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

Extending (Drábek and Takáč 2017), we investigate the Lyapunov stability of planar waves for the reaction-diffusion equation on Rn, n ≥ 2, with a α-H€older continuous (0 < α < 1), but not necessarily smooth reaction term. We first consider an initial value problem for the equation and then construct suband 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.


Introduction
We consider a reaction-diffusion equation on ℝ n where x = ðx 1 , x 2 ,⋯,x n Þ ∈ ℝ n , n ≥ 2, Δ x = ∂ 2 x 1 + ∂ 2 x 2 + ⋯ + ∂ 2 x n , t > 0, uðx, tÞ ∈ ℝ, and f : ℝ ⟶ ℝ. In particular, we are concerned with the stability of the planar wave of the form which is the most widely studied type of travelling wave to (1) with a continuously differentiable reaction term f (e.g., see [1][2][3] and the references therein). However, in the present paper, we assume the reaction term f is not necessarily Lipschitz continuous, but only α-H€ older 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 for some constants α 0 , α 1 ∈ ð0, 1Þ, c 1 ∈ ℝ, and c 2 ∈ ð0, 1Þ, 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 α 0 = α 1 = 1 in the reaction function (3), the product uð1 − uÞ represents the classical logistic growth of the population. The assumption requiring very large birth or death rate of the population leads to the formula u α 0 ð1 − uÞ α 1 with α 0 , α 1 ∈ ð0, 1Þ 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 function f 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 ℝ n , n ≥ 2, with the continuously differentiable reaction term (i.e., α 0 = α 1 = 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 ℝ n−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 ℝ n , 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 onedimensional reaction-diffusion equation for some spatial shift ζ ∈ ℝ, when the initial data u 0 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 ∞ ðℝ n Þ 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 x 1 -moving coordinate with speed c by setting so that the planar wave can be considered as a stationary solution. In the ðz, tÞ coordinates, equation (1) reads and the planar wave is then a stationary solution vðz, tÞ = Uðz 1 Þ that satisfies the profile equation The existence and monotonicity of such profiles have been studied in [1,DT2]. According to Section 2 of [6], the assumption (H1) stated in Section 1.2 guarantees C 2 -profiles U : ℝ ⟶ ½0, 1 for some speed c = c * . If Uðz 1 Þ satisfies (8), then its translate Uðz 1 − ζÞ also satisfies (8) for any constant ζ ∈ ℝ. Throughout this paper, we impose the condition for some constant s 0 ∈ ð0, 1Þ that appears in (H1), so that the C 2 -profile U : ℝ ⟶ ½0, 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 ða, bÞ ⊆ ℝ, −∞ ≤ a < 0 < b ≤ ∞, such that 0 ≤ Uðz 1 Þ ≤ 1 and U ′ ðz 1 Þ > 0 on ða, bÞ. The asymptotic behavior of the profile is determined solely by the behavior of f ðsÞ as s ⟶ 0 + and s ⟶ 1 − . It is well-known that a = −∞ and b = ∞ in the classical case, i.e., f ∈ C 1 ðℝÞ. 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 γ 0 and γ 1 such that (see a typical example (3) of f ). It is obvious that the limits of (10) are −∞ and ∞, respectively, in the case of f ∈ C 1 ðℝÞ (see [6,9] for further details). In summary, the planar wave vðz, tÞ = Uðz 1 Þ satisfies 1.2. Hypotheses on the Reaction Term f. Throughout this paper, following [6], we assume the reaction term f satisfies the following: (H1). f : ℝ ⟶ ℝ is continuous such that f ð0Þ = f ðs 0 Þ = f ð1Þ = 0 for some 0 < s 0 < 1. Moreover, f ðsÞ < 0 for any s ∈ ð0, s 0 Þ ∩ ð1, ∞Þ, f ðsÞ > 0 for all s ∈ ð−∞, 0Þ ∩ ðs 0 , 1Þ, and for any 0 < r ≤ 1,

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(H3). f is one-sided Lipschitz continuous, that is, there exists a positive number L such that (H4). There exists a positive constant satisfying that for any η ∈ ð0, η 0 there are positive constants δ ∈ ð0, η, μ and μ such that As mentioned in the previous subsection, the first assumption (H1) is needed to show the existence and monotonicity of the profile Uðz 1 Þ. The last assumption (H4), referred to as the secant conditions, holds trivially if f ∈ C 1 ðℝÞ 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).

Main Result.
We now state our main theorem. Theorem 1 says that if the equation (7) starts with an initial data v 0 ðzÞ near the planar wave Uðz 1 Þ then the solution to (7) stays near it for all time.
Theorem 1 (Lyapunov stability). Let the assumptions (H1)-(H4) hold and n ≥ 2. Suppose that a function v 0 ðzÞ ∈ L ∞ ðℝ n Þ satisfies that for all z ∈ ℝ n , where E 0 > 0 is sufficiently small. Then, there is a unique bounded classical solution vðz, tÞ to (7) with an initial data v 0 ðzÞ such that for all z ∈ ℝ n and all time t > 0, for some constant C > 0.
Remark 2. Theorem 1 is the extension of the onedimensional stability result (Proposition 4.1) of [6] to a multidimensional space. Our assumption for the profile Uð0Þ = s 0 gives ζ = 0 in their result.
The paper is organized as follows. In Section 2, we first consider the Cauchy problem with an initial data v 0 ðzÞ ∈ L ∞ ðℝ n Þ, 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 ∞ suband 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 v 0 ðzÞ is close enough to U.
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 ℝ n with n ≥ 2, which is a interesting open problem.
Moreover, even in ℝ, 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 ℝ n , n ≥ 2, for a continuously differentiable f has been studied in [3]. They also modified the planar wave

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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.

A Bounded Classical Solution
In this section, we consider the Cauchy problem on ℝ n , where z = ðz 1 , z 2 ,⋯,z n Þ, n ≥ 2, an initial data v 0 : ℝ n ⟶ ℝ is Lebesgue-measurable, and 0 ≤ v 0 ðzÞ ≤ 1 for all z ∈ ℝ n . 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 for any given 0 < τ < T < ∞ and the weak star limit vðz, tÞ * v 0 ðzÞ in L ∞ ðℝ n Þ as t ⟶ 0 + . Here, for any α ∈ ð0, 1Þ and T > 0, the norm of H€ older space C α,α/2 ðℝ n × ½0, TÞ is given by where Applying Duhamel's principle to (21) yields an integral for any ðz, tÞ ∈ ℝ n × ½0,∞Þ. Here, for any y = ðy 1 , y 2 ,⋯,y n Þ, the heat kernel G is defined by 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 solu-tion (25). The estimates of the heat potentials in H€ older norm have been established in Section 4 of [10], which lead to the following lemma.

Lemma 3 (Regularity estimates)
. Assume that f : ℝ ⟶ ℝ satisfies the condition (H2). If vðz, tÞ is a bounded mild solution of (21), then for any given 0 and we have the estimates for some constant M τ,T > 0, dependent upon τ and T.

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We notice that the inequalities (29) and (30) yield that for all ðz, tÞ, Since f is α-H€ older continuous, the estimates (32) give and similarly Finally, we have ☐ Noting that f ∘ 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).
Proof. We first consider the existence of a bounded mild solution vðz, tÞ on ℝ n × ½0, T, where We define the locally convex vector space and the closed convex subset of X By recalling a mild solution (25), let us consider an operator F : X ⟶ X defined by where We first prove FðCÞ ⊂ C. For any given w ∈ C, since Ð ℝ n Gðz − y, tÞdy = 1, v 0 ðzÞ ∈ ½0, 1, and jwðz, tÞj ≤ 1 for all ðz, tÞ ∈ ℝ n × ½0, T, it follows Thus, for any ðz, tÞ ∈ ℝ n × ½0, T which implies F maps C into itself. We now notice that for any w 1 , w 2 ∈ C, which is a bounded mild solution to (21). Therefore, we obtain the existence of a bounded mild solution v : ℝ n × ½0,∞Þ ⟶ ℝ by replacing the initial time t = 0 by t = t 0 for any t 0 > 0 and repeating the above procedure in each time interval ½t 0 , t 0 + T.☐

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.