数理解析研究所講究録
第 1924 巻 2014 年 97-114

97

ASYMPTOTIC BEHAVIOR OF ATTRACTORS FORINHOMOGENEOUS ALLEN-CAHN EQUATIONS
A. CESARONI, C. B. MURATOV, AND M. NOVAGA
ABSTRACT. We consider front propagation problems for forced mean curvature flows witha transport term and their phase field variants that take place in stratified media, i.e.,heterogeneous media whose characteristics do not vary in one direction. We provide aconvergence result relating asymptotic in time front propagation in the diﬀuse interfacecase to that in the sharp interface case, for suitably balanced nonlinearities of Allen-Cahn
type. Our results generalize previous results for forced Allen-Cahn equations.

1. INTRODUCTION
In this paper, we consider the following equation:(1.1)$\epsilon u_{t}=\epsilon\Delta u+\frac{1}{\epsilon}\mu^{2}(x)f(u)+a(x, u)$,$f(u)$ is a balanced bistable nonlinearity, with stable equilibria
whereand 1,
is a positive function that is bounded away from zero and infinit, and $a(x, u)$ is a bounded
function with $a(x, 0)=0$ for every . The reaction term on the right-hand side of (1.1) isa perturbation of a spatially modulated Allen-Cahn type balanced nonlinearity (see [2]).
Such an equation may arise, e.g., in modeling the dynamics of two co-existing phases in aphase transition with non-conserved order parameter in a medium with periodically varying
properties. When, the variations of the properties are weak and slowly changing inspace (for a more detailed discussion, see [13]).
We are interested in the behavior of the solutions of (1.1) when is a small parameter.
$\epsilon>0,$$0$$\mu(x)$$x$$\epsilon\ll 1$$\epsilon$

On formal asymptotic grounds $(see [22,$ appendix $A], [23])$ , the dynamics governed by(1.1) with some fixed initial condition is expected to converge as
to a forced mean
curvature flow with an extra transport term. The presence of the forcing term is due tothe fact that we are perturbing the balanced nonlinearity $\mu^{2}(x)f(u)$ with an unbalancing
higher order term
. The presence of the transport term is due to the -dependanceof the lower order term $\mu^{2}(x)f(u)$ in the nonlinearity. More precisely, for each $(x, t)$ fixed
the solution $u(x, t)$ is expected to converge to either or 1 everywhere except for an $(n-1)-$
dimensional evolving hypersurface
separating the regions where $u=0$ and $u=1$
in the hmit and whose equation of motion reads$\epsilonarrow 0$$\epsilon a(x, u)$

$x$

$0$

$\Gamma(t)\subset \mathbb{R}^{n}$(1.2)$V(x)=- \kappa(x)+\frac{g(x)}{c_{W}\mu(x)}-v(x)\cdot\frac{\nabla\mu(x)}{\mu(x)},$

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A. CESARONI, C. B. MURATOV, AND M. NOVAGA

where $g(x)= \int_{0}^{1}a(x, u)du$ . Here $V(x)$ is the velocity in the direction of the outward normal, is$v(x)$ $(i.e.,$ pointing into $the$ region where $u=0 in the$ limit) at a given point$u=1$
convex),
and
is
where
(positive
if
limit
set
the
the sum of the principal curvatures$x\in\Gamma(t)$(1.3)$\kappa$$c_{W}:= \int_{0}^{1}\sqrt{2W(u)}du, W(u):=-\int_{0}^{u}f(s)ds,$

where we defined the double well potential $W$ associated with , which is nonnegative andwhose only zeros are $u=0$ and $u=1$ . Such a result was rigorously established by Barles
and Souganidis, interpreting (1.2) in the viscosity sense [8] (see also [7], and [1] for rigorous
leading order asymptotic formulas). Note that, since the above mentioned results are local
in space and time, they are not suitable for making conclusions about the behavior as$tarrow+\infty$ of solutions of (1.1) for, via the analysis of (1.2).
We focus on reaction-diﬀusion equations and mean curvature flows in infinite cylinders
that describe the so-called stratified media. By a cylinder, we mean a set
is a bounded domain with suﬃciently smooth boundary. Stratified media
whereare fibered media along the cylinder, i.e., media whose features do not change along thecylinder axis, and this property can be characterized by the dependence of the nonlinearity
for reaction-diﬀusion equations and of the forcing and transport terms for mean curvature
flows only on the transverse coordinate of the cylinder. All the results obtained in this paper
remain valid in the periodic setting $(i.e.,$ when $\Omega is an (n-1)$ -dimensional parallelogramwith periodic boundary conditions), so we do not treat this case separately.
Our approach to the problem is variational: it stems from the basic observation thata rather wide class of reaction-diﬀusion systems may be viewed as gradient flows in exponentially weighted spaces when looked at in a moving reference frame (see [18], [19]).
Analogously, also the considered forced mean curvature flows can be interpreted as gradient flows of appropriate geometric functionals, given by perimeter type functionals plus
volume terms (see [12], [6]).
The purpose of this paper is to study the long-time behavior of solutions of (1.1) for
via the analysis of traveling wave solutions to (1.2). Our aim is to characterize
the long time limit of fronts in (1.1) invading the $u=0$ equilibrium in terms of uniformly
translating graphs solving (1.2). In particular we provide a convergence result, relating theasymptotic propagation speeds and the shape of the long time limit of the fronts for (1.1)
to those for (1.2) in the spirit of -convergence (as is done for stationary fronts in [17]).
In this paper we generalize previous results obtained in [13], for nonlinearities of the form$f(u)+\epsilon a(x, u)$ (i.e without-dependance on the lower order term of the nonlinearity). In
our case, with the addition of -dependance also on the lower order term of the nonlinearity,
we obtain a diﬀerent sharp interface limit, where also a transport term is appearing, so thearguments used in [13] have to be appropriately adapted.
$f$$\epsilon\ll 1$$\Sigma=\Omega\cross \mathbb{R}\subset \mathbb{R}^{n},$$\Omega\subset \mathbb{R}^{n-1}$$\epsilon\ll 1$$\Gamma$$x$$x$$BV,$ $IP,$
denote the usual spaces
1.1. Notations. Throughout the paper
functions,variation,
Lebesguecontinuous funcof Sobolev functions, functions of boundedtions with continuous derivatives, -times continuously diﬀerentiable functions with compact support, continuously diﬀerentiable functions with H\"older-continuous derivatives of$H^{1},$$k$$k$$C^{k},$$C_{c}^{k},$$C^{k,\alpha}$

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ASYMPTOTIC BEHAVIOR OF ATTRACTORS FOR INHOMOGENEOUS ALLEN-CAHN EQUATIONS

order for $\alpha\in(0,1)$ (or Lipschitz-continuous when $\alpha=1$ ), respectively. For a point, wherein the cylinder, we always write $x=(y, z)$ , whereisthe transverse coordinate andis the coordinate along the cylinder axis. The symbol$B(x, r)$ stands for the open bad inwith radius centered at , and for a set
the$|A|$ andsymbolsalways denote the closure of , the Lebesgue measure of and thecharacteristic function of , respectively. We also use the convention that $]_{J}n0=-\infty$ and$x\in\Sigma$$k$$\Omega\subset \mathbb{R}^{n-1}$$\Sigma=\Omega\cross \mathbb{R}$$y\in\Omega$$z\in \mathbb{R}$$\mathbb{R}^{n}$

$A$

$x$

$r$

$A$

$A$

$\overline{A},$$\chi_{A}$$A$$e^{-\infty}=0.$

ACKNOWLEDGEMENTS
The work of AC and MN was partially supported by the Fondazione CaRiPaRo Project“Nonlinear Partial Diﬀerential Equations: models, analysis, and control-theoretic problems.” The work of CBM was supported, in part, by NSF via grants DMS-0908279,
DMS-1119724 and DMS-1313687.

2. PROBLEM FORMULATION AND STANDING ASSUMPTIONS
and $2\leq n\leq 8$ . We
We assume that is a bounded domain with boundary of class, and in we consider the family of singularly perturbed reaction-diﬀusion
set
equations for $u=u(x, t)\in \mathbb{R}$ , with parameter:$C^{2}$$\Omega$$\Sigma$$:=\Omega\cross \mathbb{R}$$\Sigma$$\epsilon>0$(2.1)$\epsilon u_{t}=\epsilon\triangle u+\frac{1}{\epsilon}\mu^{2}(y)f(u)+a(y, u) (x, t)\in\Sigma\cross(O, +\infty)$

,

. For
with initial datum $u(x, 0)=u_{0}(x)\geq 0$ and Neumann boundary conditions on
simplicity, we also assume that $a(x, u)$ does not depend on . Nevertheless the results
for some $C>0$
remain valid after perturbing with terms that can be controlled by$\partial\Sigma$$\epsilon$$C\epsilon u$$a$

independent of$\epsilon.$

We associate to
(2.2)

$f$

and

$a$

the potentials$W(u):=- \int_{0}^{u}f(s)ds, G(y, u):=\int_{0}^{u}a(y_{\mathcal{S}})d_{\mathcal{S}}.$

We now state our assumptions on the functions
Assumption 1.

$\mu\in C^{1,\alpha}(\overline{\Omega})$

Assumption 2.

$a\in C_{loc}^{\alpha}(\overline{\Omega}\cross \mathbb{R})$, $\mu(y)>0$ for all,$\mu,$$a$and . Let $\alpha\in(0,1$ ].
$f$$y\in\overline{\Omega}.$$a_{u}\in C_{loc}^{\alpha}(\overline{\Omega}\cross \mathbb{R})$

, a

$0$

) $=0.$

Assumption 3.
, $f(O)=f(1)=0,$ $f’(O)<0,$ $f’(1)<0,$ $W(1)=W(O)=0,$$W(u)>0$ for all $u\neq 0$ , 1, and $\lim\inf W(u)>0.$$f\in C_{loc}^{1,\alpha}(\mathbb{R})$$|u|arrow\infty$

Assumption 3 implies that $W(u)$ is a balanced non-degenerate double-well potential (as amodel function one could think of $W(u)= \frac{1}{4}u^{2}(1-u)^{2}$ ). However, we do not require thathas only one other zero, which is located in $(0,1)$ , as is usually done in the literature.
Instead, we only assume that $u=0$ and $u=1$ have the same value of $W$ , and that $W$ isgreater for all other values of , including at infinity. Note that by Assumptions 2 and 3, depending only on
there exists $C,$
and , such that for every
$f$

$u$

$\delta_{0}>0$(2.3)$\mu,$$f$$a$$\epsilon\leq C^{-1}\delta_{0}$$\epsilon^{-1}\mu^{2}(y)W(u)-G(y, u)\geq 0 \forall(y, u)\in\overline{\Omega}\cross(\mathbb{R}\backslash (1-C\sqrt{\epsilon}, 1+C\sqrt{\epsilon}$

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A. CESARONI, C. B. MURATOV, AND M. NOVAGA

and
(2.4)

$\epsilon^{-1}\mu^{2}(y)W(\cdot)-G(y, \cdot)$is increasing on$[1+C\epsilon, 1+\delta_{0}]$$\forall y\in\overline{\Omega}.$

Remark 2.1. Observe that, if the initial datum
satisfies $0\leq u_{0}(x)\leq 1+\delta$ for some, then by the maximum principle and (2.4) we have $0\leq u(x, t)\leq$and all
for $aU(x, t)\in\Sigma\cross[O, +\infty$ ) and all$u_{0}$$\delta\in(0, \delta_{0})$$x\in\Sigma$$\epsilon\leq C^{-1}\delta.$$1+\delta$

We consider a family of measurable setswith regular boundary, such thatevolves according to (1.2) with forcing term
$S(t)\subseteq\Sigma$$\Gamma(t)=\partial S(t)$(2.5)$g(y):= \int_{0}^{1}a(y, s)d_{\mathcal{S}}=G(y, 1)$.

associate to this flow the following quasilinear parabolic problem for $h=h(y, t)\in \mathbb{R}$ inwhich describes the motion of the set $\{(y, z)\in\Sigma : z=h(y, t)\}$ according to (1.2):(2.6)$h_{t}= \sqrt{1+|\nabla h|^{2}}(\nabla\cdot(\frac{\nabla h}{\sqrt{1+|\nabla h|^{2}}})+\frac{g}{c_{W}\mu})+\nabla h\cdot\nabla\log\mu$in$\Omega\cross(0, +\infty)$$\Omega$,with initial datum $h(y, 0)=h_{0}(y)$ , and Neumann boumdary conditions on. In particular, the subgraph $S(t)=\{(y, z)\in\Sigma : z<h(y, t)\}$ of the solution of (2.6) coincides with thefamily of sets evolving according to (1.2), with initial datum $S_{0}=\{(y, z)\in\Sigma : z<h_{0}(y)\}.$We recall the definition of the perimeter with weight of a Lebesgue measurable set
relative to (see, e.g., [3], [5]): let$\partial\Omega$$\mu$$A\subseteq\Omega$(2.7)$\Omega$$Per_{\mu}(A, \Omega)=\sup\{\int_{A}(\nabla\cdot\phi(y)+\phi\cdot\nabla\log\mu)\mu(y)dy$:$\phi\in C_{c}^{1}(\Omega;\mathbb{R}^{n-1})$, $|\phi|\leq 1\}.$, we recover the standard definition of perimeter. We also recall the classicalWhenisoperimetric inequality [3]:$\mu\equiv 1$

Proposition 2.2. There exists(2.8)

for allPer$A\subseteq\Omega$, depending on

Assumption 4. Let$C_{\Omega}=2$$g\in C^{\alpha}(\overline{\Omega})$

$n$

and

$\Omega$, such that$(A, \Omega)\geq C_{\Omega}|A|^{\frac{n-2}{n-1}}$of finite perimeter and such that

Remark 2.3. Notice that

(2.9)

$C_{\Omega}>0$$|A| \leq\frac{1}{2}|\Omega|.$

if $n=2.$. Then there exists$A\subseteq\Omega$$\int_{A}g(y)dy>c_{W}Per_{\mu}(A, \Omega)$such that.

This assumption basically ensures that the trivial state $u=0$ is energetically less favorable
for suﬃciently small, resulting in the existence of the invasion fronts.
$\epsilon$

Remark 2.4. Notice that (2.9) implies, in particular, that $\sup_{\Omega}g>0$ , and is automaticallysatisfied if(2.10)$\int_{\Omega}g(y)dy>0.$

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ASYMPTOTIC BEHAVIOR OF ATTRACTORS FOR INHOMOGENEOUS ALLEN-CAHN EQUATIONS$g$

Throughout the rest of the paper Assumptions 1-4 are always taken to be satisfied, withdefined by (2.5).

3. TRAVELING WAVES IN THE DIFFUSE INTERFACE CASE
In this section we review some results on existence and stability of traveling wave solutions of the reaction-diﬀusion equation (2.1) for, in the form of fronts invading theequilibrium $u=0$ from above. Our main references are [19], [20] and [13].
We look for solutions to (2.1) of the form $u(x, t)=\overline{u}(y, z-ct)$ , where $c>0,$$\epsilon>0$$\overline{u}>0,$

(3.1)

and

$u$

$\overline{u}\in C^{2}(\Sigma)\cap C^{1}(\overline{\Sigma})\cap L^{\infty}(\Sigma)$(3.2)

uniformly as $zarrow+\infty,$$z)arrow 0$

solves$\epsilon\triangle\overline{u}+c\epsilon\overline{u}_{z}+\frac{1}{\epsilon}\mu^{2}(y)f(\overline{u})+a(y,\overline{u})=0 (y, z)\in\Sigma,$with Neumann boundary conditions
on . The constant is referred to as the
speed of the wave, and the profile.
Note that existence and qualitative properties of traveling fronts in a variety of settings
have been extensively studied, starting with the classical work of Berestycki and Nirenberg[9] who analyzed traveling fronts connecting zero with some equilibrium $v>0$ (i.e. which$\nu\cdot\nabla\overline{u}=0$$\partial\Sigma$$c$$\overline{u}$$\overline{u}$satisfies also
) $=v$ uniformly).
By an equilibrium for (2.1), we mean a function$\lim_{zarrow-\infty}u$

(3.3)

$z$

$v:\overline{\Omega}arrow \mathbb{R}$which solves$\epsilon\Delta v+\frac{1}{\epsilon}\mu^{2}(y)f(v)+a(y, v)=0 y\in\Omega,$with $\nu\cdot\nabla v=0$ on(3.4)$\partial\Omega$. Note that every critical point of the functional$E^{\epsilon}(v)= \int_{\Omega}(\frac{\epsilon}{2}|\nabla v|^{2}+\mu^{2}(y)\frac{W(v)}{\epsilon}-G(y, v))dy v\in H^{1}(\Omega)\cap L^{\infty}(\Omega)$

is an equilibrium for the reaction-diﬀusion equation (2.1). Moreover, we will consider stableequilibria, according to the following definition.

Definition 3.1. A function

is a stable critical point of
if it is a
critical point of the functional and the second variation ofnonnegative, $i.e.$$v\in H^{1}(\Omega)\cap L^{\infty}(\Omega)$$E^{\epsilon}$$E^{\epsilon}i_{\mathcal{S}}$(3.5)$\int_{\Omega}(\epsilon|\nabla\phi|^{2}+(\epsilon^{-1}\mu^{2}(y)W"(v)-G_{uu}(y,v))\phi^{2})dy\geq 0 \forall\phi\in H^{1}(\Omega)$

Moreover,$v$

is a nondegenerate stable critical point of$E^{\epsilon}$.

if strict inequality holds in (3.5).

By Assumption 3, $v=0$ is a non-degenerate stable critical point of the functional
for every suﬃciently small. Indeed, defining$\epsilon$(3.6)$\nu_{0}^{\epsilon} :=\int_{\Omega}\phi^{2}=1\min\int_{\Omega}(\epsilon|\nabla\phi|^{2}+(\epsilon^{-1}\mu^{2}(y)W"(0)-G_{uu}(y, 0))\phi^{2})dy,$

observe that there exists(3.7)$\epsilon_{0}>0$, depending on $W”(0)$ , $\inf\mu$ and$v_{0}^{\epsilon}>0$

for all$\epsilon<\epsilon_{0}.$$\Vert a_{u}(\cdot, 0)\Vert_{\infty}$, such that$E^{\epsilon}$

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A. CESARONI, C. B. MURATOV, AND M. NOVAGA

We point out that existence of the considered solutions (i.e. a traveling front connecting
and a prescribed equilibrium $v>0$ ) is not guaranteed in general. In particu\’iar, we have toimpose some condition on assuring the e istence of non-trivial positive equilibria, withnegative energy, as in the following proposition.
$0$

$*$

$g$

Proposition 3.2. Under Assumptions 1-4, there exist positive constants
andsuch that
there existsthat for all$v_{\epsilon}^{0}\in H^{1}(\Omega)$$\epsilon<\epsilon_{0}$$\epsilon_{0}$

and

$C$

such

$E^{\epsilon}(v_{\epsilon}^{0})<0.$$0\leq v_{\epsilon}^{0}\leq 1$

Proof. By [17] we have that
(3.8)$\Gamma-\lim_{\epsilonarrow 0}E^{\epsilon}(u)=\{\begin{array}{ll}E^{0}(A):=c_{W}Per_{\mu}(A, \Omega)-\int_{A}gdy if u=\chi_{A},+\infty otherwise,\end{array}$where the convergence is understood in the sense of -convergence infollows the same argument as in [13, Proposition 3.4].
$\Gamma$$L^{1}(\Omega)$. So the proof$\square$

Following the variational approach to front propagation problems [19] (see also [16,18,
20, 21]), for every $c>0$ we associate to the reaction-diﬀusion equation in (2.1) the energy)functional (for fixed$\epsilon>0$(3.9)$\Phi_{c}^{\epsilon}(u)=\int_{\Sigma}e^{cz}(\frac{\epsilon}{2}|\nabla u|^{2}+\frac{1}{\epsilon}\mu^{2}(y)W(u)-G(y, u))dx.$

This functional is naturally defined onweighted Sobolev space with the norm
(3.10)$H_{c}^{1}(\Sigma)\cap L^{\infty}(\Sigma)$, where$H_{c}^{1}(\Sigma)$

is an exponentially$\Vert u\Vert_{H_{c}^{1}(\Sigma)}^{2}=\int_{\Sigma}e^{cz}(|\nabla u|^{2}+|u|^{2})dx.$

Furthermore, the functional
is diﬀerentiable insatisfy the traveling wave equation (3.2) [16, 19].
$\Phi_{c}^{\epsilon}$$H_{c}^{1}(\Sigma)\cap L^{\infty}(\Sigma)$, and its critical points

Remark 3.3. Following [9, Section 4] (see also [19, Theorem 3.3(iii)]) one can show thatand is a critical point
every traveling wave $(c, u)$ to (2.1) satisfying (3.1) belongs toto$H_{c}^{1}(\Sigma)$$\Phi_{c}^{\epsilon}.$play an important role for theand, in particular, minimizers of
Critical points oflong-time behavior of the solutions of the initial value problem associated with (2.1) in thecase of front-like initial data. Indeed, in [21] it is proved under generic assumptions on theare selectedovernonlinearity (see also [18, 19]) that the non-trivial minimizers of
as long-time attractors for the initial value problem associated to (2.1) with front-like initial
data. Also, in [19] it was proved under minimal assumptions on the nonlinearity that theforspeed of the leading edge of the solution is determined by the unique value ofassumptions
to guarantee existence of
which
has a non-trivial minimizer. Appropriate
were given in [19]. In our case these conditions are verified for every
minimizers of
suﬃciently small.
$\Phi_{c}^{\epsilon}$$\Phi_{c}^{\epsilon}$$\Phi_{c}^{\epsilon}$$H_{c}^{1}(\Sigma)$$c_{\epsilon}^{\uparrow}>0$$\Phi^{\epsilon}c_{\epsilon}^{\dagger}$$\epsilon$$\Phi_{c}^{\epsilon}$

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ASYMPTOTIC BEHAVIOR OF ATTRACTORS FOR INHOMOGENEOUS ALLEN-CAHN EQUATIONS

and $M$ , deTheorem 3.4. Under Assumptions $1-4,$ there exist positive constants
there exists a uniquepending on $f,$
and , such that for all
such thatwhich satisfiesi)admits a non-trivial minimizer$\epsilon_{0},$$\overline{u}_{\epsilon}\in H^{1}c_{\epsilon}^{\dagger}\cap L^{\infty}(\Sigma)$$\Phi_{c_{\epsilon}^{\dagger}}^{\epsilon}$(3.11)
ii)
iii)$0<c_{\epsilon}^{\dagger}\leq M$$0<\epsilon<\epsilon_{0}$$\Omega$$a$$\mu,$$C$$\sup\{z\in \mathbb{R}|\sup_{y\in\Omega}\overline{u}_{\epsilon}(y, z)>\frac{1}{2}\}=0.$$\overline{u}_{\epsilon}\in C^{2}(\Sigma)\cap C^{1}(\overline{\Sigma})\cap W^{1,\infty}(\Sigma)$, andin , and$(c_{\epsilon}^{\dagger}, \overline{u}_{\epsilon})$is a traveling wave solution to (2.1).

$\overline{\Sigma}$$0< \overline{u}_{\epsilon}\leq 1+C\epsilon<\frac{3}{2},$$(\overline{u}_{\epsilon})_{z}<0$$\lim_{zarrow+\infty}\overline{u}_{\epsilon}(\cdot, z)=0$$\lim_{zarrow-\infty}\overline{u}_{\epsilon}(\cdot, z)=v_{\epsilon}$$in$$C^{1}(\overline{\Omega})$,in (3.4) withcritical point ofare translates of alongiv), and all non-trivial minimizers ofdepending only on $W,$ $G,$
, there exist $C,$v) Given
andthere holds
andsuch that, for every

where$v_{\epsilon}$is a$\mathcal{S}table$$E^{\epsilon}$$E^{\epsilon}(v_{\epsilon})<0.$$z.$$\overline{u}_{\epsilon}$$\Phi_{c_{\epsilon}^{\dagger}}^{\epsilon}(\overline{u}_{\epsilon})=0$$\Phi^{\epsilon}c_{\epsilon}^{\dagger}$$\delta\in(0, 1)$$\delta$$\overline{x}\in\overline{\Sigma}$$\overline{r},$$\epsilon\in(0,\overline{\epsilon})$(3.12)$\overline{\epsilon}>0$

and$\Omega$$\mu,$$r\in(\epsilon,$ $r$$\overline{u}_{\epsilon}(\overline{x})\geq\delta \Rightarrow \int_{B(\overline{x},r)\cap\Sigma}\overline{u}_{\epsilon}^{2}dx\geq Cr^{n},$(3.13)$\overline{u}_{\epsilon}(\overline{x})\leq 1-\delta \Rightarrow \int_{B(\overline{x},r)\cap\Sigma}(1-\overline{u}_{\epsilon})^{2}dx\geq Cr^{n}.$Proof. For the proof we refer to [13, Thm 3.5, Prop. 3.7 and Prop. 3.8].

$\square$4. TRAVELING WAVES IN THE SHARP INTERFACE CASE
In this section we consider the front propagation problem in the cylinder , for theforced mean curvature flow with transport term (1.2). Also in this case, we are interested
in traveling wave solutions with positive speed. We review and generalize some results
contained in [13] (see also [12]).
$\Sigma$

Definition 4.1 Raveling waves). A traveling wave for the forced mean curvature flowwhere $c>0$ is the speed of the wave and the graph of the junction
is a pairis the profile of the wave, such that $h(y, t)=\psi(y)+ct$ solves (2.6).
$(^{r}$$\psi\in$$(c, \psi)_{f}$$C^{2}(\Omega)\cap C^{1}(\overline{\Omega})$

Observe that to prove existence of a traveling wave solution it is suﬃcient to determine$c>0$ such that the equation in(4.1)$\Omega$$- \nabla\cdot(\frac{\nabla\psi}{\sqrt{1+|\nabla\psi|^{2}}})=\frac{1}{c_{W}\mu(y)}g(y)-\frac{c}{\sqrt{1+|\nabla\psi|^{2}}}+\frac{\nabla\psi}{\sqrt{1+|\nabla\psi|^{2}}}\cdot\nabla\log\mu(y)$,, admits a classical solution. The
with Neumann boundary condition $v\cdot\nabla\psi=0$ ongraph of this solution will be the profile of the traveling wave.
$\partial\Omega$

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A. CESARONI, C. B. MURATOV, AND M. NOVAGA

Following the variational approach for the forced mean curvature flow (see [12], [13]),
for $c>0$ we consider the family of exponentially weighted area-type functionals(4.2)$F_{c}( \psi)=\int_{\Omega}e^{c\psi(y)}(c_{W}\mu(y)\sqrt{1+|\nabla\psi(y)|^{2}}-\frac{g(y)}{c})dy,$

among all. Note that if is bounded and is a critical point of the functional, then it is a solution to (4.1). Here defines the graph $z=\psi(y)$ that represents the
sharp interface front. The functional
has a well-known geometric characterization. Let
us introduce the following exponentially weighted perimeter for:(4.3)$\psi\in C^{1}(\overline{\Omega})$$\psi$$F_{c}$$\psi$$F_{c}$$S\subseteq\Sigma$$Per_{c,\mu}(S, \Sigma)$$:= \sup\{\int_{S}[\nabla\cdot\phi+(c\hat{z}+\nabla\log\mu)\cdot\phi]\mu(y)e^{cz}dx$:$\phi\in C_{c}^{1}(\Sigma;\mathbb{R}^{n})$We then define the following geometric functional on measurable sets
weighted volume $\int_{S}e^{cz}\mu(y)dx<\infty$ :(4.4), $|\phi|\leq 1\}.$$S\subset\Sigma$

with the$\mathcal{F}_{c}(S):=c_{W}Per_{c,\mu}(S, \Sigma)-\int_{S}e^{cz}g(y)dx,$

where we noted that by Assumption 1
last term in (4.4) is well-defined in the considered, the functionalclass of sets. After the change of variable
is equivalentto$th_{l}e$$\zeta(y)$(4.5)$:= \frac{e^{c\psi(y)}}{c}\geq 0$$F_{c}$$G_{c}( \zeta)=\int_{\Omega}(c_{W}\mu(y)\sqrt{c^{2}\zeta^{2}(y)+|\nabla\zeta(y)|^{2}}-g(y)\zeta(y))dy,$

in the sense that $F_{c}(\psi)=G_{c}(\zeta)$ for all. Since the functional
is naturally defined on $BV(\Omega)$ as the lower-semicontinuous relaxation, we introduce the following
generalization to the notion of a traveling wave for (2.6).
$\zeta\in C^{1}(\overline{\Omega})[12]$$G_{c}$

Definition 4.2 (Generalized traveling waves). A generalized traveling wave for the forced
mean curvature flow is a pair $(c, \psi)$ , where $c>0$ is the speed of the wave, andIn
is the profile of the wave, where $\zeta\in BV(\Omega)$ is a non-negative critical point of , notidentically equal to zero.
$\psi=\frac{1}{c}$$c\zeta$$G_{c}$We define
to be the interior of the support of . By standard regularity ofminimizers of perimeter-type functionaJs [3], we have that solves (4.1) classically inand, therefore, we have that $h(y, t)=\psi(y)+ct$ solves (2.6) inwith
onwith Neumann boundary conditions on. In particular, if, then the above
definition implies thatis a traveling wave in the sense of Definition 4.1. In general,
however, may diﬀer from by a set of positive measure, in which case the traveling wave$\omega\subset\Omega$$\zeta$$\psi$$\nu\cdot\nabla\psi=0$$\omega$$\partial\omega\cap\partial\Omega$$\omega=\Omega$$\partial\omega\cap\partial\Omega$$\omega$$(c, \psi)$$\omega$

profile(4.6)$\psi$$\Omega$obeys the following kind of boundary condition:$\lim_{yarrow\overline{y}}\psi(y)=-\infty \forall\overline{y}\in\partial\omega\cap\Omega.$

In this situation a generalized traveling wave may have the form of one or several “fingers”
invading the cylinder from left to right with speed
$c.$

105
ASYMPTOTIC BEHAVIOR OF ATTRACTORS FOR INHOMOGENEOUS ALLEN-CAHN EQUATIONSThe next proposition explains the relation between Assumption 4 and the minimization
and, hence,
problems associated with functionals$F_{c}.$$G_{c}$

Proposition 4.3. Let Assumption 4 hold. Then there exists a unique$c^{\uparrow}>0$

such thati)$\frac{\int_{\Omega}gdy}{cW\int_{\Omega}\mu dy}\leq c^{\uparrow}\leq\frac{1}{c_{W}}\sup_{\Omega}(\frac{g}{\mu})$.

ii) If $0<c<c\dagger$ , then $\inf\{G_{c}(\zeta) : \zeta\in BV(\Omega), \zeta\geq 0\}=-\infty.$, then $\inf\{G_{c}(\zeta) : \zeta\in BV(\Omega), \zeta\geq 0\}=0$ , and $G_{c}(\zeta)>0$ for everyiii) If
non-trivialiv) If $c=c\dagger$ , then there exists a non-trivial, such that
, with$c>c^{\uparrow}$$\zeta\geq 0.$$\overline{\zeta}\in BV(\Omega)$$\overline{\zeta}\geq 0$$irJf\{G_{c}(\zeta) :$G_{c}(\overline{\zeta})=$\zeta\in BV(\Omega), \zeta\geq 0\}=0.$

Proof. The result follows from [12, Proposition 3.1 and Corollary 3.2] (see also [20, Proposition 4.1]).

$\square$Note that the same argument as in [12, Proposition 3.4, Lemma 3.5] gives that for all
such that $\zeta\in BV(\Omega)$ we have$\zeta\geq 0$(4.7)$G_{c}(\zeta)=\mathcal{F}_{c}(S_{\psi})$. Moreover if
where $S_{\psi}=\{(y, z)\in\Omega\cross \mathbb{R} : z<\psi(y)\}$ is the subgraph ofof is a minimizer, under compact
is a non trivial minimizer of , then the subgraphperturbations, of the functional
defined in (4.4). So, from Proposition 4.3 we obtainthe following result about existence of generalized traveling waves (see [13], [12]).
$\psi=\frac{1}{c}\ln c\zeta$$G_{c}$$\zeta\geq 0$$\psi$$S\psi$$\mathcal{F}_{c}$Theorem 4.4 (Existence of generalized traveling waves). Let Assumption 4 hold. Then
, which coincides with the one in Proposition 4.3, such that:
there exists a uniquei) There exist a function) such that:is a generalized traveling$:=\{(y, z)\in\Sigma|z<\psi(y)\}$ iswave for the forced mean curvature flow and the set
a minimizer ofii) The set $:=\{\psi>-\infty\}$ is open and satisfies $E^{0}(\omega)<0$ , whereis defined in (3.8).
Moreover,is a minimizer ofunder compact perturbations, and there exist$r_{0}>0$ and $\lambda>0$ such for all
and all $r\in(0, r_{0})$ the, allfollowing density estimates hold:$c\dagger>0$$\psi$$\Omegaarrow[-\infty, \infty$$(c\dagger, \psi)$$S_{\psi}$$\mathcal{F}_{c\dagger}.$$E^{0}$$\omega$$\omega\cross \mathbb{R}$$\mathcal{F}_{c\dagger}$$\overline{x}\in\overline{\omega}\cross \mathbb{R}$

(4.8)
(4.9)

iii)

$\overline{x}’\in\Sigma\backslash \omega\cross \mathbb{R}$$|\omega\cross \mathbb{R}\cap B(\overline{x}, r)| \geq \lambda r^{n-1},$$|(\Sigma\backslash \omega\cross \mathbb{R})\cap B(\overline{x}’, r)| \geq \lambda r^{n-1}.$and it is unique up to additive constants on every connected component of, in the following sense: there exists a number)
and functions$:=\{\psi_{i}>-\infty\}\neq\emptyset$ are open, connected and disjoint,
$i=1$
.
,
,
.
.
each
that
such
for
).
, for someand$\psi\in C^{2}(\omega)$$k\in \mathbb{N}$$\omega$$k$$\psi_{i}\in C^{2}(\omega_{i})$$\omega_{i}$$\psi=In(\sum_{i=1}^{k}e^{\psi_{i}+k_{i}})$$k_{i}\in[-\infty, \infty$$\psi_{i}:\Omegaarrow[-\infty, \infty$

106
A. CESARONI, C. B. MURATOV, AND M. NOVAGA

iv)$\partial S_{\psi}\cap\Sigma$

problem

is a hypersurface of class(4.10)$C^{2}.$$\partial\omega$is a$C^{2}$

solution to the prescribed curvature$on$$c_{W} \kappa=\frac{g}{\mu}-cw\nabla\log\mu\cdot\nu_{\partial\omega}$

where is the sum of the principal
conditions $\nu\cdot\nu=0$ at$\kappa$vatures of$cur^{v}$$\partial\omega\cap\Omega,$$\partial\omega\cap\Omega$, with Neumann boundary$\partial\omega\cap\Omega.$

Proof. For the proof we refer to [13, Lemma 4.4, Theorem 4.8].

$\square$

We now introduce an additional assumption, under which stronger conclusions about
the convergence of fronts can be made.
Assumption 5. Let
and assume that (2.10) holds. Then
minimizer ofunder compact perturbations among sets$:= \inf\{c>0$ :.
$g\in C^{\alpha}(\overline{\Omega})$$c\dagger$is the unique
with
and

$\Omega\cross \mathbb{R}$$S=\omega\cross \mathbb{R}$$\mathcal{F}_{c\dagger}$$\omega\subseteq\Omega$$inf\mathcal{F}_{c}\geq 0\}\in(0, \infty)$

Clearly, Assumption 5 is quite implicit. In Proposition 4.6 we give some suﬃcient conditions
for it, first in the two-dimensional case and then in every dimension.
This result of Theorem 4.4 can be strengthened under Assumption 5, as follows.

Theorem 4.5 (Existence, uniqueness and stability of traveling waves). Let Assumption 5, such that:
hold. Then there exists a uniquei) there exists a unique, andsuch thatisa traveling wave for the forced mean curvature flow (2.6). Moreover is the unique, up to additive constants,
minimizer of the functional
overii) $S=\{(y, z) \in\Sigma : z<\psi(y)\}$ is the unique minimizer ofup to translations in$c\dagger>0$$\psi\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega})$$\max_{y\in\overline{\Omega}}\psi(y)=0$$(c\dagger, \psi)$$\psi$$C^{1}(\overline{\Omega})$$F_{c\dagger}$$z.$$\mathcal{F}_{c\dagger}$

iii)
$h$

in

$t)-c^{\uparrow}t-karrow\psi$$C^{1,\alpha}(\overline{\Omega})$, as $tarrow+\infty$

where $h(y, t)$ is the unique solution to (2.6) with Neumann boundary conditions andinitial datum $h(y, 0)=h_{0}(y)\in C(\overline{\Omega})$ andis constant, depending on$k\in \mathbb{R}$$h_{0}.$

Proof. For the proof we refer to [13, Theorem 4.10, Theorem 4.11].

$\square$4.1. Suﬃcient conditions. As Assumption 5 is quite implicit, in the following proposition we list some suﬃcient conditions for it to hold.
Proposition 4.6. Let (2.10) hold and let
be the relative isoperimetric constant of(see (2.8)). Then Assumption 5 holds if one of the following conditions is verified:i) there is no embedded hypersurface
which solves the prescribed curvature problem$\Omega$$C_{\Omega}$$\partial\omega\subseteq\Omega$(4.11)$c_{W}\mu\kappa-g+c_{W}\nabla\mu\cdot\nu=0 \partial\omega\cap\Omega,$with Neumann boundary conditionsii) $n=2$ and $g(y)>c_{W}|\mu’|$ on$\Omega.$$\nu_{\partial\omega\partial\Omega}\nu=0$

on$\partial\omega\cap\partial\Omega.$

107
ASYMPTOTIC BEHAVIOR OF ATTRACTORS FOR INHOMOGENEOUS ALLEN-CAHN EQUATIONSiii) $\inf_{\Omega}(g-c_{W}|\nabla\mu|)\leq 0$ and$\sup_{\Omega}(\frac{g+c_{W}|\nabla\mu|}{\mu})-\inf_{\Omega}(\frac{g-c_{W}|\nabla\mu|}{\mu})<c_{W}C_{\Omega}2^{\frac{1}{n-1}}|\Omega|^{-\frac{1}{n-1}}.$iv) $\inf_{\Omega}(g-c_{W}|\nabla\mu|)\geq 0$ on$\Omega$and$\sup_{\Omega}(\frac{g+c_{W}|\nabla\mu|}{\mu})<c_{W}C_{\Omega}2^{\frac{1}{n-1}}|\Omega|^{-\frac{1}{n-1}}.$

Proof. We start proving (i) and (ii). (i) is actually a conseguence of Theorem 4.4. When$n=2$ , the prescribed mean curvature problem (4.11) reads $g(y)=\pm c_{W}\mu’(y)$ for some$y\in\Omega.$

Now we prove the other items. Let be a solution to (4.1), with maximal support(see (iii) in. This means that every other solution to (4.1) has support$\psi(x)arrow-\infty$
distasTheorem 4.4). Integrating (4.1) over and recalling that
we obtain$\psi$$\omega’\subset\omega$$\omega\subset\Omega$$(x;\partial\omega)arrow 0$$\omega$(4.12)

Per$( \omega, \Omega)=\int_{\omega}(\frac{g(y)}{cW\mu(y)}-\frac{c}{\sqrt{1+|\nabla\psi(y)|^{2}}}+\frac{\nabla\log\mu(y)\cdot(\nabla\psi(y))}{\sqrt{1+|\nabla\psi(y)|^{2}}})dy$

is the standard perimeter relative towhere Per, see also [3]). (4.12) gives$(\omega, \Omega)$$\Omega$(as defined in (2.7), with weight$\mu\equiv 1$(4.13)$c_{W} Per(\omega, \Omega)\leq[\sup_{\Omega}(\frac{g+c_{W}|\nabla\mu|}{\mu})]|\omega|.$. Observe that
. By Proposition 4.3, we get that is a minimizer ofLet$L^{2}(\Omega)(G_{c\dagger}(u)=+\infty$ifin (4.5) is a convex, lower semicontinuous functional on$u\in L^{2}(\Omega)\backslash BV(\Omega))$ .
So, by the general theory of subdiﬀerentials in [4, Ch.w 6] there, with $|\xi|\leq 1$ and $div(\xi)\in L^{2}(\Omega)$ , and a function:
exist a vector field, with $0\leq h\leq 1$ , such that$\zeta=\frac{e^{c}\dagger\psi}{c\dagger}$$G_{c\dagger}$$\zeta$$G_{c\dagger}$$\Omegaarrow \mathbb{R}^{n}$$\xi_{\zeta}=\xi$$h_{\zeta}=h:\Omegaarrow \mathbb{R}$(4.14)$\int_{\Omega}[-c_{W}\mu(div\xi(y)+\nabla\log\mu\cdot\xi)+c^{\uparrow}c_{W}\mu(y)h(y)-g(y)](\chi-\zeta)dy\geq 0$

for all $\chi\in BV(\Omega)$ such that$\chi\geq 0$

Moreover, for all$in\Omega,$$y\in\omega,$$h(y) = \frac{c\dagger\zeta}{\sqrt{(c\dagger)^{2}\zeta^{2}(y)+|\nabla\zeta(y)|^{2}}}$$\xi(y) = \frac{\nabla\zeta(y)}{\sqrt{(c\dagger)^{2}\zeta^{2}(y)+|\nabla\zeta(y)|^{2}}}.$

If we apply inequality (4.14) to
we obtain(4.15)$\chi(y)=\zeta(y)+\frac{\chi_{A}(y)}{\mu(y)}$, where$A\subseteq\Omega$

is a set of finite perimeter,

Per $(A, \Omega)+\int_{A}(-\nabla\log\mu\cdot\xi+c^{\dagger}h(y)-\frac{g(y)}{c_{W}\mu(y)})dy\geq 0.$

108
A. CESARONI, C. B. MURATOV, AND M. NOVAGAIn particular, (4.12) and (4.15) imply that$\omega$

is a minimum for the functional$\mathcal{G}(A)=Per_{1}(A, \Omega)+\int_{A}(-\nabla\log\mu\cdot\xi+c^{\uparrow}h(y)-\frac{g(y)}{c_{W}\mu(y)})dy.

This gives that
Proposition 4.3 and by the definition of$\int_{\Omega}(-\nabla\log\mu\cdot\xi+c\dagger h(y)-\ovalbox{\tt\small REJECT}_{c_{W}\mu y})dy\geq 0$Per$(\omega, \Omega)$$\leq$(4.16)$h,$ $\xi$

A\subseteq\Omega.$and so, by (4.12), by (i) in, we get$\int_{\Omega\backslash \omega}(-\nabla\log\mu\cdot\xi+c\dagger h(y)-\frac{g(y)}{c_{W}\mu(y)})dy$$\leq \frac{1}{c_{W}}(\sup_{\Omega}\frac{g}{\mu}-\inf_{\Omega}\frac{g-c_{W}|\nabla\mu|}{\mu})|\Omega\backslash \omega|.$

Observe that by the isoperimetric inequality (2.8) and (4.16), either$| \Omega\backslash \omega|>\frac{1}{2}|\Omega|$or(4.17)$\frac{1}{c_{W}}$$( \sup_{\Omega}\frac{g}{\mu}$-$inf\frac{g-c_{W}|\nabla\mu|}{\mu})\frac{|\Omega|^{\frac{1}{n-1}}}{2^{\frac{1}{n-1}}}\geq\frac{1}{c_{W}}(\sup_{\Omega}\frac{g}{\mu}-\inf_{\Omega}\frac{g-c_{W}|\nabla\mu|}{\mu})|\Omega\backslash \omega|^{\frac{1}{n-1}}\geq C_{\Omega}.$

In particular, if(4.18)$\frac{1}{c_{W}}(\sup_{\Omega}^{\frac{g}{\mu}-irJ}\Omega f\frac{g-c_{W}|\nabla\mu|}{\mu})<C_{\Omega}2^{\frac{1}{n-1}}|\Omega|^{-\frac{1}{n-1}}$

then necessarily(4.19)$| \omega|\leq\frac{1}{2}|\Omega|$and so by (2.8) and (4.13) we obtain$\sup_{\Omega}(\frac{g+c_{W}|\nabla\mu|}{\mu})\geq c_{W}\frac{Per(\omega,\Omega)}{|\omega|}\geq c_{W}C_{\Omega}2^{\frac{1}{n-1}}|\Omega|^{-\frac{1}{n-1}}.$We start proving (iii). Note that if$\inf_{\Omega}\frac{g-c_{W}|\nabla\mu|}{\mu}\leq 0,$$\sup_{\Omega}(\frac{g+c_{W}|\nabla\mu|}{\mu})\leq\sup_{\Omega}(^{\underline{g+c\mu|}})-\inf_{\Omega}(\frac{g-c_{W}|\nabla\mu|}{\mu})$

.

So if
$\sup_{\Omega}(\frac{g+c_{W}|\nabla\mu|}{\mu})-iry\int\Omega(\frac{g-c_{W}|\nabla\mu|}{\mu})<c_{W}C_{\Omega}2^{\frac{1}{n-1}}|\Omega|^{-\frac{1}{n-1}},$

neither (4.17) nor (4.19) are verified. Henceand (iii) is proved.
We prove (iv). If
and(4.17)
neithernor (4.19) are verified, and (iv) is proved.
$\omega=\Omega$$\inf_{\Omega}(g-c_{W}|\nabla\mu|)\geq 0$$\sup_{\Omega}(\frac{g+c_{W}|\nabla\mu|}{\mu})<C_{\Omega}c_{W}2^{\frac{1}{n-1}}|\Omega|^{-\frac{1}{n-1}}$, then$\square$

5. ASYMPROTIC BEHAVIOR AS$\epsilonarrow 0$

In this section we prove a convergence result, as, of the traveling solutions of (2.1)
to the generalized traveling waves for (2.6). For $M>0$ and $k>0$ , let us introduce thenotations that will be used throughout the proofs in this section:$\Sigma_{M}:=\Omega\cross(-M, M) , \Vert G\Vert_{k,\infty}= \max |G(y, u$(5.1)$\epsilonarrow 0$$u\in[0,k],y\in\overline{\Omega}$

109
ASYMPTOTIC BEHAVIOR OF ATTRACTORS FOR INHOMOGENEOUS ALLEN-CAHN EQUATIONS

We begin with the following basic compactness result, proved in [13, Lemma 5.1].

. Let
be such that
Lemma 5.1. Let $c>0$ andas, for some $K>0$ independent of . Assume that:and$\epsilonarrow 0$$c_{6}arrow c$(5.2)$u_{\epsilon}\in H_{c_{\epsilon}}^{1}(\Sigma)$$k,$$\Phi_{c_{\epsilon}}^{\epsilon}(u_{\epsilon})\leq K$

such that$\exists\delta\in(0,1)$$0\leq u_{\epsilon}\leq k$$\epsilon$

for all$u_{\epsilon}(y, z)\leq\delta$

Then there exists $u\in BV_{loc}(\Sigma, \{0,1\})$ such thatin$u_{\epsilon}arrow u$

$u$

$z$

$y\in\Omega$

and $z>0.$) $=0$ for all $z>0$ and$L_{loc}^{1}(\Sigma)$,

upon extraction of a sequence.
We now state our main result.

Theorem 5.2. Let Assumptions 1-4 hold. Let
letbe as in Theorem 4.4.
i) There holds$c_{\epsilon}^{\dagger},$$\overline{u}_{\epsilon}$

and$v_{\epsilon}$be as in Theorem 3.4, and$c\dagger$(5.3)$\lim_{\epsilonarrow 0}c_{\epsilon}^{\uparrow}=c^{\dagger}.$ii) For every sequencesetsuch that$\epsilon_{n}arrow 0$

there exists a subsequence (not relabeled) and an open

$S\subset\Sigma$

in$\overline{u}_{\epsilon_{n}}arrow xs$

where

$S$

is a non-trivial minimizer of. Moreover,$\mathcal{F}_{c\dagger}$,satisfying$L_{loc}^{1}(\Sigma)$$S\subseteq\Omega\cross(-\infty, 0)$and$\partial S\cap$$(\overline{\Omega}\cross\{0\})\neq\emptyset$

locally uniformly on$\overline{u}_{\epsilon_{n}}arrow\chi_{S}$$\overline{\Sigma}\backslash \partial S,$

locally uniformly
converge toand for every $\theta\in(0,1)$ the level setsin the Hausdorﬀ sense.
iii) If also Assumption 5 holds, then is the unique minimizer offrom Theorem.
Moreoversatisfying
and4.5$\partial S$$\{\overline{u}_{\epsilon_{n}}=\theta\}$$S$$S\subseteq\Omega\cross(-\infty, 0)$$v_{\epsilon}arrow 1$$\mathcal{F}_{c\dagger}$$\partial S\cap(\overline{\Omega}\cross\{0\})\neq\emptyset$$unif_{07}mly$

in St.

Proof. The proof follows the same arguments as the proof of in [13, Theorem 5.3]. So
we just go through the steps of the proof for sake of completeness, pointing out the main
diﬀerences.
Step 1: we shall prove that$\lim_{\epsilonarrow}\inf_{0}c_{\epsilon}\dagger\geq c\dagger.$

The proof follows by the standard Modica-Mortola construction of a recovery sequence [17],and is a bit diﬀerent of that in [13, Theorem 5.3], due to the presence of the pressure term$\mu.$

Let
be as in Theorem 4.4. Then the hypersurface, where $K$ is a compact subset ofon all sets of the form
$K\cross \mathbb{R}$(5.4)

is of.class, and$\partial S_{\psi}\cap\Sigma$$S_{\psi}$$c_{W} Per_{c\dagger,\mu}(S_{\psi}, \Sigma)=\int_{S\psi}e^{cz}g(y)dx\dagger.$$\Omega$$C^{2}$

uniformly

110
A. CESARONI, C. B. MURATOV, AND M. NOVAGA

We consider$d_{S_{\psi}}$to be the signed distance function from$d_{S_{\psi}}(x)$and$\gamma$:$\mathbb{R}arrow \mathbb{R}$$:=dist(x, \Sigma\backslash S_{\psi})$–dist$u_{\epsilon}\in H_{c\dagger}^{1}(\Sigma)$(5.5), i.e.,$(x, S_{\psi})$to be the unique solution to $\gamma’=\sqrt{2W(\gamma)}$ with
$u_{\epsilon}:= \gamma(\frac{\mu d_{S\psi}}{\epsilon})$then$\partial S_{\psi}$

for all$\epsilon$

suﬃciently small and$\gamma(0)=\frac{1}{2}$

. Let

,

$u_{\epsilon}arrow\chi_{S\psi}$

as$\epsilonarrow 0$in$L_{loc}^{1}(\Sigma)$. We get$| \nabla u_{\epsilon}|=\gamma’(\frac{\mu d_{S_{\psi}}}{\epsilon})|\mu\frac{\nabla d_{S_{\psi}}}{\epsilon}+\nabla\mu\frac{d_{S_{\psi}}}{\epsilon}|$$\leq\frac{1}{\epsilon}\gamma’(\frac{\mu d_{S_{\psi}}}{\epsilon})(\mu+|\nabla\mu||d_{S_{\psi}}|)$$= \frac{\mu}{\epsilon}\sqrt{2W(u_{\epsilon})}+\frac{1}{\epsilon}\gamma’(\frac{\mu d_{S\psi}}{\epsilon})|\nabla\mu||d_{S_{\psi}}|.$

Analogously(5.6)$| \nabla u_{\epsilon}|\geq\frac{1}{\epsilon}\gamma’(\frac{\mu d_{S_{\psi}}}{\epsilon})(\mu-|\nabla\mu||d_{S_{\psi}}|)$$= \frac{\mu}{\epsilon}\sqrt{2W(u_{\epsilon})}-\frac{1}{\epsilon}\gamma’(\frac{\mu d_{S_{\psi}}}{\epsilon})|\nabla\mu||d_{S_{\psi}}|.$

Hence, from (5.5) and (5.6)(5.7)$| \sqrt{\frac{\epsilon}{2}}|\nabla u_{\epsilon}|-\mu\sqrt{\frac{W(u_{\epsilon})}{\epsilon}}|\leq\frac{|\nabla\mu|}{\sqrt{2\epsilon}}\gamma’(\frac{\mu d_{S_{\psi}}}{\epsilon})|d_{S_{\psi}}|.$We recall that, thanks to Assumption 3, there exist $A,$ $B>0$ such that $|s\gamma’(s)|\leq Ae^{-B|s|}$(see [11]). It follows thatfor all$s\in \mathbb{R}$$\frac{|\nabla\mu|^{2}}{2\epsilon}\gamma’(\frac{\mu d_{S\psi}}{\epsilon})^{2}|d_{S\psi}|^{2}\leq\frac{|\nabla\mu|^{2}}{2\mu^{2}}A^{2}\epsilon e^{-2B\frac{\mu|d_{S_{\psi}}|}{\epsilon}}$which implies$( \sqrt{\frac{\epsilon}{2}}|\nabla u_{\epsilon}|-\mu\sqrt{\frac{W(u_{\epsilon})}{\epsilon}})^{2}\leq C\epsilon e^{-\frac{C|d_{S_{\psi}}|}{\epsilon}}\leq C\epsilon\min(1, e^{-\frac{Cz}{\epsilon}})$for some $C=C(\mu, W)>0$ . Therefore$\int_{\Sigma}(\frac{\epsilon}{2}|\nabla u_{\epsilon}|^{2}+\mu^{2}\frac{W(u_{\epsilon})}{\epsilon})e^{cz}dx\dagger\leq\int_{\Sigma}\sqrt{2W(u_{\epsilon})}|\nabla u_{\epsilon}|\mu(y)e^{c^{\uparrow}z}dx+C\epsilon$

111
ASYMPTOTIC BEHAVIOR OF ATTRACTORS FOR INHOMOGENEOUS ALLEN-CAHN EQUATIONS

for$\epsilon$

suﬃciently small. It then follows$\Phi_{c\dagger}^{\epsilon}(u_{\epsilon}) \leq \int_{\Sigma}\sqrt{2W(u_{\epsilon})}|\nabla u_{\epsilon}|\mu(y)e^{c}dx-\dagger_{z}\int_{\Sigma}e^{cz}G(y, u_{\epsilon})dx+C\epsilon\dagger$$= \int_{\Sigma}|\nabla\phi(u_{\epsilon})|\mu(y)e^{cz}dx-\dagger\int_{\Sigma}e^{c^{\uparrow}z}G(y, u_{\epsilon})dx+C\epsilon,$

where

(5.8)

$\phi(u):=\int_{0}^{u}\sqrt{2W(s)}ds.$

Recalling the definition of in (2.5) and observing thatuniformly in the$\delta>0$
set, we can apply the Co-area Formula [3] and obtainfor anyasfrom (5.4), as$\phi(u_{\epsilon})arrow c_{W}\chi_{S_{\psi}}$$g$$\epsilonarrow 0$$\{|ds_{\psi}|\geq\delta\}$$\epsilonarrow 0,$$\Phi_{c\dagger}^{\epsilon}(u_{\epsilon}) = \int_{0}^{\infty}Per_{c\dagger,\mu}(\{\phi(u_{\epsilon})>t\}, \Sigma)dt-\int_{\Sigma}e^{c^{\uparrow Z}}G(y, u_{\epsilon})dx+C\epsilon$(5.9)$arrow c_{W}Per_{c\dagger,\mu}(S_{\psi}, \Sigma)-\int_{S_{\psi}}e^{cz}g(y)dx=0\dagger.$

Assume now by contradiction that there exists a sequence of. A simple computation (see [13, Lemma 5.2]) gives$c_{\epsilon}^{\dagger}$converging to a constant$c<c\dagger$(5.10)$\Phi_{c\dagger}^{\epsilon}(u_{\epsilon})\geq\frac{(c\dagger)^{2}-(c_{\epsilon}^{\dagger})^{2}}{(c\dagger)^{2}}\int_{\Sigma}e^{cz}\frac{\epsilon}{2}|(u_{\epsilon})_{z}|^{2}dx\dagger,$and observe that by the definition of$u_{\epsilon}$

and the regularity of$\partial S\psi$, we get that$|\nabla\phi(u_{\epsilon})|=\epsilon|\nabla u_{\epsilon}|^{2}\leq 2\epsilon|(u_{\epsilon})_{z}|^{2},$in a ball $B(x, r)$ for some $r>0$ , where $x=(y, z)\in\partial S\psi$ and $y\in$ St is a point at which
attains its maximum. Combining these two facts yields$\psi$$\underline{\mu(y)}e^{c^{\uparrow}z}|\nabla\phi(u_{\epsilon})|dx$$\Phi_{c^{\uparrow}}^{\epsilon}(u_{\epsilon}) \geq \frac{(c^{1})^{2}-(c_{\epsilon}^{\dagger})^{2}}{4(c\dagger)^{2}}\int_{\Sigma\cap B(x,r)^{\sup_{\Omega}\mu}}$$arrow \frac{(c^{1})^{2}-c^{2}}{4(c\dagger)^{2}\sup_{\Omega}\mu}c_{W}Per_{c\dagger,\mu}(S_{\psi}, \Sigma\cap B(x, r))>0,$

which then contradicts (5.9).
Step 2: let us now prove (i). By Theorem 3.4,
is bounded from above by a constant$c\in[0,+\infty$
independent of . In particular, there exists, along) such thatasa sequence. By Step 1 we have
, so that it is enough to prove thatfor every
sequence. By Lemma 5.1 and the properties ofin Theorem 3.4 we get, up to a
subsequence, that$c_{\epsilon}^{\dagger}$$c_{\epsilon}^{\dagger}arrow c$$\epsilon$$c\geq c^{\uparrow}$$\epsilonarrow 0$(5.11)$u^{\epsilon}$$\overline{u}_{\epsilon}arrow\chi s$$\epsilonarrow 0$$c\leq c^{\uparrow}$in$L_{loc}^{1}(\Sigma)$,

112
A. CESARONI, C. B. MURATOV, AND M. NOVAGA. Moreover
where
and
and the fact that
Using the minimality of$M$ , we conclude$S\subseteq\Omega\cross(-\infty, 0)$$\chi_{S}\in BV_{loc}(\Sigma)$$S$

is not trivial by density estimates.
, for everyin$L^{1}(\Sigma_{M})$$\phi(u_{\epsilon})arrow\phi(\chi_{S})=cW\chi s$$u^{\epsilon}$(5.12)$\mathcal{F}_{c}(S)\leq 0,$

which implies that$c\leq c^{\uparrow}$

by Proposition 4.3 and following remark.

. Therefore, by
Step 3: we now prove (ii). By (5.3) it follows that (5.12) holds with
Proposition 4.3 the inequality in (5.12) is in fact an equality, and by the density estimate, satisfying all the desired properties.
(3.12) the set
is a non-trivial minimizer of) that satisfies all the:
Furthermore,is the subgraph of a function
conclusions of Theorem 4.4. The convergence in Hausdorﬀ sense of the level sets is aconsequence of the density estimates. The locally uniform convergence ofto $xs$ outside
in thetois a direct consequence of the convergence of the level sets
Hausdorﬀ sense.
$c=c\dagger$$S$$\mathcal{F}_{c\dagger}$$S$$\psi$$\Omegaarrow[-\infty, \infty$$\overline{u}_{\epsilon}$$\partial S$$\partial S$$\{\overline{u}_{e}=\theta\}$

Step 4: it remains to prove (iii). By Theorem 4.5 there exists a unique minimizer of, and soinsuch thatforandevery $M>0$ . Moreover, in this case is the subgraph of a bounded function defined in. So, so we get that $\chi_{S}(y, z)\equiv 1$ for all
uniformly inand$\Omega\cross(-\infty, M]$ for every$S$$\partial S\cap(\overline{\Omega}\cross\{0\})\neq\emptyset$$S\subseteq\Omega\cross(-\infty, 0)$$\mathcal{F}_{c^{\uparrow}}$$\overline{u}_{\epsilon}arrow xs$$S$$\psi$$y\in\Omega$$\Omega$$L^{1}(\Sigma_{M})$$\overline{u}_{\epsilon}(y, z)arrow 1$$z< \min_{\overline{\Omega}}\psi$$M< \min_{\overline{\Omega}}\psi.$$\square$

Theorem 5.2 implies that the level sets of solutions of the initial value problem withgeneral front-like linitial data for
asymptotically spread with average speed thatapproaches as. Moreover, under the stronger assumption 5, we can show that the, to a traveling wave solution to (2.6)
long-time limit of such solutions converges, as
moving with speed . For the proof we refer to [13].
$\epsilon\ll 1$$c^{\uparrow}$$\epsilonarrow 0$$\epsilonarrow 0$$c^{\uparrow}$

Corollary 5.3. Let Assumptions 2, 1, 3 and 4 hold. Let $\delta>0$ be such that$(1-u)f(u)>0$
let$u_{0}^{\epsilon}\in W^{1,\infty}(\Sigma)\cap L_{c_{\epsilon}^{\dagger}}^{2}(\Sigma)$(5. 13)for all$u\in[1-\delta, 1$) $\cup(1, 1+\delta$ ],be such that$0\leq u_{0}^{\epsilon}\leq 1+\delta$

and$\lim_{zarrow}\underline{\inf_{\infty}}u_{0}^{\epsilon}(y, z)\geq 1-\delta$

and letbe the solution of (2.1) with initial datumi) For every $\theta\in(0, 1)$

uniformly in$\Omega,$$u^{\epsilon}$(5.14)$u_{0}^{\epsilon}.$$\lim_{\epsilonarrow 0}\lim_{tarrow\infty}\frac{\sup\{z\in \mathbb{R}:u^{\epsilon}(y,z,t)>\theta forsomey\in\Omega\}}{t}=c\dagger.$ii) If also Assumption 5 holds, then there exists(5.15)$R_{\infty}\in \mathbb{R}$such that, for all $M>0,$$\lim_{\epsilonarrow 0}\lim_{tarrow\infty}\Vert u^{\epsilon}(y, z+c_{\epsilon}\dagger t+R_{\infty}, t)-\chi s_{\psi}(y, z)\Vert_{L^{1}(\Sigma_{M})}=0,$

where is given by Theorem 4.5. Moreover, the convergence as
to the limit
is locally uniform in$\psi$$tarrow\infty$$\overline{\Sigma}\backslash \partial S_{\psi}.$$\epsilonarrow 0$

afler passing

113
ASYMPTOTIC BEHAVIOR OF ATTRACTORS FOR INHOMOGENEOUS ALLEN-CAHN EQUATIONSREFERENCES[1] M. Alfaro, H. Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and
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$\Gamma$

A. CESARONI, DIPARTIMENTO DI MATEMATICA, UNIVERSIT\‘A DI PADOVA, VIA TRIESTE 63, 35121
PADOVA, ITALY-mar l address: acesar@math.unipd.it
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A. CESARONI, C. B. MURATOV, AND M. NOVAGA

C. B. MURATOV, DEPARTMENT OF MATHEMATICAL SCIENCES, NEW JERSEY INSTITUTE OF TECHNOL, NEWARK, NJ 07102, USA$OGY$-mail address: muratovOnj it. edu$E$

M. NOVAGA, DIPARTIMENTO$DI$

MATEMATICA, UNIVERSIT\‘A

56127 PISA, ITALY-mail address: novagaOdm. unipi. it

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PISA, LARGO BRUNO PONTECORVO 5,