(W^{\prime\prime})}{\varepsilon^{2}}\,+\frac{{\rm
Lip}(V_{\delta}^{\prime\prime})}{\varepsilon}\,\,|\lambda_{\varepsilon,\delta}[v(t)]\Big{\\}}\,|\nabla
v(t)|\,|u(t)-v(t)|$ $\displaystyle\hskip
113.81102pt+\Big{|}\lambda_{\varepsilon,\delta}[u(t)]-\lambda_{\varepsilon,\delta}[v(t)]\Big{|}\,|\nabla
v(t)|\,\frac{|V_{\delta}^{\prime\prime}(u(t))|}{\varepsilon}$
$\displaystyle\leq
C(\varepsilon,\delta)\,\Big{\\{}\max\big{\\{}1,\big{|}\lambda_{\varepsilon,\delta}[u(t)]\big{|}\big{\\}}\,\big{\\{}|\nabla
v(t)|\,|u(t)-v(t)|+|\nabla u(t)-\nabla v(t)|\big{\\}}$ (4.23)
$\displaystyle\hskip
227.62204pt+\Big{|}\lambda_{\varepsilon,\delta}[u(t)]-\lambda_{\varepsilon,\delta}[v(t)]\Big{|}\,|\nabla
v(t)|\Big{\\}}\,,$

where we have made crucial use of the regularization $V_{\delta}$ of $V$ to
assert that ${\rm Lip}(V_{\delta}^{\prime\prime})\leq C(\delta)$. By (4.11),
(4.12), (4.23), and $\sigma<1$,

$\displaystyle\|\nabla(F_{\delta}[u](t))-\nabla(F_{\delta}[v](t))\|_{L^{2}}\leq
C(\varepsilon,\delta,u_{0})\,\|u(t)-v(t)\|_{X}\,,\qquad\forall
t\in[0,\tau)\,.$ (4.24)

By combining (4.22) and (4.24) with (4.20) we conclude that

$\|T_{\delta}[u]-T_{\delta}[v]\|_{Y}\leq
C(\varepsilon,\delta,u_{0})\,\tau\,\|u-v\|_{Y}\,.$

In particular, up to further decrease $\tau$ depending on $\varepsilon$,
$\delta$ and $u_{0}$, we can ensure that ${\rm Lip}(T_{\delta};Y)<1$, and that
$T_{\delta}$ is a contraction of $Y$.

Step two, regularity and global-in-time existence for (DFδ): Given
$\varepsilon>0$, $\delta\in(0,\delta_{0}]$ and $u_{0}$ as in (4.4), by step
five we can define $\tau_{*}\in(0,\infty]$ as the supremum of those $\tau>0$
such that there exists $u\in C^{0}([0,\tau);X)$ such that $u(0)=u_{0}$ and
(4.9) holds for every $t\in(0,\tau)$ and

$\sup_{0<t<\tau}\big{\\{}\mathcal{AC}_{\varepsilon}(u(t)),\mathcal{V}_{\delta}(u(t))\big{\\}}<\infty\,,\qquad\inf_{0<t<\tau}\mathcal{V}_{\delta}(u(t))>0\,.$
(4.25)

In this step we prove that $\tau_{*}=+\infty$, that
$\mathcal{AC}_{\varepsilon}(u(t))$ is Lipschitz continuous and decreasing on
$[0,\infty)$, that $\mathcal{V}_{\delta}(u(t))=\mathcal{V}_{\delta}(u_{0})$
for every $t\in[0,\infty)$, and that $u(t)\in W^{3,p}(\mathbb{R}^{n})$ and
$\partial_{t}u(t)\in W^{1,p}(\mathbb{R}^{n})$ for every $t>0$ and $p\geq 2$
with

$\displaystyle\max\big{\\{}\|u(t)\|_{W^{3,p}(\mathbb{R}^{n})},\|\partial_{t}u(t)\|_{W^{1,p}}\big{\\}}$
(4.26) $\displaystyle\leq
C(\varepsilon,p)\,M\big{(}\mathcal{AC}_{\varepsilon}(u_{0}),\mathcal{V}_{\delta}(u_{0}),1/\mathcal{V}_{\delta}(u_{0}),\|\nabla
u_{0}\|_{W^{2,p}},t\big{)}\,,$

where $M$ denotes a generic constant which is increasing and continuous in its
arguments.

We first notice that by combining $0\leq u\leq 1$, (3.3), (3.5) and (4.25) we
find that

$\sup_{0<s<t}\|u(s)\|_{L^{p}}<\infty\,,\qquad\sup_{0<s<t}\big{|}\lambda_{\varepsilon,\delta}[u(s)]\big{|}<\infty$
(4.27)

for every $t\in(0,\tau_{*})$ and $p\geq 2$. Next, setting $G(t)=G(\cdot,t)$
(recall (4.1)), we notice that for every $t>0$ we have

$\|G(t)\|_{L^{1}}=1\,,\qquad\max\big{\\{}t^{1/2}\|\nabla
G(t)\|_{L^{1}},\,t^{3/2}\|\nabla^{2}G(t)\|_{L^{1}},\,t\|\partial_{t}G(t)\|_{L^{1}}\big{\\}}\leq
C(n)\,.$ (4.28)

Combining $u(t)=T_{\delta}[u(t)]$ with (4.28), (4.13) and standard
applications of Fubini’s theorem and Hölder’s inequality, we find that, if
$p\geq 2$, then, for every $t\in(0,\tau_{*})$,

$\displaystyle\|\nabla u(t)\|_{L^{p}}$ $\displaystyle\leq$
$\displaystyle\|\nabla u_{0}\|_{L^{p}}\,\|G(t)\|_{L^{1}}$
$\displaystyle+C(\varepsilon)\,\sup_{0<s<t}\,\max\big{\\{}1,\big{|}\lambda_{\varepsilon,\delta}[u(s)]\big{|}\big{\\}}\,\|u(s)\|_{L^{p}}\,\,\int_{0}^{t}\|\nabla
G(t-s)\|_{L^{1}}\,ds$ $\displaystyle\leq$ $\displaystyle\|\nabla
u_{0}\|_{L^{p}}+C(\varepsilon)\,t^{1/2}\,\sup_{0<s<t}\,\max\big{\\{}1,\big{|}\lambda_{\varepsilon,\delta}[u(s)]\big{|}\big{\\}}\,\|u(s)\|_{L^{p}}\,,$

so that (4.27) gives

$\sup_{0<s<t}\|\nabla u(s)\|_{L^{p}}<\infty\,,\qquad\forall
t\in(0,\tau_{*})\,,p\geq 2\,.$ (4.30)

Similarly, again from $u(t)=T_{\delta}[u(t)]$ and (4.28), we obtain that

$\displaystyle\|\nabla^{2}u(t)\|_{L^{p}}$ $\displaystyle\leq$
$\displaystyle\|\nabla^{2}u_{0}\|_{L^{p}}\,\|G(t)\|_{L^{1}}$
$\displaystyle+C(\varepsilon)\,\sup_{0<s<t}\,\max\big{\\{}1,\big{|}\lambda_{\varepsilon,\delta}[u(s)]\big{|}\big{\\}}\,\|\nabla
u(s)\|_{L^{p}}\,\,\int_{0}^{t}\|\nabla G(t-s)\|_{L^{1}}\,ds$
$\displaystyle\leq$
$\displaystyle\|\nabla^{2}u_{0}\|_{L^{p}}+C(\varepsilon)\,t^{1/2}\,\sup_{0<s<t}\,\max\big{\\{}1,\big{|}\lambda_{\varepsilon,\delta}[u(s)]\big{|}\big{\\}}\,\|\nabla
u(s)\|_{L^{p}}\,$

which, combined with (4.11), (4.27), and (4.30), gives

$\sup_{0<s<t}\|u(s)\|_{W^{2,p}}<\infty\,,\qquad\forall
t\in(0,\tau_{*})\,,p\geq 2\,.$ (4.32)

In particular, $u\in L^{\infty}_{\rm loc}((0,\tau);W^{2,2}(\mathbb{R}^{n}))$,
so that, by standard properties of mild solutions (see e.g. [CH98, Proposition
4.1.9]) (4.9) implies that

$u\in W^{1,1}_{\rm loc}((0,\tau_{*});X)\,,$ (4.33)

with

$\partial_{t}u=2\,\Delta u+f_{\delta}\qquad\mbox{a.e. on
$\mathbb{R}^{n}\times(0,\tau_{*})$}\,,$ (4.34)

where we have set $f_{\delta}(x,t)=F_{\delta}[u](x,t)$, and where, by (4.13)
and (4.15),

$\displaystyle\|f_{\delta}(t)\|_{L^{p}}\\!\\!$ $\displaystyle\leq$
$\displaystyle\\!\\!C(\varepsilon)\,\max\big{\\{}1,|\lambda_{\varepsilon,\delta}[u(t)]\big{\\}}\,\|u(t)\|_{L^{p}}\,,$
(4.35) $\displaystyle\|\nabla f_{\delta}(t)\|_{L^{p}}\\!\\!$
$\displaystyle\leq$
$\displaystyle\\!\\!C(\varepsilon)\,\max\big{\\{}1,|\lambda_{\varepsilon,\delta}[u(t)]\big{\\}}\,\|\nabla
u(t)\|_{L^{p}}\,,\qquad\forall p\geq 2\,.$ (4.36)

Moreover, by (4.34), we see that for every $t\in(0,\tau_{*})$ and $p\geq 2$,

$\|\partial_{t}u(t)\|_{L^{p}}\leq
C(\varepsilon)\,\Big{\\{}\|\nabla^{2}u(t)\|_{L^{p}}+\|f_{\delta}(t)\|_{L^{p}}\Big{\\}}\,.$
(4.37)

We now differentiate the flow in space to obtain $L^{p}$-estimates on
$\nabla^{3}u$ and on $\nabla(\partial_{t}u)$. Given $e\in\mathbb{R}^{n}$ with
$|e|=1$ and $v=v(x,t)$ we set $e_{h}v(x,t)=(v(x+h\,e,t)-v(x,t))/h$ for the
(spatial) incremental ratio of $v$ in the direction $v$ of step $h$. In this
way (4.34) implies that $e_{h}u$ solves

$\displaystyle\partial_{t}(e_{h}u)-2\,\Delta(e_{h}u)=e_{h}f_{\delta}\qquad\mbox{on
$\mathbb{R}^{n}\times(0,\infty)$},\qquad\mbox{$e_{h}u(0)=e_{h}u_{0}$ on
$\mathbb{R}^{n}$}\,.$ (4.38)

Next we consider the decomposition $e_{h}u=u_{1,h}+u_{2,h}$ where
$u_{1,h}(t)=S_{t}[e_{h}u_{0}]$ and thus

$\displaystyle\partial_{t}u_{2,h}-2\,\Delta
u_{2,h}=e_{h}f_{\delta}\qquad\mbox{on
$\mathbb{R}^{n}\times(0,\infty)$},\qquad\mbox{$u_{2,h}(0)=0$ on
$\mathbb{R}^{n}$}\,.$ (4.39)

Since $u_{2,h}(0)=0$ on $\mathbb{R}^{n}$ we can apply [Lie96, Corollary 7.31]
to deduce that for every $(a,b)\subset\subset(0,\tau_{*})$ and $p\geq 2$

$\int_{a}^{b}\,dt\int_{\mathbb{R}^{n}}|\partial_{t}u_{2,h}(t)|^{p}+|\nabla^{2}u_{2,h}(t)|^{p}\leq
C(p)\int_{a}^{b}\,dt\int_{\mathbb{R}^{n}}|e_{h}f_{\delta}(t)|^{p}\,.$ (4.40)

Since $\|e_{h}f_{\delta}(t)\|_{L^{p}}\leq\|\nabla f_{\delta}(t)\|_{L^{p}}$
and, thanks to (4.36), (4.27), and (4.30),

$\sup_{0<s<t}\|\nabla f_{\delta}(s)\|_{L^{p}}<\infty\,,\qquad\forall
t\in(0,\tau_{*})\,,$

we conclude that, for every $(a,b)\subset(0,t)$, $t<\tau_{*}$, and $p\geq 2$,

$\int_{a}^{b}\,dt\int_{\mathbb{R}^{n}}|\partial_{t}u_{2,h}(t)|^{p}+|\nabla^{2}u_{2,h}(t)|^{p}\leq
C(p)(b-a)\,\sup_{a<s<b}\|\nabla f_{\delta}(s)\|_{L^{p}}^{p}\,,$ (4.41)

so that, by arbitrariness of $(a,b)$, for every
$(a,b)\subset\subset(0,\tau_{*})$ and $p\geq 2$,

$\sup_{a<s<b}\big{\\{}\|\partial_{t}u_{2,h}(s)\|_{L^{p}},\|\nabla^{2}u_{2,h}(s)\|_{L^{p}}\big{\\}}\leq
C(p)\,\sup_{a<s<b}\|\nabla f_{\delta}(s)\|_{L^{p}}\,.$ (4.42)

At the same time, recalling that $u_{1,h}(s)=S_{s}(e_{h}u_{0})$, by (4.28) we
find that for every $s>0$

$\displaystyle\|\partial_{t}u_{1,h}(s)\|_{L^{p}}\leq\|\partial_{t}G(s)\|_{L^{1}}\,\|e_{h}u_{0}\|_{L^{p}}\leq\frac{C}{s}\,\|\nabla
u_{0}\|_{L^{p}}\,,$
$\displaystyle\|\nabla^{2}u_{1,h}(s)\|_{L^{p}}\leq\|\nabla^{2}G(s)\|_{L^{1}}\,\|e_{h}u_{0}\|_{L^{p}}\leq\frac{C}{s^{3/2}}\,\|\nabla
u_{0}\|_{L^{p}}\,.$

By combining these last two estimates with (4.42) and
$e_{h}u=u_{1,h}+u_{2,h}$, by the uniformity in $h>0$ and $e$ with $|e|=1$, we
conclude that for every $p\geq 2$ and $(a,b)\subset\subset(0,\tau_{*})$ we
have

$\displaystyle\sup_{a<s<b}\big{\\{}\|\nabla(\partial_{t}u)(s)\|_{L^{p}},\|\nabla^{3}u(s)\|_{L^{p}}\big{\\}}\leq
C(p)\sup_{a<s<b}\Big{\\{}\frac{\|\nabla u_{0}\|_{L^{p}}}{a^{3/2}},\|\nabla
f_{\delta}(s)\|_{L^{p}}\Big{\\}}\,.$ (4.43)

In combination with (4.32), (4.37), (4.35), and (4.27) this implies that, for
every $p\geq 2$ and $(a,b)\subset\subset(0,\tau_{*})$ we have

$\displaystyle\sup_{a<t<b}\big{\\{}\|\partial_{t}u(t)\|_{W^{1,p}},\|u(t)\|_{W^{3,p}}\big{\\}}<\infty\,.$
(4.44)

In particular, $u(t)\in C^{2,\alpha}(\mathbb{R}^{n})$ and $\partial_{t}u(t)\in
C^{0,\alpha}(\mathbb{R}^{n})$ for every $\alpha\in(0,1)$ and
$t\in(0,\tau_{*})$.

The regularity of $u(t)$ established in (4.44) is sufficient to prove that

$\displaystyle\mathcal{V}_{\delta}(u(t))=\mathcal{V}_{\delta}(u_{0})\,,\qquad\forall
t\in[0,\tau_{*})\,,$ (4.45) $t\mapsto\mathcal{AC}_{\varepsilon}(u(t))$ is
decreasing, continuous on $[0,\tau_{*})$, (4.46) $\displaystyle\hskip
73.97733pt\mbox{and locally Lipschitz on $(0,\tau_{*})$}\,.$

Indeed, let us set $u_{\eta}=(\psi_{\eta}\,u)\star\rho_{\eta}$, where, for
each $\eta>0$, $\rho_{\eta}$ is a mollifying kernel on
$\mathbb{R}^{n}\times\mathbb{R}$, and where $\psi_{\eta}\in
C^{\infty}_{c}(\mathbb{R}^{n}\times(0,\infty))$ is monotonically converging to
$1$ as $\eta\to 0^{+}$. If we set $D(t)=\int_{\mathbb{R}^{n}}|\nabla
u(t)|^{2}$ and $D_{\eta}(t)=\int_{\mathbb{R}^{n}}|\nabla u_{\eta}(t)|^{2}$,
then by (4.30) we find that, for every $\varphi\in C^{\infty}_{c}(0,\tau)$,

$\displaystyle\int_{0}^{\tau}D(t)\,\varphi^{\prime}(t)\,dt=\lim_{\eta\to
0^{+}}\int_{0}^{\tau}\,D_{\eta}(t)\,\varphi^{\prime}(t)\,dt$
$\displaystyle=-2\,\lim_{\eta\to
0^{+}}\int_{0}^{\tau}\varphi(t)\,dt\int_{\mathbb{R}^{n}}\nabla
u_{\eta}(t)\cdot\nabla(\partial_{t}u_{\eta}(t))$
$\displaystyle=-2\,\int_{0}^{\tau}\,\varphi(t)\,dt\int_{\mathbb{R}^{n}}\,\nabla
u(t)\cdot\nabla(\partial_{t}u(t))=2\,\int_{0}^{\tau}\,\varphi(t)\,dt\int_{\mathbb{R}^{n}}\Delta
u(t)\,\partial_{t}u(t)\,,$

where, in computing the second limit, we have used (4.44). We have thus proven
that $D$ is locally Lipschitz continuous on $(0,\tau_{*})$, with

$D^{\prime}(t)=\frac{d}{dt}\int_{\mathbb{R}^{n}}|\nabla
u(t)|^{2}=-\int_{\mathbb{R}^{n}}2\,\Delta u(t)\,\partial_{t}u(t)\,.$ (4.47)

for a.e. $t\in(0,\tau_{*})$. By an analogous approximation argument we see
that, for a.e. $t\in(0,\tau_{*})$,

$\frac{d}{dt}\int_{\mathbb{R}^{n}}W(u(t))=\int_{\mathbb{R}^{n}}W^{\prime}(u)\,\partial_{t}u\,,\qquad\frac{d}{dt}\int_{\mathbb{R}^{n}}V_{\delta}(u(t))=\int_{\mathbb{R}^{n}}V_{\delta}^{\prime}(u)\,\partial_{t}u\,.$
(4.48)

In particular, by (4.34), for a.e. $t\in(0,\tau_{*})$,

$\displaystyle\int_{\mathbb{R}^{n}}V_{\delta}^{\prime}(u)\,\partial_{t}u=\int_{\mathbb{R}^{n}}\Big{(}2\,\Delta
u-\frac{W^{\prime}(u)}{\varepsilon^{2}}+\lambda_{\varepsilon,\delta}[u(t)]\,\frac{V_{\delta}^{\prime}(u)}{\varepsilon}\Big{)}\,V_{\delta}^{\prime}(u)$
$\displaystyle=-\int_{\mathbb{R}^{n}}2\,\nabla
u\cdot\nabla(V_{\delta}^{\prime}(u))-\frac{1}{\varepsilon^{2}}\int_{\mathbb{R}^{n}}W^{\prime}(u)\,V_{\delta}^{\prime}(u)+\frac{\lambda_{\varepsilon,\delta}[u(t)]}{\varepsilon}\,\int_{\mathbb{R}^{n}}V_{\delta}^{\prime}(u)^{2}=0\,,$

where the last identity follows from the definition (3.2) of
$\lambda_{\varepsilon,\delta}[u(t)]$. This proves that
$\mathcal{V}_{\delta}(u(t))$ is constant on $(0,\tau_{*})$, and since
$t\mapsto\mathcal{V}_{\delta}(u(t))$ is continuous on $[0,\tau_{*})$, we
deduce (4.45). Finally, by (4.47) and (4.48) we have that
$t\mapsto\mathcal{AC}_{\varepsilon}(u(t))$ is locally Lipschitz continuous on
$(0,\tau_{*})$ with

$\displaystyle\frac{d}{dt}\mathcal{AC}_{\varepsilon}(u(t))$ $\displaystyle=$
$\displaystyle\varepsilon\,\int_{\mathbb{R}^{n}}\partial_{t}u\,\Big{\\{}-2\,\Delta
u+\frac{W^{\prime}(u)}{\varepsilon^{2}}\Big{\\}}$ $\displaystyle=$
$\displaystyle\varepsilon\,\int_{\mathbb{R}^{n}}\partial_{t}u\,\Big{\\{}-\partial_{t}u+\lambda_{\varepsilon,\delta}[u(t)]\,\frac{V_{\delta}^{\prime}(u)}{\varepsilon}\Big{\\}}=-\varepsilon\,\int_{\mathbb{R}^{n}}(\partial_{t}u)^{2}\,.$

where we have taken into account
$\int_{\mathbb{R}^{n}}V_{\delta}^{\prime}(u)\,\partial_{t}u=0$. The continuity
of $t\mapsto\mathcal{AC}_{\varepsilon}(u(t))$ on $[0,\tau_{*})$ is of course
immediate from $u\in C^{0}([0,\tau_{*});X)$. This proves (4.46).

We now prove that $\tau_{*}=+\infty$. We argue by contradiction, and assume
that $\tau_{*}<\infty$. By combining (4.45) and (4.46) (which implies
$\mathcal{AC}_{\varepsilon}(u(t))\leq\mathcal{AC}_{\varepsilon}(u_{0})$ for
every $t\in(0,\tau_{*})$) with (3.3) and (3.5) we deduce that, for every
$p\geq 2$,

$\displaystyle\sup_{0<t<\tau_{*}}\big{\\{}\|u(t)\|_{L^{p}},|\lambda_{\varepsilon,\delta}[u(t)]|\big{\\}}\leq
C(\varepsilon)\,M\big{(}\varepsilon,\mathcal{AC}_{\varepsilon}(u_{0}),\mathcal{V}_{\delta}(u_{0}),1/\mathcal{V}_{\delta}(u_{0})\big{)}\,.$
(4.49)

Then, by (4.49) and (4) we find that, for every $p\geq 2$,

$\sup_{0<t<\tau_{*}}\|\nabla u(t)\|_{L^{p}}\leq
C(\varepsilon)\,M\big{(}\mathcal{AC}_{\varepsilon}(u_{0}),\mathcal{V}_{\delta}(u_{0}),1/\mathcal{V}_{\delta}(u_{0}),\|\nabla
u_{0}\|_{L^{p}},\tau_{*}\big{)}\,,$ (4.50)

which, combined with (4) gives that, for every $p\geq 2$,

$\sup_{0<t<\tau_{*}}\|\nabla^{2}u(t)\|_{L^{p}}\leq
C(\varepsilon,p)\,M\big{(}\varepsilon,\mathcal{AC}_{\varepsilon}(u_{0}),\mathcal{V}_{\delta}(u_{0}),1/\mathcal{V}_{\delta}(u_{0}),\|\nabla
u_{0}\|_{W^{1,p}},\tau_{*}\big{)}\,.$ (4.51)

By combining (4.35) and (4.36) with (4.49) and (4.50) we find that, for every
$p\geq 2$,

$\sup_{0<t<\tau_{*}}\|f_{\delta}(t)\|_{W^{1,p}}\leq
C(\varepsilon)\,M\big{(}\varepsilon,\mathcal{AC}_{\varepsilon}(u_{0}),\mathcal{V}_{\delta}(u_{0}),1/\mathcal{V}_{\delta}(u_{0}),\|\nabla
u_{0}\|_{L^{p}},\tau_{*}\big{)}\,,$ (4.52)

so that by (4.37) and (4.43) with (4.51) and (4.52) we find

$\displaystyle\sup_{\tau_{*}/2<t<\tau_{*}}\big{\\{}\|\nabla^{3}u(t)\|_{L^{p}},\|\partial_{t}u(t)\|_{W^{1,p}}\big{\\}}$
$\displaystyle\hskip 85.35826pt\leq
C(\varepsilon,p)\,M\big{(}\mathcal{AC}_{\varepsilon}(u_{0}),\mathcal{V}_{\delta}(u_{0}),1/\mathcal{V}_{\delta}(u_{0}),\|\nabla
u_{0}\|_{W^{2,p}},\tau_{*},1/\tau_{*}\big{)}\,,$

which combined with (4.49), (4.50) and (4.51) finally gives, for every $p\geq
2$,

$\displaystyle\sup_{\tau_{*}/2<t<\tau_{*}}\big{\\{}\|u(t)\|_{W^{3,p}},\|\partial_{t}u(t)\|_{W^{1,p}}\big{\\}}$
(4.53) $\displaystyle\hskip 56.9055pt\leq
C(\varepsilon,p)\,M\big{(}\mathcal{AC}_{\varepsilon}(u_{0}),\mathcal{V}_{\delta}(u_{0}),1/\mathcal{V}_{\delta}(u_{0}),\|\nabla
u_{0}\|_{W^{2,p}},\tau_{*},1/\tau_{*}\big{)}\,.$

By the $W^{1,2}$-estimate on $\partial_{t}u$ contained in (4.53), we can
deduce that for every $t,s\in(\tau_{*}/2,\tau_{*})$ it holds

$\|u(t)-u(s)\|_{X}\leq C(\varepsilon,\delta,u_{0},\tau_{*})\,|t-s|\,.$ (4.54)

Since $X$ is a Banach space, this means that there is $u_{*}\in X=C^{0}\cap
W^{1,2}(\mathbb{R}^{n};[0,1])$ such that

$\lim_{t\to\tau_{*}^{-}}\|u(t)-u_{*}\|_{X}=0\,.$

By combining this last fact with (4.53) we see that $u_{*}\in
W^{3,p}(\mathbb{R}^{n})$ for every $p\geq 2$, with

$\mathcal{V}_{\delta}(u_{*})=\lim_{t\to\tau_{*}^{-}}\mathcal{V}_{\delta}(u(t))=\mathcal{V}_{\delta}(u_{0})>0\,.$

Therefore, for some $\eta>0$, we can repeat the argument of step five to
extend $u$ as an element of $C^{0}([0,\tau_{*}+\eta);X)$ such that (4.9) holds
for every $t\in(0,\tau_{*}+\eta)$. We can also repeat the proof of (4.45) and
(4.46) to show that $\mathcal{AC}_{\varepsilon}(u(t))$ is decreasing on
$[\tau_{*},\tau_{*}+\eta)$ and $\mathcal{V}_{\delta}(u(t))$ is constant on
$t\in[\tau_{*},\tau_{*}+\eta)$. Since $\tau_{*}$ was introduced as the
supremum of those $\tau>0$ such that there exists $u\in C^{0}([0,\tau);X)$
with $u(0)=u_{0}$, solving (4.9) for every $t\in(0,\tau)$, and such that

$\sup_{0<t<\tau}\big{\\{}\mathcal{AC}_{\varepsilon}(u(t)),\mathcal{V}_{\delta}(u(t))\big{\\}}<\infty\,,\qquad\inf_{0<t<\tau}\mathcal{V}_{\delta}(u(t))>0\,.$
(4.55)

and since we have just proved that the bounds stated in (4.55) holds with
$\tau=\tau_{*}+\eta$, thus reaching a contradiction with the maximality of
$\tau_{*}$. This proves that $\tau_{*}=+\infty$. The monotonicity of
$\mathcal{AC}_{\varepsilon}(u(t))$ and constancy of
$\mathcal{V}_{\delta}(u(t))$ then hold on $[0,\infty)$, and the validity of
(4.26) is deduced by arguing as in the proof of (4.53).

Step three: We prove, for an initial datum $u_{0}$ as in (4.4), the existence
of $u\in C^{0}(\mathbb{R}^{n}\times[0,\infty))$ with $u(0)=u_{0}$ which is a
classical solution of (DF) on $\mathbb{R}^{n}\times(0,\infty)$, with
$\mathcal{V}(u(t))=\mathcal{V}(u_{0})=1$ for every $t\geq 0$ and such that
$t\mapsto\mathcal{AC}_{\varepsilon}(u(t))$ is continuous and decreasing on
$[0,\infty)$.

Given $\varepsilon>0$ and $\delta\in(0,\delta_{0}]$, let us denote by
$u^{\delta}$ the unique global solution of (DFδ) with $u^{\delta}(0)=u_{0}$
constructed in step six. Recalling that $u^{\delta}$ satisfies (4.26), and
keeping in mind that $\mathcal{V}_{\delta}(u_{0})\to\mathcal{V}(u_{0})=1$ by
(4.5), up to make $\delta_{0}$ depend on $u_{0}$ too, we can ensure that
$1/2\leq\mathcal{V}_{\delta}(u_{0})\leq 2$ for every
$\delta\in(0,\delta_{0}]$, and thus deduce from (4.26) that for every
$(a,b)\subset(0,\infty)$ and $p\geq 2$, uniformly on
$\delta\in(0,\delta_{0}]$,

$\displaystyle\sup_{a<s<b}\big{\\{}\|u^{\delta}(s)\|_{W^{3,p}},\|\partial_{t}u^{\delta}(s)\|_{W^{1,p}}\big{\\}}\leq
C\big{(}\varepsilon,p,\mathcal{AC}_{\varepsilon}(u_{0}),\|\nabla
u_{0}\|_{W^{2,p}},a,b\big{)}\,.\hskip 28.45274pt$ (4.56)

By Morrey’s embedding theorem we can obtain a $C^{0,1/2}$-estimate on
$\partial_{t}u^{\delta}$ from (4.56), which combined with the
$W^{1,2}$-estimate on $\partial_{t}u^{\delta}$ contained in (4.56) leads to
prove that, for every $(a,b)\subset(0,\infty)$,

$\displaystyle\|u^{\delta}(r)-u^{\delta}(s)\|_{X}\leq
C(\varepsilon,u_{0},a,b)\,|r-s|,\qquad\forall r,s\in(a,b)\,,$ (4.57)

uniformly on $\delta\in(0,\delta_{0}]$.

If we now consider a sequence $\delta_{j}\to 0^{+}$, then, up to extracting a
subsequence, we deduce from (4.57) and (4.56) that there is $u\in{\rm
Lip}_{\rm loc}((0,\infty);X)$ with

$\displaystyle\sup_{r,s\in(a,b)}\frac{\|u(r)-u(s)\|_{X}}{|r-s|}\\!\\!$
$\displaystyle\leq$ $\displaystyle\\!\\!C(\varepsilon,u_{0},a,b)\,,$ (4.58)
$\displaystyle\sup_{a<s<b}\big{\\{}\|u(s)\|_{W^{3,p}},\|\partial_{t}u(s)\|_{W^{1,p}}\big{\\}}\\!\\!$
$\displaystyle\leq$
$\displaystyle\\!\\!C(\varepsilon,p,\mathcal{AC}_{\varepsilon}(u_{0}),\|\nabla
u_{0}\|_{W^{2,p}},a,b\big{)}\,,$ (4.59)

for every $p\geq 2$ and $(a,b)\subset(0,\infty)$, and such that

$\displaystyle\lim_{j\to\infty}\sup_{a<s<b}\big{\\{}\|u^{\delta_{j}}(s)-u(s)\|_{W^{2,p}},\|\partial_{t}u^{\delta_{j}}(s)-\partial_{t}u(s)\|_{L^{p}\cap
C^{0}}\big{\\}}=0\,,$ (4.60)

for every $p\geq 2$ and $(a,b)\subset(0,\infty)$. By (4.59), $u(t)\in
C^{2,\alpha}(\mathbb{R}^{n})$, $\partial_{t}u(t)\in
C^{0,\alpha}(\mathbb{R}^{n})$ for every $\alpha\in(0,1)$ and $t>0$. By (4.60),
and since

$\varepsilon^{2}\,\partial_{t}u^{\delta_{j}}=2\varepsilon^{2}\,\Delta
u^{\delta_{j}}-W^{\prime}(u^{\delta_{j}})+\varepsilon\,\lambda_{\varepsilon,\delta_{j}}[u^{\delta_{j}}(t)]\,V^{\prime}_{\delta_{j}}(u^{\delta_{j}})\,,$

holds in classical sense on $\mathbb{R}^{n}\times(0,\infty)$, we can deduce
that

$\varepsilon^{2}\,\partial_{t}u=2\,\varepsilon^{2}\,\Delta
u-W^{\prime}(u)+\varepsilon\,\lambda_{\varepsilon}[u(t)]\,V^{\prime}(u)\,,$
(4.61)

also holds in classical sense on on $\mathbb{R}^{n}\times(0,\infty)$, provided
we show that

$\lim_{j\to\infty}\lambda_{\varepsilon,\delta_{j}}[u^{\delta_{j}}(t)]=\lambda_{\varepsilon}[u(t)]\,,\qquad\forall
t>0\,.$ (4.62)

To prove (4.62) we first notice that, by $0\leq V_{\delta}(r)\leq C\,r^{2}$
for $r\in[0,1]$ any by dominated convergence, for every $t>0$ it holds that
$\mathcal{V}_{\delta_{j}}(u^{\delta_{j}}(t))\to\mathcal{V}(u(t))$ as
$j\to\infty$. At the same time, for every $t>0$,
$\mathcal{V}_{\delta_{j}}(u^{\delta_{j}}(t))=\mathcal{V}_{\delta_{j}}(u_{0})\to\mathcal{V}(u_{0})$
as $j\to\infty$ so that, in summary,

$\mathcal{V}(u(t))=\mathcal{V}(u_{0})=1\,,\qquad\forall t>0\,.$ (4.63)

Now, by (3.7), (3.8),
$\mathcal{V}_{\delta}(u^{\delta}(s))=\mathcal{V}_{\delta}(u_{0})$, (4.63), and
(4.59), we have that

$\displaystyle|\lambda_{\varepsilon,\delta}[u^{\delta}(t)]-\lambda_{\varepsilon,\delta}[u(t)]|\\!\\!$
$\displaystyle\leq$
$\displaystyle\\!\\!C(\varepsilon,u_{0},b)\,\frac{\|u^{\delta}(t)-u(t)\|_{X}^{\gamma(n)}}{\min\\{1,\mathcal{V}_{\delta}(u_{0})^{2n}\\}\,\min\\{1,\mathcal{V}_{\delta}(u(t))\\}^{2n}}\,,$
$\displaystyle|\lambda_{\varepsilon,\delta}[u(t)]-\lambda_{\varepsilon}[u(t)]|\\!\\!$
$\displaystyle\leq$
$\displaystyle\\!\\!C(\varepsilon,u_{0},a,b)\,\frac{\|V_{\delta}-V\|_{C^{2}[0,1]}}{\min\\{1,\mathcal{V}_{\delta}(u(t))^{2n}\\}}\,,$

for all $t\in(a,b)\subset(0,\infty)$. Since
$\mathcal{V}_{\delta}(u(t))\in(0,\infty)$ for every $t>0$ and
$\delta\in(0,\delta_{0}]$, by letting $\delta=\delta_{j}$ and $j\to\infty$ in
the above two estimates we obtain (4.62), and thus (4.61). The constancy of
$\mathcal{V}(u(t))$ and the monotonicity of $\mathcal{AC}_{\varepsilon}(u(t))$
are then immediate to prove.

We are left to prove that $u(0)=u_{0}$. To begin with, we notice that, by
(3.5), (4.45) and (4.46), for all $t>0$ we have

$|\lambda_{\varepsilon,\delta}[u^{\delta}(t)]|\leq
C(\varepsilon)\,\frac{\mathcal{AC}_{\varepsilon}(u^{\delta}(t))^{2n+2}}{\mathcal{V}_{\delta}(u^{\delta}(t))^{2n}}\leq
C(\varepsilon)\,\frac{\mathcal{AC}_{\varepsilon}(u_{0})^{2n+2}}{\mathcal{V}_{\delta}(u_{0})^{2n}}\,,$

so that (4.62) and (4.5) imply

$|\lambda_{\varepsilon}[u(t)]|\leq C(\varepsilon,u_{0})\,,\qquad\forall
t>0\,.$ (4.64)

Next, if we set

$\displaystyle F[u](t)$ $\displaystyle=$
$\displaystyle-\frac{W^{\prime}(u(t))}{\varepsilon^{2}}+\lambda_{\varepsilon}[u(t)]\,\frac{V^{\prime}(u(t))}{\varepsilon}\,,$
(4.65) $\displaystyle T[u](t)$ $\displaystyle=$ $\displaystyle
S_{t}u_{0}+\int_{0}^{t}S_{t-s}[F[u](s)]\,ds\,,$ (4.66)

then (4.61) implies $u(t)=T[u(t)]S_{t}u_{0}+\int_{0}^{t}S_{t-s}[F[u](s)]\,ds$
for every $t>0$, and by the contraction properties of the heat flow, (4.16),
(4.17), (4.18), and (4.64), we find that

$\displaystyle\|u(t)-u_{0}\|_{X}$ $\displaystyle\leq$
$\displaystyle\|S_{t}u_{0}-u_{0}\|_{X}+C(\varepsilon)\,\int_{0}^{t}\big{(}1+|\lambda_{\varepsilon}[u(s)]\big{)}\,\|u(s)\|_{X}\,ds$
$\displaystyle\leq$
$\displaystyle\|S_{t}u_{0}-u_{0}\|_{X}+C(\varepsilon,u_{0})\,t\,\sup_{0<s<t}\|u(s)\|_{X}\,.$

By $u_{0}\in X$ and (4.59) we find that $\|u(t)-u_{0}\|_{X}\to 0$ as $t\to
0^{+}$, and thus that $u\in C^{0}(\mathbb{R}^{n}\times[0,\infty))$ with
$u(0)=u_{0}$, as claimed. This completes the proof of step seven.

Step four: We now prove that

$|\lambda_{\varepsilon}[u(t)]|\leq
C\,\mathcal{AC}_{\varepsilon}(u_{0})^{2n+2}\,,\qquad\forall t>0\,,$ (4.67)

and that, for every $t_{0}>0$ and $p\geq 2$ we have

$\displaystyle\sup_{t\geq
0}\big{\\{}\|u(t)\|_{W^{2,p}},\|\partial_{t}u(t)\|_{L^{p}}\big{\\}}\leq
C(\varepsilon,p,u_{0})\,,$ (4.68) $\displaystyle\sup_{t\geq
t_{0}}\max\big{\\{}\|u(t)\|_{W^{3,p}},\|\partial_{t}u(t)\|_{W^{2,p}},\|\partial_{tt}u(t)\|_{L^{p}}\big{\\}}\leq
C(\varepsilon,p,u_{0},t_{0})\,.$ (4.69)

Indeed, recalling that in step seven we have proved $\mathcal{V}(u(t))=1$ and
$\mathcal{AC}_{\varepsilon}(u(t))\leq\mathcal{AC}_{\varepsilon}(u_{0})$ for
every $t\geq 0$, and recalling that the constant $C(\varepsilon)$ in (3.5) can
be taken independent from $\varepsilon$ when (3.5) is applied with $\delta=0$,
we find that, for all $t>0$,

$|\lambda_{\varepsilon}[u(t)]|\leq
C\,\frac{\mathcal{AC}_{\varepsilon}(u(t))^{2n+1}}{\mathcal{V}(u(t))^{2n}}\,\int_{\mathbb{R}^{n}}|\nabla
u|^{2}\leq\frac{C}{\varepsilon}\,\mathcal{AC}_{\varepsilon}(u_{0})^{2n+2}\,,$

that is (4.67). Similarly, we deduce from (3.3) (with $\delta=0$) and $0\leq
u\leq 1$ that

$\sup_{t\geq 0}\|u(t)\|_{L^{p}}\leq C(\varepsilon,u_{0})\,,\qquad\forall p\geq
2\,,$ (4.70)

while by combining (4) with (4.67) and (4.70) we get

$\sup_{0\leq t\leq 1}\|\nabla u(t)\|_{L^{p}}\leq
C(\varepsilon,p,u_{0})\,,\qquad\forall p\geq 2\,.$ (4.71)

Now, using the semigroup property of the heat flow we see that for every
$t>s\geq 0$ we have

$u(t)=S_{t-s}u(s)+\int_{s}^{t}S_{t-r}\big{[}F[u](r)\big{]}\,dr\,.$ (4.72)

By differentiating (4.72), and by using (4.28), (4.67) and (4.70), we find
that

$\displaystyle\|\nabla u(t)\|_{L^{p}}$ $\displaystyle\leq$
$\displaystyle\|\nabla
G(t-s)\|_{L^{1}}\,\|u(s)\|_{L^{p}}+\sup_{s<r<t}\|F[u](r)\|_{L^{p}}\,\int_{s}^{t}\|\nabla
G(t-r)\|_{L^{1}}\,dr$ (4.73) $\displaystyle\leq$
$\displaystyle\frac{C(\varepsilon,u_{0})}{(t-s)^{1/2}}+C(\varepsilon,u_{0})\,(t-s)^{1/2}\,,$

where we have used the analog to (4.13) with $\delta=0$ in estimating
$\|F[u](r)\|_{L^{p}}$. If $t\geq 1$, then we can apply (4.73) with $s=t-1\geq
0$ to deduce that $\|\nabla u(t)\|_{L^{p}}\leq C(\varepsilon,u_{0})$, which,
combined with (4.71), gives

$\sup_{t\geq 0}\|\nabla u(t)\|_{L^{p}}\leq
C(\varepsilon,p,u_{0})\,,\qquad\forall p\geq 2\,.$ (4.74)

Similarly, combining (4) with (4.67) and (4.74) we see that, on the one hand

$\sup_{0\leq t\leq 1}\|\nabla^{2}u(t)\|_{L^{p}}\leq
C(\varepsilon,p,u_{0})\,,\qquad\forall p\geq 2\,;$ (4.75)

on other hand, using again (4.72), (4.28), and (4.67), this time in
combination with (4.70), (4.15), and (4.74), we find that, if $t>s\geq 0$,
then

$\displaystyle\|\nabla^{2}u(t)\|_{L^{p}}$ $\displaystyle\leq$
$\displaystyle\|\nabla^{2}G(t-s)\|_{L^{1}}\,\|u(s)\|_{L^{p}}+\sup_{s<r<t}\|\nabla
F[u](r)\|_{L^{p}}\,\int_{s}^{t}\|\nabla G(t-r)\|_{L^{1}}\,dr$
$\displaystyle\leq$
$\displaystyle\frac{C(\varepsilon,u_{0})}{(t-s)^{3/2}}+C(\varepsilon,u_{0})\,(t-s)^{1/2}\,,$

and using this last estimate for $t\geq 1$ (and with $s=t-1\geq 0$), we find
$\|\nabla^{2}u(t)\|_{L^{p}}\leq C(\varepsilon,u_{0})$ for every $t\geq 1$ and
$p\geq 2$. By combining this last fact with (4.75) we have thus proved

$\sup_{t>0}\big{\\{}\|u(t)\|_{W^{2,p}},\|\partial_{t}u(t)\|_{L^{p}}\big{\\}}\leq
C(\varepsilon,p,u_{0})\,,\qquad\forall p\geq 2\,.$ (4.76)

Here the $L^{p}$-estimate on $\partial_{t}u(t)$ has been obtained by combining
the $W^{2,p}$-estimate for $u(t)$ with $\partial_{t}u=2\,\Delta u+f$, where
$f(t)=F[u(t)]$, and thus $\|f(t)\|_{L^{p}}\leq C(\varepsilon,p,u_{0})$ thanks
to (4.13) and (4.70). In fact, thanks also to (4.15), (4.67), and (4.74) for
every $p\geq 2$, we have

$\sup_{t\geq 0}\|f(t)\|_{W^{1,p}}\leq C(\varepsilon,p,u_{0})\,.$

We can now repeat the argument based on the incremental ratios method and on
the parabolic Calderon–Zygmund theorem used in the proof of (4.43) to deduce
that

$\sup_{t>t_{0}}\big{\\{}\|\nabla(\partial_{t}u)(t)\|_{L^{p}},\|\nabla^{3}u(t)\|_{L^{p}}\big{\\}}\leq
C(p)\,\max\Big{\\{}\frac{\|\nabla
u_{0}\|_{L^{p}}}{t_{0}^{3/2}},\sup_{t>t_{0}}\|\nabla
f(t)\|_{L^{p}}\Big{\\}}\,,$

and conclude, in summary, that, for every $p\geq 2$ and $t_{0}>0$,

$\sup_{t>t_{0}}\big{\\{}\|u(t)\|_{W^{3,p}},\|\partial_{t}u(t)\|_{W^{1,p}}\big{\\}}\leq
C(\varepsilon,p,u_{0},t_{0})\,.$ (4.77)

The $W^{1,2}$-estimate on $\partial_{t}u(t)$ allows one to deduce by
elementary means that

$\|u(t)-u(s)\|_{W^{1,2}(\mathbb{R}^{n})}\leq
C(\varepsilon,u_{0},t_{0})\,|t-s|\,,\qquad\forall t,s>t_{0}\,.$ (4.78)

Combining (4.78) with (3.9) (with $\delta=0$) and (4.76) with $p=2$ we
conclude that

$|\lambda_{\varepsilon}[u(t)]-\lambda_{\varepsilon}[u(s)]|\leq
C(\varepsilon,u_{0},t_{0})\,|t-s|\,,\qquad\forall t,s>t_{0}\,,$ (4.79)

that is $t\mapsto\lambda_{\varepsilon}[u(t)]$ is Lipschitz continuous on
$(t_{0},\infty)$ for every $t_{0}>0$.

To obtain $L^{p}$-estimates for $\partial_{tt}u(t)$ and
$\nabla^{2}(\partial_{t}u)$ we need to differentiate $\partial_{t}u=2\,\Delta
u+f$ in time. To this end, given $t_{0}>0$, we introduce that incremental
ratio operator $T_{h}$ that acts on $v=v(x,t)$ by taking
$T_{h}v(t)=(v(t_{0}+t+h)-v(t_{0}+t))/h$ for every $t\geq 0$ and
$h\in(-t_{0},t_{0})\setminus\\{0\\}$. With this notation,
$\partial_{t}u=2\,\Delta u+f$ on $\mathbb{R}^{n}\times(0,\infty)$ implies that

$\displaystyle\partial_{t}(T_{h}u)-2\,\Delta(T_{h}u)=T_{h}f\,,\qquad\mbox{on
$\mathbb{R}^{n}\times(0,\infty)$}\,,$ (4.80) $\displaystyle
T_{h}u(0)=\frac{u(t_{0}+h)-u(t_{0})}{h}\,,\qquad\mbox{on $\mathbb{R}^{n}$}\,.$

Setting

$u_{1,h}=S_{t}\Big{[}\frac{u(t_{0}+h)-u(t_{0})}{h}\Big{]}\,,\qquad
u_{2,h}=T_{h}u-u_{1,h}\,,$

we find that $u_{2,h}$ satisfies

$\partial_{t}u_{2,h}-2\,\Delta u_{2,h}=T_{h}f\,,\qquad\mbox{on
$\mathbb{R}^{n}\times(0,\infty)$}\,,\qquad\mbox{$u_{2,h}(0)=0$ on
$\mathbb{R}^{n}$}\,.$ (4.81)

By [Lie96, Corollary 7.31], for every $(a,b)\subset(0,\infty)$ we have

$\displaystyle\int_{a}^{b}\,dt\int_{\mathbb{R}^{n}}|\nabla^{2}u_{2,h}(t)|^{p}+|\partial_{t}u_{2,h}(t)|^{p}\leq
C(p)\,\int_{a}^{b}\,dt\int_{\mathbb{R}^{n}}|T_{h}f(t)|^{p}\,.$

Setting

$g(t)=\Big{\\{}-\frac{W^{\prime\prime}(u)}{\varepsilon^{2}}+\lambda_{\varepsilon}[u(t)]\,\frac{V^{\prime\prime}(u)}{\varepsilon}\Big{\\}}\,v+\frac{V^{\prime}(u)}{\varepsilon}\,\frac{d}{dt}\,\lambda_{\varepsilon}[u(t)]\,,$
(4.82)

by (4.67) and (4.79) we find that

$\displaystyle\int_{a}^{b}\,dt\,\int_{\mathbb{R}^{n}}|T_{h}f(t)|^{p}\leq\int_{a-|h|}^{b+|h|}dt\,\int_{\mathbb{R}^{n}}|g(t)|^{p}\leq
C(\varepsilon,u_{0},a)\,(b-a+2\,|h|)\,,$

so that, in summary,

$\displaystyle\int_{a}^{b}\,dt\int_{\mathbb{R}^{n}}|\nabla^{2}u_{2,h}(t)|^{p}+|\partial_{t}u_{2,h}(t)|^{p}\leq
C(\varepsilon,p,u_{0},a)\,(b-a+2\,|h|)\,.$ (4.83)

At the same time, thanks to (4.76), we have, for $|h|<t_{0}/2$ and $t>t_{0}$,

$\displaystyle\|\nabla^{2}u_{1,h}(t)\|_{L^{p}}$ $\displaystyle\leq$
$\displaystyle\|\nabla^{2}G(t)\|_{L^{1}}\,\Big{\|}\frac{u(t_{0}+h)-u(t_{0})}{h}\Big{\|}_{L^{p}}$
$\displaystyle\leq$
$\displaystyle\frac{C}{t^{3/2}}\,\sup_{t_{0}-|h|<s<t_{0}+|h|}\|\partial_{t}u(s)\|_{L^{p}}\leq\frac{C(\varepsilon,p,u_{0})}{t_{0}^{3/2}}\,,$
$\displaystyle\|\partial_{t}u_{1,h}(t)\|_{L^{p}}$ $\displaystyle\leq$
$\displaystyle\|\partial_{t}G(t)\|_{L^{1}}\Big{\|}\frac{u(t_{0}+h)-u(t_{0})}{h}\Big{\|}_{L^{p}}\leq\frac{C(\varepsilon,p,u_{0})}{t_{0}}\,,$

and thus, by $T_{h}u=u_{1,h}+u_{2,h}$ and (4.83)

$\displaystyle\int_{a}^{b}\,dt\int_{\mathbb{R}^{n}}|\nabla^{2}(T_{h}u)(t)|^{p}+|\partial_{t}(T_{h}u)(t)|^{p}\leq
C(\varepsilon,p,u_{0},a)\,(b-a+2\,|h|)\,.$ (4.84)

Letting $h\to 0^{+}$ in (4.84) we obtain

$\displaystyle\int_{a}^{b}\,dt\int_{\mathbb{R}^{n}}|\nabla^{2}(\partial_{t}u)(t)|^{p}+|\partial_{tt}u(t)|^{p}\leq
C(\varepsilon,p,u_{0},a)\,(b-a)\,,$

from which we easily conclude that, for every $t_{0}>0$ and $p\geq 2$,

$\sup_{t\geq
t_{0}}\big{\\{}\|\nabla^{2}(\partial_{t}u)(t)\|_{L^{p}},\|\partial_{tt}u(t)\|_{L^{p}}\big{\\}}\leq
C(\varepsilon,p,u_{0},t_{0})\,.$ (4.85)

This last estimate, combined with (4.77) concludes the proof of (4.69).

Step five: We prove the uniqueness of the solution $u$ of (DF) constructed in
step seven. Indeed, let $v$ be another solution of (DF) with $v(0)=u_{0}$. By
combining (3.9) with
$\mathcal{V}(u(t))=\mathcal{V}(v(t))=\mathcal{V}(u_{0})=1$ and with (4.76)
with $p=2$ for both $u$ and $v$, we find that, for all $t>0$,

$|\lambda_{\varepsilon}[u(t)]-\lambda_{\varepsilon}[v(t)]|\leq
C(\varepsilon,u_{0})\,\|u(t)-v(t)\|_{W^{1,2}}\,.$ (4.86)

Since $w=u-v$ satisfies $\partial_{t}w-2\,\Delta w=F[u]-F[v]$ with $w(0)=0$,
by the Duhamel formula (4), for all $t\geq 0$ we can represent $w(t)$ as

$w(t)=\int_{0}^{t}S_{t-s}\big{[}F[u](s)-F[v](s)\big{]}\,ds\,.$ (4.87)

Notice that by using, in the order, (4.67), $0\leq V^{\prime}(r)\leq C\,r$ for
$r\in[0,1]$, and (4.86), we find

$\displaystyle|F[u](s)-F[v](s)|$ $\displaystyle\leq$ $\displaystyle
C(\varepsilon)\,\Big{\\{}{\rm
Lip}(W^{\prime})+|\lambda_{\varepsilon}[u(s)]|\,{\rm
Lip}(V^{\prime})\Big{\\}}\,|w(s)|$
$\displaystyle+V^{\prime}(v(s))\,|\lambda_{\varepsilon}[u(s)]-\lambda_{\varepsilon}[v(s)]|$
$\displaystyle\leq$ $\displaystyle
C(\varepsilon,u_{0})\,\big{\\{}|w(s)|+\|w(s)\|_{W^{1,2}}\,|v(s)|\big{\\}}\,,$

so that, by (4.74) with $p=2$ applied to $v$,

$\|F[u](s)-F[v](s)\|_{L^{2}}\leq
C(\varepsilon,u_{0})\,\|w(s)\|_{W^{1,2}}\,,\qquad\forall s>0\,.$ (4.88)

Setting

$a(t)=\sup_{0<s<t}\|w(s)\|_{W^{1,2}}$

and combining (4.88) with (4.28) we deduce from (4.87) that, if $0<s<t$, then

$\displaystyle\|w(s)\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\int_{0}^{s}\,\|F[u](r)-F[v](r)\|_{L^{2}}\,dr\leq
C(\varepsilon,u_{0})\,\int_{0}^{s}\|w(r)\|_{W^{1,2}}\,dr$ $\displaystyle\leq$
$\displaystyle C(\varepsilon,u_{0})\,t\,a(t)\,,$ $\displaystyle\|\nabla
w(s)\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\sup_{0<r<s}\|F[u](r)-F[v](r)\|_{L^{2}}\,\int_{0}^{s}\|\nabla
G(s-r)\|_{L^{1}}\,ds$ $\displaystyle\leq$ $\displaystyle
C(\varepsilon,u_{0})\,\sqrt{s}\,\sup_{0<r<s}\|w(r)\|_{W^{1,2}}\leq
C(\varepsilon,u_{0})\,\sqrt{t}\,a(t)\,.$

Combining this last two estimates, we find that $a(t)\leq
C(\varepsilon,u_{0})\,\sqrt{t}\,a(t)$ for every $t>0$. In particular, setting
$t_{0}=1/4\,C(\varepsilon,u_{0})^{2}$, if $a(t)>0$ for some $t\in(0,t_{0})$,
then a contradiction follows. We have thus proved that $u(t)=v(t)$ on
$\mathbb{R}^{n}$ for every $t\in(0,t_{0})$. The argument can of course be
iterated (using $t_{0}$ and then integer multiples of $t_{0}$ as initial
times) to prove that $u(t)=v(t)$ on $\mathbb{R}^{n}$ for every $t>0$.

Conclusion: Statement (i) was proved in (1.15) and (1.16). Statement (ii) was
proved in (4.67) and (4.79). Statement (iii) was proved in (4.63). Moreover,
we have

$\displaystyle\frac{d}{dt}\mathcal{AC}_{\varepsilon}(u(t))$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}\Big{\\{}2\,\varepsilon\,\nabla
u\cdot\partial_{t}(\nabla
u)+\frac{W^{\prime}(u)}{\varepsilon}\,\partial_{t}u\Big{\\}}(t)$
$\displaystyle=$
$\displaystyle-\int_{\mathbb{R}^{n}}\partial_{t}u(t)\,\Big{\\{}2\,\varepsilon\,\Delta
u-\frac{W^{\prime}(u)}{\varepsilon}\Big{\\}}(t)$ $\displaystyle=$
$\displaystyle-\int_{\mathbb{R}^{n}}\partial_{t}u(t)\,\Big{\\{}\varepsilon\,\partial_{t}u+\lambda_{\varepsilon}[u(t)]\,V^{\prime}(u)\Big{\\}}(t)$
$\displaystyle=$
$\displaystyle-\varepsilon\,\int_{\mathbb{R}^{n}}(\partial_{t}u(t))^{2}+\lambda_{\varepsilon}[u(t)]\,\int_{\mathbb{R}^{n}}\partial_{t}u(t)\,V^{\prime}(u(t))=-\varepsilon\,\int_{\mathbb{R}^{n}}(\partial_{t}u(t))^{2}\,,$

where we have used the regularity properties of $u$ to differentiate in time
and to apply the divergence theorem, (DF), and, in the last identity,
statement (iii), to deduce that
$0=(d/dt)\mathcal{V}(u(t))=\int_{\mathbb{R}^{n}}\partial_{t}u(t)\,V^{\prime}(u(t))$.
This proves statement (iv).

We prove that $0<u<1$ on $\mathbb{R}^{n}\times(0,\infty)$, i.e. statement (v).
Since $V,W\in C^{2}[0,1]$ with $V^{\prime}(1)=W^{\prime}(1)=0$ and
$|\lambda_{\varepsilon}[u(t)]|\leq C(\varepsilon,u_{0})$ for all $t>0$, we can
find a positive constant $K=K(\varepsilon,u_{0})$ such that $r\mapsto
K\,r-W^{\prime}(r)+\varepsilon\lambda_{\varepsilon}[u(t)]\,V^{\prime}(r)$ is
strictly increasing on $[0,1]$. Correspondingly,

$K\,u-W^{\prime}(u)+\varepsilon\lambda_{\varepsilon}[u(t)]\,V^{\prime}(u)\leq
K\,,\qquad\mbox{on $\mathbb{R}^{n}\times(0,\infty)$}\,,$

which, combined with (DF), implies

$\varepsilon^{2}\partial_{t}u=2\,\varepsilon^{2}\,\Delta
u-W^{\prime}(u)+\varepsilon\,\lambda_{\varepsilon}[u(t)]\,V^{\prime}(u)\leq
2\,\varepsilon^{2}\,\Delta u-K\,u+K$

that is, $v=1-u$ is a non-negative solution of
$\varepsilon^{2}(\partial_{t}-2\,\Delta)v+K\,v\geq 0$ on
$\mathbb{R}^{n}\times(0,\infty)$. By the strong maximum principle, either
$v\equiv 0$ or $v>0$ on $\mathbb{R}^{n}\times(0,\infty)$, where the first
option is excluded a priori since $V(1)=1$ and $\mathcal{V}(u(t))$ is finite
for every $t>0$. We conclude that $u<1$ on $\mathbb{R}^{n}\times(0,\infty)$,
and argue analogously for proving $u>0$ on $\mathbb{R}^{n}\times(0,\infty)$.

We finally prove statement (vi). The argument of step eight shows that
$v=\partial_{t}u$ satisfies $\partial_{t}v-2\,\Delta v=g$ for $g$ as in
(4.82). By testing this equation with $\partial_{t}u$ and integrating by
parts,

$\displaystyle\varepsilon^{2}\,\int_{\mathbb{R}^{n}}\big{(}\partial_{t}u\,\partial_{tt}u\big{)}(t)\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!-2\,\varepsilon^{2}\,\int_{\mathbb{R}^{n}}|\nabla(\partial_{t}u)|^{2}-\int_{\mathbb{R}^{n}}\Big{\\{}W^{\prime\prime}(u)-\varepsilon\,\lambda_{\varepsilon}[u(t)]\,V^{\prime\prime}(u)\Big{\\}}(t)\,(\partial_{t}u(t))^{2}$
(4.89)
$\displaystyle+\varepsilon\,\Big{(}\frac{d}{dt}\,\lambda_{\varepsilon}[u(t)]\Big{)}\int_{\mathbb{R}^{n}}V^{\prime}(u(t))\,\partial_{t}u(t)\,,$

where the last integral is equal to zero since $\mathcal{V}(u(t))=0$ for every
$t>0$. This implies the validity of (1.18). Setting
$b(t)=\int_{\mathbb{R}^{n}}(\partial_{t}u(t))^{2}$, (1.18) combined with
(1.16) implies that $b^{\prime}\in L^{1}(t_{0},\infty)$ for every $t_{0}>0$.
Since (1.17) and the monotonicity of $\mathcal{AC}_{\varepsilon}(u(t))$ imply
$b\in L^{1}(0,\infty)$, we have proved that $b\in W^{1,1}(t_{0},\infty)$ for
every $t_{0}>0$. This completes the proof of the theorem. ∎

##  5\. Subsequential bubbling resolution (Proof of Theorem 1.3)

###### Proof of Theorem 1.3.

Let $u_{0}\in W^{2,p}(\mathbb{R}^{n};[0,1])$ for all $p\geq 2$, with
$\mathcal{V}(u_{0})=1$. By Theorem 1.2, there is a unique solution $u$ to the
diffused VPMCF (DF) with initial datum $u_{0}$, satisfying the various
statements (i)–(vi) listed therein.

Given $\\{t_{j}\\}_{j\in\mathbb{N}}$ with $t_{j}\to\infty$ as $j\to\infty$, we
now want to prove that, up to extracting a subsequence, there are
$M\in\mathbb{N}$, $\ell_{\varepsilon}>0$ such that

$\ell_{\varepsilon}=\lim_{j\to\infty}\lambda_{\varepsilon}[u(t_{j})]\,,$ (5.1)

sequences $\\{x^{i}_{j}\\}_{j\in\mathbb{N}}$ ($i=1,...,M$) satisfying
$|x^{i}_{j}-x^{k}_{j}|\to\infty$ as $j\to\infty$ ($i\neq k$), and strictly
radially decreasing solutions $\\{\xi_{i}\\}_{i=1}^{M}$ of

$2\,\varepsilon^{2}\,\Delta\xi_{i}=W^{\prime}(\xi_{i})-\varepsilon\,\ell_{\varepsilon}\,V^{\prime}(\xi_{i})\quad\mbox{on
$\mathbb{R}^{n}$}\,,$ (5.2)

with

$\displaystyle\sum_{i=1}^{M}\mathcal{V}(\xi_{i})=1\,,\qquad\sum_{i=1}^{M}\mathcal{AC}_{\varepsilon}(\xi_{i})\leq\mathcal{AC}_{\varepsilon}(u_{0})\,.$
(5.3)

such that, for all $p>2$,

$\displaystyle\lim_{j\to\infty}\Big{\|}u(t_{j})-\sum_{i=1}^{M}\tau_{x_{i}^{j}}[\xi_{i}]\Big{\|}_{(W^{2,p}\cap
W^{1,2})(\mathbb{R}^{n})}=0\,.$ (5.4)

This will be proved in step one through five. Finally, in step six, we shall
prove that, if ${\rm spt}\,u_{0}\subset\subset\mathbb{R}^{n}$, then $M=1$ and
$x_{j}^{1}\to x_{*}$ as $j\to\infty$ for some $x_{*}\in\mathbb{R}^{n}$.

Step one: We start by proving that, if $\xi\in
W^{1,2}(\mathbb{R}^{n};[0,1])\setminus\\{0\\}$ and $\ell\in\mathbb{R}$ satisfy

$2\,\varepsilon^{2}\,\Delta\xi=W^{\prime}(\xi)-\varepsilon\,\ell\,V^{\prime}(\xi)\quad\mbox{on
$\mathbb{R}^{n}$},$ (5.5)

then $\ell>0$, $\xi$ is strictly radially decreasing with respect to some
$x_{0}\in\mathbb{R}^{n}$, and

$\mathcal{V}(\xi)\geq\frac{1}{C\,\ell^{n}}\,.$ (5.6)

Indeed, $\xi\in W^{2,p}(\mathbb{R}^{n})$ for every $p\geq 2$ thanks to the
Calderon–Zygmund theorem, $\|\xi\|_{L^{p}}\leq\|\xi\|_{L^{2}}<\infty$ (as
$0\leq\xi\leq 1$ on $\mathbb{R}^{n}$), and
$|W^{\prime}(r)|,|V^{\prime}(r)|\leq C\,t$ for all $r\in[0,1]$. Since $\xi\in
W^{2,p}(\mathbb{R}^{n})$ we have enough regularity to test (5.5) with
$\varphi=X\cdot\nabla\xi$ for $X(x)=\eta(x/R)\,x$, $\eta$ a cut-off function
between $B_{1}$ and $B_{2}$, and $R>0$, and deduce that (see [LM89], or [MR24,
Step five, Proof of Theorem 2.1], for the details)

$n\,\ell\,\mathcal{V}(\xi)=n\,\mathcal{AC}_{\varepsilon}(\xi)-2\,\varepsilon\,\int_{\mathbb{R}^{n}}|\nabla\xi|^{2}\,.$
(5.7)

Since $\xi\neq 0$ and $n\geq 2$, the right-hand side of (5.7) is strictly
positive, thus proving that $\ell>0$. Moreover, since $\xi\in
W^{2,p}(\mathbb{R}^{n})$, we have that $\xi(x)\to 0$ as $|x|\to\infty$, so
that the moving planes method of [GNN81] can be applied (see for example
[MR24, Theorem 6.2-(i)]) to deduce that $\xi$ is strictly radially decreasing
with respect to some $x_{0}\in\mathbb{R}^{n}$.

We finally prove (5.6). If $n\geq 3$, then we can combine (5.7) with (1.5) to
find

$n\,\ell\,\mathcal{V}(\xi)\geq(n-2)\,\mathcal{AC}_{\varepsilon}(\xi)>(n-2)\,n\,\omega_{n}^{1/n}\,\mathcal{V}(\xi)^{(n-1)/n}\,,$

which immediately gives (5.6). In the case $n=2$, we argue as follows. We test
(5.5) with $\varphi_{k}\,\xi$ for $\varphi_{k}\in
C^{\infty}_{c}(B_{k+2};[0,1])$ with $\varphi_{k}=1$ on $B_{k}$ and
$|\nabla\varphi_{k}|\leq 1_{B_{k+2}\setminus B_{k}}$ for each
$k\in\mathbb{N}$. In this way we find

$2\,\varepsilon^{2}\,\int_{\mathbb{R}^{n}}\varphi_{k}\,|\nabla\xi|^{2}\leq
2\,\varepsilon^{2}\,\int_{\mathbb{R}^{n}}\xi\,|\nabla\xi|\,|\nabla\varphi_{k}|+\int_{\mathbb{R}^{n}}\varphi_{k}\,\xi\,W^{\prime}(\xi)+\varepsilon\,\ell\,\varphi_{k}\,\xi\,V^{\prime}(\xi)\,.$

Letting $k\to\infty$ (and using monotone convergence for all the integrals but
the one involving $W^{\prime}$, which may be negative, and is dealt with by
dominated convergence), we find that

$2\,\varepsilon^{2}\,\int_{\mathbb{R}^{n}}|\nabla\xi|^{2}\leq\int_{\mathbb{R}^{n}}\xi\,W^{\prime}(\xi)+\varepsilon\,\ell\,\xi\,V^{\prime}(\xi)\,.$

Thanks to (1.2) there is $\delta_{0}>0$ such that $W^{\prime}(r)\leq 0$ for
all $r\in[1-\delta_{0},1]$ and $r\,|W^{\prime}(r)|\leq C\,r^{2}\leq C\,W(r)$
for all $r\in[0,1]$. Therefore, $r\,W^{\prime}(r)\leq C\,W(r)$ for all
$r\in[0,1]$, and, taking also into account that $r\,V^{\prime}(r)\leq C\,V(r)$
for all $r\in[0,1]$, we deduce that

$2\,\varepsilon^{2}\,\int_{\mathbb{R}^{n}}|\nabla\xi|^{2}\leq
C\,\Big{\\{}\varepsilon\,\ell\,\mathcal{V}(\xi)+\int_{\mathbb{R}^{n}}W(\xi)\Big{\\}}\,.$
(5.8)

Since, when $n=2$, (5.7) boils down to
$\varepsilon\,\ell\,\mathcal{V}(\xi)=\int_{\mathbb{R}^{n}}W(\xi)$, we have
finally proved $C\,\ell\,\mathcal{V}(\xi)\geq\mathcal{AC}_{\varepsilon}(\xi)$.
Again by (1.5), as in the case $n\geq 3$, we deduce (5.6).

Step two: We now begin the proof of (5.1), (5.2), (5.3), and (5.4) by
discussing a compactness argument aimed at “extracting one bubble” from
$u(t_{j})$ as $t_{j}\to\infty$. In later steps we will of course discuss the
iteration of this argument.

We begin by recalling that, by Theorem 1.2-(i,ii), for all $p\geq 2$ and
$\alpha\in(0,1)$ we have

$\displaystyle\sup_{t\geq
0}\big{\\{}\|u(t)\|_{W^{2,p}(\mathbb{R}^{n})},\|u(t)\|_{C^{1,\alpha}(\mathbb{R}^{n})},|\lambda_{\varepsilon}[u(t)]|\big{\\}}<\infty\,,$
(5.9) $\displaystyle\sup_{t\geq
1}\big{\\{}\|u(t)\|_{W^{3,p}(\mathbb{R}^{n})},\|u(t)\|_{C^{2,\alpha}(\mathbb{R}^{n})}\big{\\}}<\infty\,.$
(5.10)

By (5.9), $\sup_{\mathbb{R}^{n}}u(t)$ is achieved for every $t>0$, and we
actually claim that

$\max_{\mathbb{R}^{n}}u(t)\geq\min\Big{\\{}\frac{1}{2},\Big{(}\frac{1}{C\,\varepsilon\,\mathcal{AC}_{\varepsilon}(u_{0})}\Big{)}^{(n-1)/2}\Big{\\}}\,,\qquad\forall
t>0\,.$ (5.11)

Indeed, setting for brevity $\beta(t)=\max_{\mathbb{R}^{n}}u(t)$, if
$\beta(t)\leq 1/2$, then we can use the elementary estimate

$V(r)\leq C\,r^{2n/(n-1)}\leq C\,r^{2/(n-1)}\,r^{2}\leq
C\,r^{2/(n-1)}\,W(r)\,,\qquad\forall r\in[0,1/2]\,,$

to deduce from Theorem 1.2-(iii,iv) that, for all $t>0$,

$\displaystyle 1$ $\displaystyle=$ $\displaystyle\mathcal{V}(u(t))\leq
C\,\int_{\mathbb{R}^{n}}u(t)^{2/(n-1)}\,W(u(t))\leq
C\,\beta(t)^{2/(n-1)}\,\varepsilon\,\mathcal{AC}_{\varepsilon}(u(t))$
$\displaystyle\leq$ $\displaystyle
C\,\beta(t)^{2/(n-1)}\,\varepsilon\,\mathcal{AC}_{\varepsilon}(u_{0})\,.$

This proves (5.11). Combining (5.11) with (5.9) we find
$\beta_{0}=\beta_{0}(\varepsilon,u_{0})\in(0,1/2]$ and
$r_{0}=r_{0}(\varepsilon,u_{0})>0$ such that for each $t>0$ there is
$x_{t}\in\mathbb{R}^{n}$ with the property that

$u(t)\geq\beta_{0}\qquad\mbox{on $B_{r_{0}}(x_{t})$}\,.$ (5.12)

In particular, given $t_{j}\to\infty$ as $j\to\infty$, then, thanks to (5.9),
(5.10) and (5.12), and up to extracting subsequences, we can find
$\ell_{\varepsilon}\in\mathbb{R}$, $\xi_{1}\in\cap_{p\geq
2}W^{3,p}(\mathbb{R}^{n};[0,1])\setminus\\{0\\}$, and a sequence
$(x_{j}^{1})_{j}$ in $\mathbb{R}^{n}$ such that, as $j\to\infty$,

$\displaystyle\lim_{j\to\infty}\lambda_{\varepsilon}[u(t_{j})]=\ell_{\varepsilon}\,,$
(5.13)
$\displaystyle\mbox{$\tau_{(-x_{j}^{1})}[u(t_{j})]\rightharpoonup\xi_{1}$
weakly in $W^{3,p}(\mathbb{R}^{n})$ as $j\to\infty$}\,,$ (5.14)
$\displaystyle\lim_{j\to\infty}\big{\|}\tau_{(-x_{j}^{1})}[u(t_{j})]-\xi_{1}\big{\|}_{W^{2,p}(B_{R})}=0\,,\qquad\forall
p\geq 2\,,\forall R>0\,.$ (5.15)

By Theorem 1.2-(vi), if we first take $t=t_{j}$ in

$\partial_{t}u(t)=2\,\varepsilon^{2}\,\Delta
u(t)-W^{\prime}(u(t))+\varepsilon\,\lambda_{\varepsilon}[u(t)]\,V^{\prime}(u(t))\quad\mbox{on
$\mathbb{R}^{n}$}\,,$ (5.16)

and then let $j\to\infty$, we deduce that

$2\,\varepsilon^{2}\,\Delta\xi_{1}=W^{\prime}(\xi_{1})-\varepsilon\,\ell_{\varepsilon}\,V^{\prime}(\xi_{1})\quad\mbox{on
$\mathbb{R}^{n}$}\,.$

By step one, we conclude that $\ell_{\varepsilon}>0$, that $\xi_{1}$ is
strictly radially decreasing, and that

$\mathcal{V}(\xi_{1})\geq\frac{1}{C\,\ell_{\varepsilon}^{n}}\,.$ (5.17)

Step three: We now iterate the construction of step two. Let us consider the
following statement, depending on $k\in\mathbb{N}$:

$(S)_{k}$: There are $\\{(x^{i}_{j})_{j}\\}_{i=1}^{k}$ sequences in
$\mathbb{R}^{n}$ with $|x^{i}_{j}-x_{j}^{\ell}|\to\infty$ as $j\to\infty$
($i\neq\ell$), and there are $\\{\xi_{i}\\}_{i=1}^{k}$ radially symmetric
decreasing functions solving (5.2), such that

$\sum_{i=1}^{k}\mathcal{V}(\xi_{i})\leq 1\,,$ (5.18)

and, for each $i=1,...,k$,

$\displaystyle\mbox{$\tau_{(-x_{j}^{i})}[u(t_{j})]\rightharpoonup\xi_{i}$
weakly in $W^{3,p}(\mathbb{R}^{n})$ as $j\to\infty$}\,,$
$\displaystyle\lim_{j\to\infty}\big{\|}\tau_{(-x_{j}^{i})}[u(t_{j})]-\xi_{i}\big{\|}_{W^{2,p}(B_{R})}=0\,,\qquad\forall
R>0\,,\forall p\geq 2\,.$ (5.19)

Notice that if $(S)_{k}$ holds with equality in (5.18) for some
$k\in\mathbb{N}$, then the theorem is proved (up to showing the validity of
(5.4)) with $M=k$. We also notice that, in step two, we have proved that
$(S)_{1}$ holds. In this step we prove that if $(S)_{k}$ holds for some
$k\in\mathbb{N}$ with strict sign in (5.18), that is, with

$\sum_{i=1}^{k}\mathcal{V}(\xi_{i})<1\,,$

then, up to extracting a subsequence in $j$, we can find $(x_{j}^{k+1})_{j}$
and $\xi_{k+1}$ such that $(S)_{k+1}$ holds. Since, by step one,
$\mathcal{V}(\xi_{i})\geq(1/C\,\ell_{\varepsilon})^{n}$ for each
$i=1,...,k+1$, we conclude that, for some $M\in\mathbb{N}$, $(S)_{M}$ must
hold with equality in (5.18).

So let us consider $k\in\mathbb{N}$ such that $(S)_{k}$ holds with strict sign
in (5.18), and let $m=\sum_{i=1}^{k}\mathcal{V}(\xi_{i})$. Given
$h\in\mathbb{N}$, let

$A_{j}^{h}=\bigcup_{i=1}^{k}B_{h}(x_{j}^{i})\,.$

By (5.19), for every $h\in\mathbb{N}$ there is $j(h)\in\mathbb{N}$ such that

$\mathcal{V}(u(t_{j});B_{h}(x_{j}^{i}))\geq\mathcal{V}(\xi_{i};B_{h})-\frac{1-m}{2k}\,,\qquad\forall
i=1,...,k,\forall j\geq j(h)\,,$

so that if, $j\geq j(h)$,

$\displaystyle 1-m$ $\displaystyle\leq$ $\displaystyle
1-\sum_{i=1}^{k}\mathcal{V}(\xi_{i};B_{h})\leq\frac{1-m}{2}+1-\sum_{i=1}^{k}\mathcal{V}\big{(}u(t_{j});B_{h}(x_{j}^{i})\big{)}$
$\displaystyle=$
$\displaystyle\frac{1-m}{2}+\mathcal{V}(u(t_{j});\mathbb{R}^{n}\setminus
A_{j}^{h})\,,$

that is

$\frac{1-m}{2}\leq\mathcal{V}(u(t_{j});\mathbb{R}^{n}\setminus
A_{j}^{h})\,,\qquad\forall h\in\mathbb{N}\,,\forall j\geq j(h)\,,$

If we now set

$\beta_{j}^{h}=\max_{\mathbb{R}^{n}\setminus A_{j}^{h}}u(t_{j})\,,$

then, either $\beta_{j}^{h}\geq 1/2$ or, by arguing as in step two, for every
$h\in\mathbb{N}$ and $j\geq j(h)$,

$\displaystyle\frac{1-m}{2}\leq\mathcal{V}(u(t_{j});\mathbb{R}^{n}\setminus
A_{j}^{h})\leq C\,(\beta_{j}^{h})^{2/(n-1)}\,\int_{\mathbb{R}^{n}\setminus
A_{j}^{h}}W(u(t_{j}))\leq
C\,(\beta_{j}^{h})^{2/(n-1)}\,\varepsilon\,\mathcal{AC}_{\varepsilon}(u_{0})\,,$

that is, for every $h\in\mathbb{N}$ and $j\geq j(h)$,

$\beta_{j}^{h}\geq\min\Big{\\{}\frac{1}{2},\Big{(}\frac{1-m}{C\,\varepsilon\,\mathcal{AC}_{\varepsilon}(u_{0})}\Big{)}^{(n-1)/2}\Big{\\}}=:\beta_{*}>0\,.$

In particular, up to extracting a subsequence in $j$, for each $j$ we can find
$x_{j}^{k+1}\in\mathbb{R}^{n}$ such that

$\inf_{i=1,...,k}\,|x_{j}^{k+1}-x_{j}^{i}|\geq j\,,\qquad
u_{j}(x_{j}^{k+1},t_{j})\geq\beta_{*}\,,\qquad\forall j\,;$

and then, thanks to (5.9), we can find $r_{*}>0$ such that

$u(t_{j})\geq\frac{\beta_{*}}{2}\quad\mbox{on
$B_{r_{*}}(x_{j}^{k+1})$}\,,\qquad\forall j\,.$

Hence, by (5.10), there exists $\xi_{k+1}\in\cap_{p\geq
2}W^{3,p}(\mathbb{R}^{n};[0,1])\setminus\\{0\\}$ such that

$\displaystyle\mbox{$\tau_{(-x_{j}^{k+1})}[u(t_{j})]\rightharpoonup\xi_{k+1}$
weakly in $W^{3,p}(\mathbb{R}^{n})$ as $j\to\infty$}\,,$ (5.20)
$\displaystyle\lim_{j\to\infty}\big{\|}\tau_{(-x_{j}^{k+1})}[u(t_{j})]-\xi_{k+1}\big{\|}_{W^{2,p}(B_{R})}=0\,,\qquad\forall
R>0\,,\forall p\geq 2\,,$ (5.21)

and by the same argument of step two we see that $\xi_{k+1}$ satisfies

$2\,\varepsilon^{2}\,\Delta\xi_{k+1}=W^{\prime}(\xi_{k+1})-\varepsilon\,\ell_{\varepsilon}\,V^{\prime}(\xi_{k+1})\quad\mbox{on
$\mathbb{R}^{n}$}\,,$

and thus

$\mathcal{V}(\xi_{k+1})\geq\frac{1}{C\,\ell_{\varepsilon}^{n}}\,.$ (5.22)

We have thus proved the desired induction step.

Step four: We have so far proved the existence of $M\in\mathbb{N}$,
$\ell_{\varepsilon}>0$, $\\{(x_{j}^{i})_{j}\\}_{i=1}^{M}$ sequences in
$\mathbb{R}^{n}$, and $\\{\xi_{i}\\}_{i=1}^{M}$ radially symmetric decreasing
functions on $\mathbb{R}^{n}$ such that, up to extracting a subsequence,
(5.1), and (5.2) hold, together with part of (5.3) (i.e.,
$1=\sum_{i=1}^{M}\mathcal{V}(\xi_{i})$), and

$\displaystyle\mbox{$\tau_{(-x_{j}^{i})}[u(t_{j})]\rightharpoonup\xi_{i}$
weakly in $W^{3,p}(\mathbb{R}^{n})$ as $j\to\infty$}\,,$
$\displaystyle\lim_{j\to\infty}\big{\|}\tau_{(-x_{j}^{i})}[u(t_{j})]-\xi_{i}\big{\|}_{W^{2,p}(B_{R})}=0\,,\qquad\forall
R>0\,,p\geq 2\,,i=1,...,M\,.$ (5.23)

To complete the proof of (5.3) we just need to notice that, by (5.23), for
every $R>0$

$\displaystyle\sum_{i=1}^{M}\mathcal{AC}_{\varepsilon}(\xi_{i};B_{R})$
$\displaystyle\leq$
$\displaystyle\liminf_{j\to\infty}\sum_{i=1}^{M}\mathcal{AC}_{\varepsilon}\big{(}\tau_{-x_{j}^{i}}[u(t_{j})];B_{R}\big{)}=\liminf_{j\to\infty}\sum_{i=1}^{M}\mathcal{AC}_{\varepsilon}\big{(}u(t_{j});B_{R}(x_{j}^{i})\big{)}$
$\displaystyle=$
$\displaystyle\liminf_{j\to\infty}\mathcal{AC}_{\varepsilon}\Big{(}u(t_{j});\bigcup_{i=1}^{M}B_{R}(x_{j}^{i})\Big{)}\leq\mathcal{AC}_{\varepsilon}(u_{0})\,,$

where in the last equality we have used $|x_{j}^{i}-x_{j}^{\ell}|\to\infty$ as
$j\to\infty$ if $i\neq\ell$, and in the last inequality we have used the
monotonicity of $t\mapsto\mathcal{AC}_{\varepsilon}(u(t))$. By arbitrariness
of $R$, we have completed the proof of (5.3). We are thus left to prove (5.4).
In this step, we shall prove a slightly weaker statement, namely