Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 10, 2003, pp ??–??. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Uniform stability of multidimensional travelling waves for the nonlocal Allen-Cahn equation ^{∗}

Fengxin Chen

Abstract In this paper, we study the uniform stability of mutidimensional planar

travelling waves for the nonlocal Allen-Cahn equation.

1 Introduction

The main concern of this paper is the stability of planar travelling wave solutions of the multidimensional nonlocal Allen-Cahn equation

u_{t} = J ∗ u − u + f(u).  (1.1)

Here J ∈ C^{1}(ℝ^{n}) is a nonnegative function with ∫_{ℝ^{n}} J(y)dy = 1 and J(0) ≠ 0; J ∗ u = ∫_{ℝ^{n}} J(x − y)u(y)dy is the convolution of J and u; f is a smooth bistable function with three zeros, ±1 and a ∈ (−1, 1) satisfying f′(±1) < 0 and f′(a) > 0. A typical example is f(u) = (u − a)(1 − u^{2}) for some a ∈ (−1, 1).

Travelling wave solutions of the nonlocal Allen-Cahn equation in one spatial dimension have been extensively studied. It is well known that there exists a travelling wave solution of the form u(x, t) = ϕ(x − c_{0}t) satisfying

c_{0}ϕ′ + J ∗ ϕ − ϕ + f(ϕ) = 0, ϕ(±∞) = ±1,  (1.2)

where ϕ is a monotone function; If ϕ is continuous,

c_{0} = ∫_{−1}^{1} f(u)du/∫_{−∞}^{∞}(ϕ′(z))^{2}dz;

if c_{0} ≠ 0, the travelling wave solution is smooth and unique modulo a spatial shift; and it is uniformly and asymptotically stable (see [4], [5] and [6]). If the unique speed c_{0} = 0, the wave may be discontinuous but monotone waves are still unique up to a spatial shift.

A planar travelling wave solutions of (1.1) is a solution of the form ϕ(ξ) = ϕ(k · x − ct) and ϕ(±∞) = ±1, where k ∈ S^{n−1} is a unit vector. Without loss of generality, we assume k = (1, 0, ⋯, 0). Then ϕ(k · x − ct) = ϕ(x_{1} − ct) satisfies (1.2) with J being replaced by J_{1}(·) = ∫_{ℝ^{n−1}} J(·, x′)dx′. Notice that ∫_{ℝ} J_{1}(x)dx = 1. Therefore the existence of such planar travelling wave solutions can be derived from the one dimensional case. Therefore, in this paper, we assume that ϕ(x_{1} − c_{0}t) is a planar travelling satisfying ϕ′(x) > 0 for all x ∈ ℝ; and ϕ(±∞) = lim_{x→∞} ϕ(±x) = ±1, with wave speed c_{0}. Our main concern is the multidimensional stability for the planar travelling wave ϕ(x_{1} − c_{0}t). We have the following theorem.

∗Mathematics Subject Classifications: 35K55, 35Q99.

Key words: nonlocal phase transition, travelling waves, continuation.

\begin{matrix}c \\ ◯ \end{matrix}2003 Southwest Texas State University.

Published ...

Theorem 1.1 (Uniform Stability) Let u(x, t) = ϕ(x_{1} − c_{0}t) be a travelling wave solution satisfying ϕ′(x) > 0 for all x ∈ ℝ and ϕ(±∞) = ±1. Then ϕ(x_{1} − c_{0}t) is uniformly stable, that is, for any ϵ > 0 there is δ(ϵ) > 0 such that for any u_{0} ∈ L^{∞}(ℝ^{n}) with ∥u_{0}(·) − ϕ(·)∥_{L^{∞}(ℝ^{n})} < δ(ϵ), one has

∥u(·, t; u_{0}) − ϕ(· − c_{0}t)∥_{L^{∞}(ℝ^{n})} < ϵ

for all t > 0, where u(·, t; u_{0}) is the solution of (1.1) with initial data u(·, 0; u_{0}) = u_{0}.

The global exponential stability in one space dimension is due to the spectral gap [2]. In the multidimensional case, however, the gap disappears due to the effects of the transverse diffusion along the planar wave front and there may exist continuous spectrum all the way up to zero. The global asymptotic stability for the multidimensional case is studied in [2] for special kernel J. For general case the asymptotic stability is still open.

2 Proof of the main Theorem

In this section, we will use super-and sub- solution method to prove the theorem. First we have the following comparison principle.

Lemma 2.1 (Comparison Principle) Suppose R_{1} is an open set in ℝ^{n} and R_{2} = ℝ^{n} \ R_{1} is the complement of R_{1}. Suppose u ∈ C^{1}([τ, t_{0}], L^{∞}(ℝ^{n})) and u(x, t) ≥ 0 for almost all x ∈ R_{2} and t ∈ [τ, t_{0}]. Assume u(x, t) satisfies

u_{t} − K_{0}(x, t)u − (J ∗ u)(x, t) ≥ 0  (2.1)

for almost all (x, t) ∈ R_{1}×(τ, t_{0}], where K_{0}(x, t) ∈ L^{∞}(ℝ^{n}×[τ, t_{0}]). If u(x, τ) ≥ 0 for almost all x ∈ ℝ^{n}, then u(x, t) ≥ 0 for almost all x ∈ ℝ^{n}, and t ∈ [τ, t_{0}]. If, furthermore, u ∈ C_{unif}(ℝ^{n} × [τ, t_{0}]) and u(x, τ) ≢ 0, then u(x, t) > 0 for x ∈ R_{1}, and t ∈ (τ, t_{0}].

Proof The proof is similar to that of one dimensional case(see[5] and [6]). We may assume τ = 0. By assumption, ess inf_{x∈ℝ^{n}}u(x, t) is continuous. If the conclusion of the lemma is not true, then there exist constants ϵ > 0, T > 0 such that u(x, t) > −ϵe^{2Kt} for almost all x ∈ ℝ^{n}, 0 < t < T and ess inf_{x∈ℝ}u(x, T) = −ϵe^{2KT}, where

K = ∥K_{0}∥_{L^{∞}(ℝ^{n}×[τ,t_{0}])} + 1.  (2.2)

Let z(x) be a smooth function such that min_{x∈ℝ^{n}} z(x) = z(0) = 1, sup_{x∈ℝ^{n}} z(x) = z(±∞) = 3, and |z_{x_{i}}(x)| ≤ 1 for i = 1, ⋯, n. Define w_{σ}(x, t) = −ϵ (\frac{3}{4} + σz(x)) e^{2Kt}, for σ ∈ [0, 1]. Since w_{1}(x, t) < u(x, t) for almost all x ∈ ℝ^{n}, and 0 ≤ t ≤ T, and w_{0}(x, t) = −\frac{3}{4}ϵe^{2Kt}, there is a minimum σ^{∗} ∈ (\frac{1}{8}, \frac{1}{4}] such that w_{σ^{∗}}(x, t) ≤ u(x, t) for almost all x ∈ ℝ^{n}, and t ∈ [0, T]. Since w_{σ^{∗}}(±∞, t) ≤ −\frac{9}{8}ϵe^{2Kt} < u(x, t) and u(x, t) > w_{σ^{∗}}(x, t) for almost all x ∈ R_{2}, and t ∈ (0, T], there exist (x_{n}, t_{n}) ∈ R_{1} × (0, T] and (x̄, t̄) such that lim_{n→∞}(x_{n}, t_{n}) = (x̄, t̄), lim_{n→∞}{u(x_{n}, t_{n}) − w_{σ^{∗}}(x_{n}, t_{n})} = 0, the infimum of u(x, t) − w_{σ^{∗}}(x, t) on ℝ × [0, T], and lim_{n→∞}(u − w_{σ^{∗}})_{t}(x_{n}, t_{n}) ≤ 0. Therefore,

0 ≥ lim_{n→∞}(u − w_{σ^{∗}})_{t}(x_{n}, t_{n})

≥ lim_{n→∞}\{(J ∗ u)(x_{n}, t_{n}) + K_{0}(x_{n}, t_{n})u(x_{n}, t_{n})\} + 2Kϵe^{2Kt̄}(σ^{∗}z(x̄) + \frac{3}{4})

≥ lim_{n→∞}{K_{0}(x_{n}, t_{n})(u − w_{σ^{∗}})(x_{n}, t_{n}) + K_{0}(x_{n}, t_{n})w_{σ^{∗}}(x_{n}, t_{n})

+ J ∗ (u − w_{σ^{∗}})(x_{n}, t_{n}) + J ∗ w_{σ^{∗}}(x_{n}, t_{n})\} + 2Kϵe^{2Kt̄}(σ^{∗}z(x̄) + \frac{3}{4})

≥ ϵe^{2Kt̄}[\frac{7}{4}K − \frac{3}{2}∥K_{0}∥ − \frac{3}{2}] > 0.

by the choice of K in (2.2), which is a contradiction. Therefore u(x, t) ≥ 0 for almost all x ∈ ℝ^{n} and t ∈ [τ, t_{0}].

Let v(x, t) = e^{Kt}u(x, t). Then we have v_{t}(x, t) ≥ J ∗ v(x, t) for x ∈ R_{1} and t ∈ (τ, t_{0}] since u(x, t) ≥ 0. Therefore, v(x, t) ≥ tJ ∗v(x, 0). After N^{th} iteration, we have v(x, t) ≥ \frac{t^{N}}{N!}J ∗⋯∗J ∗u(x, 0). If u ∈ C_{unif}(ℝ^{n} ×[τ, t_{0}]) and u(x, 0) ≢ 0, we can choose N large enough such that J ∗ ⋯ ∗ J ∗ u(x, 0) > 0. Therefore, we have v(x, t) > 0. This completes the proof. □

Lemma 2.2 Suppose u_{1}(x, t) and u_{2}(x, t) are super-solution and sub-solution of (1.1), respectively, with u_{1}(x, τ) ≥ u_{2}(x, τ), for all x ∈ ℝ^{n} and for some τ ∈ ℝ^{n}. Then u_{1}(x, t) ≥ u_{2}(x, t) for all x ∈ ℝ^{n} and t > τ. Moreover, if u_{1}(x, τ) ≢ u_{2}(x, τ), then u_{1}(x, t) > u_{2}(x, t) for all x ∈ ℝ^{n} and t > τ.

Proof Let v(x, t) = u_{1}(x, t) − u_{2}(x, t). Then v(x, τ) ≥ 0 for all x ∈ ℝ^{n} and v(x, t) satisfies

v_{t} − K_{0}(x, t)v − (J ∗ v)(x, t) ≥ 0  (2.3)

for all x ∈ ℝ^{n} and t ≥ τ, where

K_{0}(x, t) = ∫_{0}^{1} f_{u}(u_{2} + θ(u_{1} − u_{2}))dθ − 1.  (2.4)

The result follows from Lemma 2.2. □

We use the super- and sub-solution method employed in [6] to prove the stability in one dimensional case. To that end, we first develop the following lemma.

Lemma 2.3 Let ϕ(x−c_{0}t) be as in Theorem 1.1 and β_{1} = −\frac{1}{2} max\{f(−1), f(1)\}. There exist δ_{1} > 0 and σ_{1} > 0 such that, for any δ ∈ (0, δ_{1}), ξ_{0} ∈ ℝ and w^{±}(x, t) are super- and sub-solutions of (1.1) on (0, ∞), respectively, where

w^{±}(x, t) = ϕ(x_{1} + ξ_{0} ± σ_{1}δ(1 − e^{−β_{1}t}) − c_{0}t) ± δe^{−β_{1}t}  (2.5)

for x ∈ ℝ^{n}, t ∈ (0, ∞).

Proof We prove only that w^{+}(x, t) is a super-solution. The other can be proved similarly.

\begin{matrix}Lw^{+} := & w_{t}^{+} − (J ∗ w^{+} − w^{+}) − f(w^{+}) \\ = & [σ_{1}β_{1}ϕ′(η_{+}(x, t)) − β_{1} − K_{0}(x, t)]δe^{−β_{1}t} \end{matrix}  (2.6)

where

K_{0}(x, t) = ∫_{0}^{1} f_{u}(ϕ(η_{+}(x, t)) + θδe^{−β_{1}t})dθ,

and η_{+}(x, t) = x_{1} + ξ_{0} + σ_{1}δ(1 − e^{−β_{1}t}) − c_{0}t. Since lim_{x→∞} ϕ(±x) = ±1, K_{0}(x, t) → f_{u}(±1) uniformly in t ∈ [0, ∞) as η_{+}(x, t) → ±∞ and δ → 0. So, there exist m̄ > 0 and δ_{1} > 0 such that for x ∈ ℝ^{n} with |η_{+}(x, t)| ≥ m̄ and 0 < δ < δ_{1},

K_{0}(x, t) < −β_{1},  (2.7)

that is, −β_{1} − K_{0}(x, t) ≥ 0 for x ∈ ℝ^{n} and t ∈ ℝ^{+} with |η_{+}(x, t)| ≥ m̄. Therefore, Lw^{+} ≥ 0 for x ∈ ℝ^{n} and t ∈ ℝ^{+} with |η_{+}(x, t)| ≥ m̄.

For |η_{+}(x, t))| ≤ m̄, choose

σ_{1} = \frac{β_{1} + K}{β_{1}α(m̄)},  (2.8)

where K = sup{|f_{u}(u)| : u ∈ [−2, +2]} and α(m̄) = min{ϕ(x) : x ∈ [−m̄, m̄]}. We know that α(m̄) > 0 since ϕ(x) > 0 for all x ∈ ℝ. Then, for t ≥ 0, x ∈ ℝ^{n} with |η_{+}(x, t)| ≤ m̄ and any 0 < δ ≤ δ_{1}, we have Lw^{+} ≥ 0.

Therefore Lw^{+} ≥ 0 for all x ∈ ℝ^{n} and t ∈ (0, ∞). That is, with the above choices of δ_{1} and σ_{1}, the function w^{+}(x, t) is a super-solution for (1.1). □

Proof of Theorem 1.1 For ϵ > 0 given, since ϕ is uniformly continuous, there exists k_{0} > 0 such that, for all |k| ≤ k_{0},

|ϕ(x_{1} + k) − ϕ(x_{1})| < \frac{ϵ}{2}  (2.9)

for all x_{1} ∈ ℝ. Let β_{1}, σ_{1} and δ_{1} be as in Lemma 2.3. Choose δ > 0 such that δ < min \{\frac{ϵ}{2}, \frac{k_{0}}{σ_{1}} , δ_{1}\}. Then by Lemma 2.2, the condition

ϕ(x_{1}) − δ < u_{0}(x) < ϕ(x_{1}) + δ

implies

ϕ(x_{1} − σ_{1}δ(1 − e^{−β_{1}t}) − c_{0}t) − δe^{−β_{1}t}
≤ u(x, t)
≤ ϕ(x_{1} + σ_{1}δ(1 − e^{−β_{1}t}) − c_{0}t) + δe^{−β_{1}t}.  (2.10)

By the choice of δ and (2.9) - (2.10), we have

|u(x, t) − ϕ(x_{0} − c_{0}t)| < ϵ

for all x ∈ ℝ^{n} and t > 0. That completes the proof.

References

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[2] P. W. Bates and F. Chen, Spectral analysis and multidimensional stability of travelling waves for nonlocal Allen-Cahn equation, J. Math. Anal. Appl. 273 (2002) 45-57.

[3] P. W. Bates, F. Chen, and J. Wang, Global existence and uniqueness of solutions to a nonlocal phase-field syatem, US-Chinese Conference on Differential Equations and Applications, P. W. Bates, S-N. Chow, K. Lu and X. Pan, Eds,International Press, Cambridge MA,, 1997 pp 14-21.

[4] P. W. Bates, P. C. Fife, X. Ren, and X. Wang, Traveling waves in a nonlocal model of phase transitions, Arch. Rat. Mech. Anal. 138 (1997), 105-136.

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Fengxin Chen Division of Mathematics and Statistics University of Texas at San Antonio San Antonio, TX 78249, USA e-mail: feng@sphere.math.utsa.edu