# Lyapunov Stability of Planar Waves to the Reaction-Diffusion Equation with a Non-Lipschitzian Reaction Term

Authors: Soyeun Jung; Eunkyung Ko
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

Extending (Drábek and Takáč 2017), we investigate the Lyapunov stability of planar waves for the reaction-diffusion equation onℝn, n≥2, with a α-Ho¨lder continuous (0<α<1), but not necessarily smooth reaction term. We first consider an initial value problem for the equation and then construct sub- and supersolutions to the problem by a subtle modification of the planar wave. Our main result states that a bounded classical solution to the problem stays near the planar wave for all time whenever an initial data is close enough to the planar wave.


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## Body

## 1. Introduction

We consider a reaction-diffusion equation onℝn(1)ut=Δxu+fu,where x=x1,x2,⋯,xn∈ℝn, n≥2, Δx=∂x12+∂x22+⋯+∂xn2, t>0, ux,t∈ℝ, and f:ℝ⟶ℝ. In particular, we are concerned with the stability of the planar wave of the form
(2)ux,t=Ux1−ct,U−∞=0,U+∞=1,which is the most widely studied type of travelling wave to (1) with a continuously differentiable reaction term f (e.g., see [1–3] and the references therein). However, in the present paper, we assume the reaction term f is not necessarily Lipschitz continuous, but only α-Ho¨lder continuous (0<α<1) and one-sided Lipschitz continuous (see (H3) in Section 1.2). A typical example for a non-Lipschitzian reaction term f is
(3)fu=c1uα01−uα1u−c2,0≤u≤1,for some constants α0,α1∈0,1, c1∈ℝ, and c2∈0,1, which has the singular derivatives at u=0 and u=1. Equation (1) with this type of f is more realistic and has been used extensively as biological models, in particular, Fisher’s model for population genetics. If α0=α1=1 in the reaction function (3), the product u1−u represents the classical logistic growth of the population. The assumption requiring very large birth or death rate of the population leads to the formula uα01−uα1 with α0,α1∈0,1 in (3), which gives the restriction on the differentiability of f (see the classical work of population genetics [4, 5], or [6, 7] for the derivation of f).The restriction on the reaction functionf makes us unable to linearize equation (1) about the planar wave (2) and to use the spectral analysis which is a standard method to study of stability of travelling waves. Instead, we construct sub- and supersolutions to (1) by an appropriate modification of the planar wave U, and then we show that both the planar wave U and a unique solution to a Cauchy problem (1) with an initial data near U are trapped by the sub- and supersolutions whose difference is sufficiently small. This method has also been used in [3] to prove the stability of planar waves (2) in the Allen-Cahn equation on ℝn, n≥2, with the continuously differentiable reaction term (i.e., α0=α1=1 in (3)). They studied that the planar wave is asymptotically stable under any initial perturbations that decay at space infinity or almost periodic perturbations. In their works, the continuous differentiability of the reaction term is necessary to construct sub- and supersolutions and to obtain the convergence rate by using the idea of mean curvature flow on ℝn−1 (see [8] for the idea of mean curvature). The main purpose in our project is to study the stability of the planar wave in ℝn, n≥2, without the differentiability of the reaction term.Our analysis is totally motivated by the results of Drábek and Takáč [6], showing that the long-time asymptotic behavior of solutions to an initial value problem of a one-dimensional reaction-diffusion equation
(4)ut=uxx+fuforx,t∈ℝ×ℝ+;ux,0=u0xforx∈ℝ,with a non-Lipschitzian reaction term f defined by (3). By constructing sub- and supersolutions without the differentiability of f, they established the convergence of a solution ux,t to a travelling wave solution Ux−ct, that is,
(5)supx∈ℝux,t−Ux−ct+ζ⟶0ast⟶∞,for some spatial shift ζ∈ℝ, when the initial data u0 is close enough to U as x goes to ±∞. However, they restricted the model to the simple case of one space variable by assuming a habitat of a population is a one-dimensional space, for example, a long thin strip along a straight shoreline. In order to make the model to be more realistic, we extend their method to a multidimensional space and prove the planar wave is stable under small initial perturbations in L∞ℝn with n≥2.The purpose of this introduction is to provide information of the profile of the planar wave (2) and the precise assumptions on the reaction term f and to state our main result.


### 1.1. The Profile of Planar Waves

In order to study of stability of the planar wave (2), we first introduce the x1-moving coordinate with speed c by setting
(6)ux,t=vz,t,z=z1,z2,⋯,zn=x1−ct,x2,⋯,xn,so that the planar wave can be considered as a stationary solution. In the z,t coordinates, equation (1) reads
(7)vt=Δzv+cvz1+fv,and the planar wave is then a stationary solution vz,t=Uz1 that satisfies the profile equation
(8)0=U″z1+cU′z1+fUz1,limz1⟶−∞Uz1=0andlimz1⟶∞Uz1=1.The existence and monotonicity of such profiles have been studied in [1, DT2]. According to Section2 of [6], the assumption (H1) stated in Section 1.2 guarantees C2-profiles U:ℝ⟶0,1 for some speed c=c∗. If Uz1 satisfies (8), then its translate Uz1−ζ also satisfies (8) for any constant ζ∈ℝ. Throughout this paper, we impose the condition
(9)U0=s0,for some constant s0∈0,1 that appears in (H1), so that the C2-profile U:ℝ⟶0,1 satisfying (8) is unique for some unique speed c=c∗. Moreover, under the assumption (H1), the profile is nondecreasing on ℝ and there is an open interval a,b⊆ℝ, −∞≤a<0<b≤∞, such that 0≤Uz1≤1 and U′z1>0 on a,b. The asymptotic behavior of the profile is determined solely by the behavior of fs as s⟶0+ and s⟶1−. It is well-known that a=−∞ and b=∞ in the classical case, i.e., f∈C1ℝ. However, in our case of f being non-Lipschitzian at the points s=0 and s=1, one has −∞<a<0<b<∞. More precisely, a non-Lipschitzian f satisfies that there exist positive constants γ0 and γ1 such that
(10)lims⟶0+fssα0=−γ0<0andlims⟶1−fs1−sα1=γ1>0,α0,α1∈0,1,(see a typical example (3) of f). It is obvious that the limits of (10) are −∞ and ∞, respectively, in the case of f∈C1ℝ (see [6, 9] for further details).In summary, the planar wavevz,t=Uz1 satisfies
(11)0=U″z1+cU′z1+fUz1,U0=s0,and there is an open interval a,b⊂ℝ, −∞<a<0<b<∞, such that
(12)Uz1=0for−∞<z1≤a,U′z1>0fora<z1<b,Uz1=1forb≤z1<∞.


### 1.2. Hypotheses on the Reaction Termf

Throughout this paper, following [6], we assume the reaction term f satisfies the following:(H1).f:ℝ⟶ℝ is continuous such that f0=fs0=f1=0 for some 0<s0<1. Moreover, fs<0 for any s∈0,s0∩1,∞, fs>0 for all s∈−∞,0∩s0,1, and for any 0<r≤1,
(13)Fr≔∫0rfsds<0.(H2).f is α-Ho¨lder continuous (0<α<1).(H3).f is one-sided Lipschitz continuous, that is, there exists a positive number L such that
(14)fs′−fs≤Ls′−sforanys,s′∈ℝwiths<s′.(H4). There exists a positive constant(15)η0<14mins0,1−s0,satisfying that for any η∈0,η0 there are positive constants δ∈0,η, μ¯ and μ¯ such that
(16)inf0≤s≤δfs−fs+q≥μ¯qforallq∈0,s0−2η,(17)inf1−δ≤s≤1fs−q−fs≥μ¯qforallq∈0,1−s0−2η.As mentioned in the previous subsection, the first assumption (H1) is needed to show the existence and monotonicity of the profileUz1. The last assumption (H4), referred to as the secant conditions, holds trivially if f∈C1ℝ is a classical bistable type and δ>0 is small enough. We can also assume (H4) even for a non-Lipschitzian reaction term f. The reader can see Figure 1 in [6] for an example satisfying (H4).Figure 1
Planar waveUx1−ct on ℝ2.


### 1.3. Main Result

We now state our main theorem. Theorem1 says that if the equation (7) starts with an initial data v0z near the planar wave Uz1 then the solution to (7) stays near it for all time.Theorem 1 (Lyapunov stability).
Let the assumptions (H1)–(H4) hold andn≥2. Suppose that a function v0z∈L∞ℝn satisfies that for all z∈ℝn,
(18)0≤v0z≤1andv0z−Uz1<E0,where E0>0 is sufficiently small. Then, there is a unique bounded classical solution vz,t to (7) with an initial data v0z such that for all z∈ℝn and all time t>0,
(19)0≤vz,t≤1andvz,t−Uz1<CE0,for some constant C>0.Remark 2.
Theorem1 is the extension of the one-dimensional stability result (Proposition 4.1) of [6] to a multidimensional space. Our assumption for the profile U0=s0 gives ζ=0 in their result.The paper is organized as follows. In Section2, we first consider the Cauchy problem
(20)vt=Δzv+cvz1+fv,vz,0=v0z,with an initial data v0z∈L∞ℝn, n≥2. Under the assumption (H2), we establish regularity estimates so that a bounded mild solution to (20) is well-defined, and it becomes a bounded classical solution to (20). In Section 3, we define weak sub- and supersolutions to (20) and discuss the weak comparison principle for them under the assumption (H3). This section will also show the uniqueness of the solution to (20). Finally, in Section 4, using the assumption (H4), we modify the planar wave U to construct L∞ sub- and supersolutions to (20) and prove Theorem 1 by showing both the solution to (20) and the planar wave U stay between sub- and supersolutions when the initial data v0z is close enough to U.


### 1.4. Discussion and Open Problems

Theorem1 does work if E0 is sufficiently small that 0<E0<η0<1/4mins0,1−s0. We use the assumption (H4) to prove suitable modifications of the planar wave Uz1−ξt−qt and Uz1+ξt+qt are sub- and supersolutions of (20) for some functions ξt and qt. By setting q0=2E0 and η=E0 in (H4), we can apply the inequalities (16) and (17) to the proof of Theorem 1.The main idea of our work follows [6], but in the present paper, we do not impose one of their assumptions, saying (H5). The assumption (H5) in [6] means that the initial perturbation is small enough only at space infinity, so the assumption (18) for an initial data implies their assumption (H5). Indeed, they proved the travelling wave is asymptotically stable by showing the solution of (20) converges to the travelling wave as t goes to infinity when the initial data satisfies (H5) (see (5)). So the initial perturbation does not need to be small in their work. However, unfortunately, the assumption (H5) is not enough to prove the convergence, even the stability, for our multidimensional case. It seems that we need more assumptions on the initial perturbation or the reaction term f to prove the asymptotic stability of the planar wave on ℝn with n≥2, which is a interesting open problem.Moreover, even inℝ, the convergence rate for a non-Lipschitzian reaction term f has not been proven yet. The asymptotic stability with the convergence rate of the planar wave (2) on ℝn, n≥2, for a continuously differentiable f has been studied in [3]. They also modified the planar wave to construct sup- and supersolutions by using more delicate phase functions thanks to the differentiability of f. The study of convergence rate on ℝ for a non-Lipschitzian f would be another very interesting direction to carry out.


## 1.1. The Profile of Planar Waves

In order to study of stability of the planar wave (2), we first introduce the x1-moving coordinate with speed c by setting
(6)ux,t=vz,t,z=z1,z2,⋯,zn=x1−ct,x2,⋯,xn,so that the planar wave can be considered as a stationary solution. In the z,t coordinates, equation (1) reads
(7)vt=Δzv+cvz1+fv,and the planar wave is then a stationary solution vz,t=Uz1 that satisfies the profile equation
(8)0=U″z1+cU′z1+fUz1,limz1⟶−∞Uz1=0andlimz1⟶∞Uz1=1.The existence and monotonicity of such profiles have been studied in [1, DT2]. According to Section2 of [6], the assumption (H1) stated in Section 1.2 guarantees C2-profiles U:ℝ⟶0,1 for some speed c=c∗. If Uz1 satisfies (8), then its translate Uz1−ζ also satisfies (8) for any constant ζ∈ℝ. Throughout this paper, we impose the condition
(9)U0=s0,for some constant s0∈0,1 that appears in (H1), so that the C2-profile U:ℝ⟶0,1 satisfying (8) is unique for some unique speed c=c∗. Moreover, under the assumption (H1), the profile is nondecreasing on ℝ and there is an open interval a,b⊆ℝ, −∞≤a<0<b≤∞, such that 0≤Uz1≤1 and U′z1>0 on a,b. The asymptotic behavior of the profile is determined solely by the behavior of fs as s⟶0+ and s⟶1−. It is well-known that a=−∞ and b=∞ in the classical case, i.e., f∈C1ℝ. However, in our case of f being non-Lipschitzian at the points s=0 and s=1, one has −∞<a<0<b<∞. More precisely, a non-Lipschitzian f satisfies that there exist positive constants γ0 and γ1 such that
(10)lims⟶0+fssα0=−γ0<0andlims⟶1−fs1−sα1=γ1>0,α0,α1∈0,1,(see a typical example (3) of f). It is obvious that the limits of (10) are −∞ and ∞, respectively, in the case of f∈C1ℝ (see [6, 9] for further details).In summary, the planar wavevz,t=Uz1 satisfies
(11)0=U″z1+cU′z1+fUz1,U0=s0,and there is an open interval a,b⊂ℝ, −∞<a<0<b<∞, such that
(12)Uz1=0for−∞<z1≤a,U′z1>0fora<z1<b,Uz1=1forb≤z1<∞.


## 1.2. Hypotheses on the Reaction Termf

Throughout this paper, following [6], we assume the reaction term f satisfies the following:(H1).f:ℝ⟶ℝ is continuous such that f0=fs0=f1=0 for some 0<s0<1. Moreover, fs<0 for any s∈0,s0∩1,∞, fs>0 for all s∈−∞,0∩s0,1, and for any 0<r≤1,
(13)Fr≔∫0rfsds<0.(H2).f is α-Ho¨lder continuous (0<α<1).(H3).f is one-sided Lipschitz continuous, that is, there exists a positive number L such that
(14)fs′−fs≤Ls′−sforanys,s′∈ℝwiths<s′.(H4). There exists a positive constant(15)η0<14mins0,1−s0,satisfying that for any η∈0,η0 there are positive constants δ∈0,η, μ¯ and μ¯ such that
(16)inf0≤s≤δfs−fs+q≥μ¯qforallq∈0,s0−2η,(17)inf1−δ≤s≤1fs−q−fs≥μ¯qforallq∈0,1−s0−2η.As mentioned in the previous subsection, the first assumption (H1) is needed to show the existence and monotonicity of the profileUz1. The last assumption (H4), referred to as the secant conditions, holds trivially if f∈C1ℝ is a classical bistable type and δ>0 is small enough. We can also assume (H4) even for a non-Lipschitzian reaction term f. The reader can see Figure 1 in [6] for an example satisfying (H4).Figure 1
Planar waveUx1−ct on ℝ2.


## 1.3. Main Result

We now state our main theorem. Theorem1 says that if the equation (7) starts with an initial data v0z near the planar wave Uz1 then the solution to (7) stays near it for all time.Theorem 1 (Lyapunov stability).
Let the assumptions (H1)–(H4) hold andn≥2. Suppose that a function v0z∈L∞ℝn satisfies that for all z∈ℝn,
(18)0≤v0z≤1andv0z−Uz1<E0,where E0>0 is sufficiently small. Then, there is a unique bounded classical solution vz,t to (7) with an initial data v0z such that for all z∈ℝn and all time t>0,
(19)0≤vz,t≤1andvz,t−Uz1<CE0,for some constant C>0.Remark 2.
Theorem1 is the extension of the one-dimensional stability result (Proposition 4.1) of [6] to a multidimensional space. Our assumption for the profile U0=s0 gives ζ=0 in their result.The paper is organized as follows. In Section2, we first consider the Cauchy problem
(20)vt=Δzv+cvz1+fv,vz,0=v0z,with an initial data v0z∈L∞ℝn, n≥2. Under the assumption (H2), we establish regularity estimates so that a bounded mild solution to (20) is well-defined, and it becomes a bounded classical solution to (20). In Section 3, we define weak sub- and supersolutions to (20) and discuss the weak comparison principle for them under the assumption (H3). This section will also show the uniqueness of the solution to (20). Finally, in Section 4, using the assumption (H4), we modify the planar wave U to construct L∞ sub- and supersolutions to (20) and prove Theorem 1 by showing both the solution to (20) and the planar wave U stay between sub- and supersolutions when the initial data v0z is close enough to U.


## 1.4. Discussion and Open Problems

Theorem1 does work if E0 is sufficiently small that 0<E0<η0<1/4mins0,1−s0. We use the assumption (H4) to prove suitable modifications of the planar wave Uz1−ξt−qt and Uz1+ξt+qt are sub- and supersolutions of (20) for some functions ξt and qt. By setting q0=2E0 and η=E0 in (H4), we can apply the inequalities (16) and (17) to the proof of Theorem 1.The main idea of our work follows [6], but in the present paper, we do not impose one of their assumptions, saying (H5). The assumption (H5) in [6] means that the initial perturbation is small enough only at space infinity, so the assumption (18) for an initial data implies their assumption (H5). Indeed, they proved the travelling wave is asymptotically stable by showing the solution of (20) converges to the travelling wave as t goes to infinity when the initial data satisfies (H5) (see (5)). So the initial perturbation does not need to be small in their work. However, unfortunately, the assumption (H5) is not enough to prove the convergence, even the stability, for our multidimensional case. It seems that we need more assumptions on the initial perturbation or the reaction term f to prove the asymptotic stability of the planar wave on ℝn with n≥2, which is a interesting open problem.Moreover, even inℝ, the convergence rate for a non-Lipschitzian reaction term f has not been proven yet. The asymptotic stability with the convergence rate of the planar wave (2) on ℝn, n≥2, for a continuously differentiable f has been studied in [3]. They also modified the planar wave to construct sup- and supersolutions by using more delicate phase functions thanks to the differentiability of f. The study of convergence rate on ℝ for a non-Lipschitzian f would be another very interesting direction to carry out.


## 2. A Bounded Classical Solution

In this section, we consider the Cauchy problem onℝn,
(21)vt=Δzv+cvz1+fv,vz,0=v0z,where z=z1,z2,⋯,zn, n≥2, an initial data v0:ℝn⟶ℝ is Lebesgue-measurable, and 0≤v0z≤1 for all z∈ℝn. Especially, under the assumption (H2), we prove the existence of a bounded classical solution to (21) in the sense that a mild solution v, defined in (25), of (21) satisfies
(22)vt,vz1,Δzv∈Cα,α/2ℝn×τ,T,for any given 0<τ<T<∞ and the weak star limit vz,t∗v0z in L∞ℝn as t⟶0+. Here, for any α∈0,1 and T>0, the norm of Ho¨lder space Cα,α/2ℝn×0,T is given by
(23)gCα,α/2ℝn×0,T≔gL∞ℝn×0,T+gα,α/2,where
(24)gα,α/2≔supz,z′∈ℝn,0<z−z′≤1,t∈0,Tgz,t−gz′,tz−z′α+supz∈ℝn,0≤t<t′≤Tgz,t−gz,t′t−t′α/2.Applying Duhamel’s principle to (21) yields an integral equation of v:
(25)vz,t=∫ℝnGz−y,tv0ydy+∫0t∫ℝnGz−y,t−sfvy,sdyds≕G1tv0z+G2f∘vz,t,for any z,t∈ℝn×0,∞. Here, for any y=y1,y2,⋯,yn, the heat kernel G is defined by
(26)Gz−y,t≔14πtn/2e−z1−y1+ct2+z2−y22+⋯+zn−yn2/4t.It is well known that if the initial value problem (21) has a solution, this solution is given by (25), referred to as the mild solution of (21). However, it is not trivial that every mild solution is a classical solution. So the first step is to establish the regularity estimates (22) for any bounded mild solution and then we will consider well-posedness of a bounded mild solution (25). The estimates of the heat potentials in Ho¨lder norm have been established in Section 4 of [10], which lead to the following lemma.Lemma 3 (Regularity estimates).
Assume thatf:ℝ⟶ℝ satisfies the condition (H2). If vz,t is a bounded mild solution of (21), then for any given 0<τ<T<∞(27)vt,vz1,Δzv∈Cα,α/2ℝn×τ,T,and we have the estimates
(28)vt,vz1,Δzvz,tCα,α/2ℝn×τ,T≤Mτ,TvL∞ℝn×0,T+f∘vL∞ℝn×0,T,for some constant Mτ,T>0, dependent upon τ and T.Proof.
We first fix anyτ and T with 0<τ<T<∞. A direct calculation gives that for all z,t∈ℝn×0,T,
(29)∂k+m1+m2+⋯+mn∂tk∂z1m1∂z2m2⋯∂znmnG1tv0z≤Mk,m1,⋯,mnt−k−m1+m2+⋯+mn/2v0L∞ℝn,and for all z,t,z,t′∈ℝn×0,T with t≠t′ and all i=1,2,⋯,n,
(30)∂∂ziG2f∘vz,t,G2f∘vz,t′−G2f∘vz,tt′−t1/2≤MTf∘vL∞ℝn×0,T.
The second inequality of (30) is proved in Chapter 4 of [10]. We now fix t0 with 0<t0<τ and replace the initial time t=0 by t=t0. By (29) and the Ho¨lder estimates in [10], we have for all i=1,2,⋯,n,
(31)G1tv0t,G1tv0z1,G1tv0ziziz,tCα,α/2ℝn×τ,T≤Mt0,τ,Tv·,t0L∞ℝnandG2f∘vt,G2f∘vz1,G2f∘vziziz,tCα,α/2ℝn×τ,T≤Mτ,Tf∘vCα,α/2ℝn×t0,T.
Sincev·,t0L∞ℝn≤vL∞ℝn×0,T, to obtain the estimate (28), it is enough to show that
(32)f∘vCα,α/2ℝn×t0,Tf∘vL∞ℝn×0,T.
We notice that the inequalities (29) and (30) yield that for all z,t,z,t′∈ℝn×t0,T with t≠t′,
(33)∇vz,t,vz,t′−vz,tt′−t1/2≤Mt0,Tv·,t0L∞ℝn+f∘vL∞ℝn×0,T.
Sincef is α-Ho¨lder continuous, the estimates (32) give
(34)sup0<z−z′≤1,t∈t0,Tf∘vz,t−f∘vz′,tz−z′α=sup0<∣z−z′∣≤1,t∈t0,Tf∘vz,t−f∘vz′,tvz,t−vz′,tα·vz,t−vz′,tαz−z′α≤Mτ,TvL∞ℝn×0,T+f∘vL∞ℝn×0,T,and similarly
(35)supz∈ℝn,t0≤t<t′≤Tf∘vz,t−f∘vz,t′t−t′α/2=supz∈ℝn,t0≤t<t′≤Tf∘vz,t−f∘vz,t′vz,t−vz,t′α·vz,t−vz,t′αt−t′α/2≤Mτ,TvL∞ℝn×0,T+f∘vL∞ℝn×0,T.
Finally, we have(36)vt,vz1,Δvz,tCα,α/2ℝn×τ,T≤Mτ,TvL∞ℝn×0,T+f∘vL∞ℝn×0,T.☐Noting thatf∘v is the source term of the inhomogeneous initial value problem (21), we now apply the regularity estimates (28) and Tikhonov fixed point theorem to show the well-posedness of a bounded mild solution (25).Lemma 4.
Assume thatf:ℝ⟶ℝ satisfies the condition (H2). If an initial data v0z∈L∞ℝn satisfies 0≤v0z≤1, then the initial value problem (21) has a bounded mild solution vz,t∈L∞ℝn×0,∞ such that 0≤vz,t≤1 for all z,t∈ℝn×0,∞.Proof.
We first consider the existence of a bounded mild solutionvz,t on ℝn×0,T, where
(37)T≔1sup−1≤r≤2fr>0.
We define the locally convex vector space(38)X≔L∞ℝn×0,T∩Cℝn×0,T,and the closed convex subset of X(39)C≔w∈X:wz,t≤1forallz,t∈ℝn×0,T.
By recalling a mild solution (25), let us consider an operator F:X⟶X defined by
(40)Fwz,t≔∫0t∫ℝnGz−y,t−sfvy,sdyds,∀z,t∈ℝn×0,T,where
(41)vz,t=∫ℝnGz−y,tv0ydy+wz,t.
We first proveFC⊂C. For any given w∈C, since ∫ℝnGz−y,tdy=1, v0z∈0,1, and wz,t≤1 for all z,t∈ℝn×0,T, it follows
(42)−1≤vz,t≤∫ℝnGz−y,tv0ydy+wz,t≤2.
Thus, for anyz,t∈ℝn×0,T(43)Fwz,t≤∫0t∫ℝnGz−y,t−sfvy,sdyds≤sup−1≤v≤2fvT=1,which implies F maps C into itself. We now notice that for any w1, w2∈C,
(44)Fw1z,t−Fw2z,t≤M∫0t∫ℝnGz−y,t−sw1y,s−w2y,sαdyds≤MTsupz∈ℝn,t∈0,Tw1x,t−w2x,tα,for some constant M>0 satisfying fr−fr′≤Mr−r′α for any r, r′∈ℝ, which means that the operator F is continuous from C into itself. Moreover, the image FC is relatively compact because FC is uniformly bounded for all w∈C and FC is equicontinuous by the estimates (30). Hence, by Tikhonov’s fixed point theorem, there is w^z,t∈C such that Fw^=w^. Recalling (39) and (40) gives that for all z,t∈ℝn×0,T,
(45)v^z,t≔∫ℝnGz−y,tv0ydy+w^z,t=∫ℝnGz−y,tv0ydy+∫0t∫ℝnGz−y,t−sfv^y,sdyds,which is a bounded mild solution to (21). Therefore, we obtain the existence of a bounded mild solution v:ℝn×0,∞⟶ℝ by replacing the initial time t=0 by t=t0 for any t0>0 and repeating the above procedure in each time interval t0,t0+T.☐


## 3. A Bounded Weak Solution and a Weak Comparison Principle

In the previous section, we have proved that a mild solution to (21) enjoys regularity estimates for a classical solution. We now prove the uniqueness of the classical solution by showing that (21) possesses at most one weak solution. As a starting point, we define a weak L∞ sub- and supersolution of (21), and we then establish a comparison principle for them. The weak comparison principle concludes the uniqueness of a weak solution and plays an important role in the proof of stability in the next section.Definition 5 (WeakL∞ sub- and supersolution).
Letv0z∈L∞ℝn. By a weak L∞ supersolution of (21), we mean a function v¯z,t∈L∞ℝn×0,∞ satisfying the following three conditions:
(i)
For all nonnegative test functionϕz∈W01,1ℝn,
(46)limsupt⟶0+∫ℝnv¯z,t−v0zϕzdz≥0.(ii)
v¯ is Lipschitz-continuous in every set ℝn×τ,T whenever 0<τ<T<∞, that is, ∂v¯/∂zi, ∂v¯/∂t∈L∞ℝn×τ,T for all i=1,⋯,n(iii)
For all nonnegative test functionϕz∈W01,1ℝn,
(47)∫ℝnv¯tz,tϕzdz+∫ℝn∇v¯z·∇ϕzdz−c∫ℝnv¯z1z,tϕzdz≥∫ℝnfv¯z,tϕzdz.
By a weakL∞ subsolution of (21), we mean a function v¯x,t∈L∞ℝn×0,∞ satisfying the above three conditions (i), (ii), and (iii) replaced the reverse inequality in (45) and (46). We say that a function v∈L∞ℝn×0,∞ is a weak L∞ solution of (21) if v is a weak L∞ sub- and supersolution to (21).We now establish a weak comparison principle for a weakL∞ sub- and supersolution under the crucial hypothesis (H3) which is one-sided Lipschitz continuity of f.Theorem 6 (Weak comparison principle).
Letf:ℝ⟶ℝ be continuous with the one-sided Lipschitz condition (H3). Assume that v¯,v¯∈L∞ℝn×0,∞ are weak L∞ sub- and supersolutions to (21), respectively, such that v¯z,0≤v¯z,0 a.e. in ℝn. Then, we have v¯z,t≤v¯z,t a.e. in ℝn×0,∞.Proof.
Subtracting (46) from the reverse inequality of (46) for a subsolution yields
(48)∫ℝnv¯−v¯tϕdz+∫ℝn∇v¯−v¯·∇ϕdz−c∫ℝnv¯−v¯z1ϕdz≤∫ℝnfv¯−fv¯ϕdz,for any nonnegative function ϕ∈W01,1ℝn. Let v¯−v¯+≔maxv¯−v¯,0, and we then prove that v¯−v¯+=0 a.e. in ℝn×0,∞. For any given nonnegative ϕ∈W01,1ℝn, since v¯−v¯+ϕ is also a nonnegative test function, replacing ϕ by v¯−v¯+ϕ in (47) gives
(49)∫ℝnv¯−v¯tv¯−v¯+ϕdz+∫ℝn∇v¯−v¯·∇v¯−v¯+ϕdz−c∫ℝnv¯−v¯z1v¯−v¯+ϕdz≤∫ℝnfv¯−fv¯v¯−v¯+ϕdz.
From a simple calculation, we evaluate that(50)∫ℝnv¯−v¯tv¯−v¯+ϕdz=12ddt∫ℝnv¯−v¯+2ϕdz,(51)∫ℝn∇v¯−v¯·∇v¯−v¯+ϕdz=∫ℝn∇v¯−v¯+2ϕdz+12∫ℝn∇v¯−v¯+2·∇ϕdz,(52)∫ℝnv¯−v¯z1v¯−v¯+ϕdz=12ddz1∫ℝnv¯−v¯+2ϕdz,and
(53)∫ℝnfv¯−fv¯v¯−v¯+ϕdz≤L∫ℝnv¯−v¯+2ϕdz,where the last inequality is from the one-sided Lipschitz condition (H3). Hence, we find
(54)12ddt∫ℝnv¯−v¯+2ϕdz+12∫ℝn∇v¯−v¯+2·∇ϕdz+∫ℝn∇v¯−v¯+2ϕdz−c2ddz1∫ℝnv¯−v¯+2ϕdz≤L∫ℝnv¯−v¯+2ϕdz.
By multiplying each term of (50) by 2e−2Lt and rearranging, we obtain
(55)e−2Ltddt∫ℝnv¯−v¯+2ϕdz−2Le−2Lt∫ℝnv¯−v¯+2ϕdz+e−2Lt∫ℝn∇v¯−v¯+2·∇ϕdz−ce−2Ltddz1∫ℝnv¯−v¯+2ϕdz+2e−2Lt∫ℝn∇v¯−v¯+2ϕdz≤0.
By settingWz,t=e−2Ltv¯−v¯+2, it follows that
(56)∫ℝnWtϕdz+∫ℝn∇W·∇ϕdz−c∫ℝnWz1ϕdz+2∫ℝn∇W2ϕdz≤0.
Since∫ℝn∇W·∇ϕdz=−∫ℝnΔWϕdz and ∫ℝn∇W2ϕdz≥0, we find
(57)∫ℝnWtϕdz−∫ℝnΔWϕdz−c∫ℝnWz1ϕdz≤0,for all nonnegative ϕ∈W01,1ℝn. As v¯z,0≤v¯z,0 a.e. in ℝn, Wz,0=0 a.e. in ℝn and thus Wz,t satisfies the initial value problem
(58)Wt−ΔzW−cWz1≤0,Wz,0=0.
By settingVz1,z2,⋯,zn,t=Wz1−ct,z2,⋯,zn,t,
(59)Vt−ΔV≤0,Vz,0=0.
The weak maximum principle for the heat equation concludes thatWz,t≤0 a.e. in ℝn×0,∞, which implies v¯−v¯+≡0 a.e. in ℝn×0,∞. Therefore,
(60)v¯z,t≤v¯z,ta.e.inℝn×0,∞.☐Since any bounded weak solution is a bounded sub- and supersolution simultaneously, as a consequence of the weak comparison principle together with Lemmas3 and 4, we obtain the following corollary.Corollary 7.
Letf:ℝ⟶ℝ be continuous satisfying (H2) and (H3). Then the initial value problem (21) has a unique bounded classical solution.


## 4. Lyapunov Stability of the Planar WaveU

In this section, we give a proof of Theorem1. We prove the planar wave U is stable in the sense that the unique solution vz,t to the initial value problem (21) stays near U for all time t>0 when the initial data v0z at t=0 is close enough to U.We first construct, by an appropriate modification of the planer waveUz1, weak sub- and supersolutions v¯z,t and v¯z,t, respectively, to (21) satisfying
(61)v¯z,0≤v0z≤v¯z,0and0≤v¯z,t≤v¯z,t≤1,for all z,t∈ℝn×0,∞. The assumption (H4) plays a significant role in the construction of the sub- and supersolution. We notice that the constant functions vz,t=0 and vz,t=1 are sub- and supersolutions, respectively, to (21) because the initial data v0z satisfies 0≤v0z≤1 for all z∈ℝn.Lemma 8.
Suppose thatf satisfies the assumptions (H1)–(H4) and the initial data v0z of (21) satisfies for all z∈ℝn,
(62)0≤v0z≤1andv0z−Uz1<E0,where E0>0 is sufficiently small. Then, there exists a constant ν>0 such that the functions defined by
(63)v¯z,t=maxUz1−ξt−qt,0andv¯z,t=minUz1+ξt+qt,1,where
(64)qt=2E0e−μt,μ=minμ¯,μ¯andξt=2νE01−e−μtare sub- and supersolutions, respectively, to (21) satisfying (60) (see Figure 2).Proof.
To begin, we recall the profileUz1 in (12) and let E0 be sufficiently small that
(65)0<E0<η0<14mins0,1−s0,where s0 is a constant such that U0=s0. Then, by the assumption (61) for an initial data v0, one can say that for all z∈ℝn,
(66)0≤maxUz1−2E0,0≤v0z≤minUz1+2E0,1≤1,which implies that
(67)v¯z,0≤v0z≤v¯z,0.
We notice thatqt is decreasing to 0 and ξt is increasing to 2νE0 as t⟶∞, where the positive constant ν will be determined later. Moreover, by the monotonicity of the profile Uz1 and the inequality (66), the functions v¯ and v¯ satisfy (60).
By settingη≔E0<η0 in the assumption (H4), we prove v¯z,t is a supersolution to (21). In order to prove v¯z,t:ℝn×0,∞⟶ℝ is a supersolution to (21), we decompose its domain as
(68)ℝn×0,∞=Ω¯1∪Ω¯2∪Ω¯3,where
(69)Ω¯1=z,t∈ℝn×0,∞∣0≤Uz1+ξt≤δ,Ω¯2=z,t∈ℝn×0,∞∣1−δ≤Uz1+ξt≤1,Ω¯3=z,t∈ℝn×0,∞∣δ≤Uz1+ξt≤1−δ,with the positive constant δ≤E0 in the assumption (H4). That is, the decomposition is depending only on the z1-coordinate and t.
For the regionΩ¯1, since qt is decreasing and by (64),
(70)Uz1+ξt+qt≤Uz1+ξt+q0≤δ+2E0≤3E0<1,and the profile equation U″+cU′+fU=0 gives
(71)v¯t−Δv¯−cv¯z1−fv¯=U′z1+ξtνq′t+q′t−U″z1+ξt−cU′z1+ξt−fUz1+ξt+qt=U′z1+ξtνq′t+q′t+fUz1+ξt−fUz1+ξt+qt.
Since0≤U≤δ and 0<qt≤q0=2E0<s0−2E0=s0−2η for all z,t∈Ω¯1, applying the inequality (16) to the last two terms on the right-hand side of (73) and recalling U′≥0 and μ=minμ¯,μ¯ yield
(72)v¯t−Δv¯−cv¯z1−fv¯≥U′z1+ξtνq′t+q′t+μ¯qt=U′z1+ξtνμqt−μqt+μ¯qt=qtνμU′−μ+μ¯≥0,which implies v¯z,t is a supersolution to (21) on Ω¯1.
For the regionΩ¯2∪Ω¯3, by the definition of v¯ and the fact that v≡1 is a supersolution to (21), it is enough to prove v¯ is a supersolution only when Uz1+ξt+qt<1. Since 0<qt≤q0=2E0<1−s0−2E0=1−s0−2η for all z,t∈Ω¯2, applying (17) to the last two terms on the right-hand side of (73) gives
(73)v¯t−Δv¯−cv¯z1−fv¯≥U′z1+ξtνq′t+q′t+μ¯qt=qtνμU′−μ+μ¯≥0.
For allz,t∈Ω¯3, applying the assumption (H3) to the last two terms on the right-hand side of (73) yields
(74)v¯t−Δv¯−cv¯z1−fv¯≥U′z1+ξtνq′t+q′t−Lqt=qtνμU′−μ−L.
Since the profileU is a strictly increasing C2-function for δ≤U≤1−δ, there exists some ω>0 such that U′≥ω for all z,t∈Ω¯3. If we choose ν satisfying ν≥μ+L/μω, we obtain
(75)v¯t−Δv¯−cv¯z1−fv¯≥qtνμω−μ−L≥0,which implies v¯ is a supersolution to (21). The proof of the subsolution v¯z,t follows similarly by decomposing ℝn×0,∞ as
(76)ℝn×0,∞=Ω¯1∪Ω¯2∪Ω¯3,where
(77)Ω¯1=z,t∈ℝn×0,∞∣1−δ≤Uz1−ξt≤1,Ω¯2=z,t∈ℝn×0,∞∣0≤Uz1−ξt≤δ,Ω¯3=z,t∈ℝn×0,∞∣δ≤Uz1−ξt≤1−δ,and using (17), (16), and (H3) for Ω¯1, Ω¯2, and Ω¯3, respectively.☐Remark 9.
Similarly as in [6], one can also construct sub- and supersolutions satisfying (60) even if the initial data is close enough to U only at space infinity in ℝn (possibly large initial perturbations). In that case, ξ0 might be strictly positive, while ξ0=0 in our case. However, the initial perturbation in [6] is not enough to obtain stability results of the planar wave in ℝn. Thus, we prove the stability under small initial perturbations (61) and so simply set ξ0=0.Figure 2
The subsolutions (red) and supersolutions (green) given in (62)–(63) by a modification of the planar wave U (yellow) in ℝ2. If the initial data v0z is close enough to U so that v0z is between sub- and supersolutions at t=0, the solution vz,t stays between them for all t>0. The fact that both v and U are trapped by the sub- and supersolutions whose difference is sufficiently small for all time t>0 guarantees the stability of U.
(a)
t=0(b)
t>0We now prove the main theorem.Proof of Theorem 1.
LetE0 be sufficiently small that
(78)0<E0<η0<14mins0,1−s0.
By Lemma8, the sub- and supersolutions v¯z,t and v¯z,t to (21) defined by (62)–(63) satisfy
(79)v¯z,t≤Uz1≤v¯z,t.
Moreover, since the initial datav0 satisfies v¯z,0≤v0z≤v¯z,0, by the weak comparison principle in the previous section, the unique bounded classical solution vz,t to (21) also satisfies
(80)v¯z,t≤vz,t≤v¯z,t.
Therefore, recalling (63) and U is a C2-profile satisfying (12), the inequalities (78) and (79) yield that for all z,t∈ℝn×0,∞,
(81)vz,t−Uz1≤v¯z,t−v¯z,t=Uz1+ξt+qt−Uz1−ξt+qt≤2ξtsupz1∈ℝU′z1+2qt≤4νE01−e−μtsupz1∈ℝU′z1+4E0e−μt≤CE0,for some constant C>0.



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*Source: 7756150-2021-10-15.xml*