A multi-dimensional bistable nonlinear diffusion equation in a periodic medium

Abstract

In this work we study the existence of wave solutions for a scalar reaction-diffusion equation of bistable type posed in a multi-dimensional periodic medium. Roughly speaking our result states that bistability ensures the existence of waves for both balanced and unbalanced reaction term. Here the term wave is used to describe either pulsating travelling wave or standing transition solution. As a special case we study a two-dimensional heterogeneous Allen–Cahn equation in both cases of slowly varying medium and rapidly oscillating medium. We prove that bistability occurs in these two situations and we conclude to the existence of waves connecting $u = 0$ and $u = 1$. Moreover in a rapidly oscillating medium we derive a sufficient condition that guarantees the existence of pulsating travelling waves with positive speed in each direction.

Appendices

Appendix A: Weak limit of rapidly oscillating functions

When $g\in L^\infty (\mathbb {T}^N)$ then setting ${\mathcal {M}}(g)=\int _{\mathbb {T}^N}g(x)dx$, it is well known that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0{^{+}}}\int _{\mathbb {R}^N} g\left( \frac{x}{\varepsilon }\right) \phi (x)dx={\mathcal {M}}(g)\int _{\mathbb {R}^N}\phi (x)dx,\quad \forall \phi \in L^1(\mathbb {R}^N). \end{aligned}$$

(52)

In this appendix we aim to show that under some regularity conditions, the above convergence is in some sense uniform with respect to $\mathbb {T}^N$-translations. Our result reads as:

Lemma 6.1

Let $g\in C(\mathbb {T}^N)$ be a given function. Then the following convergence holds true: For all $\phi \in L^1(\mathbb {R}^N)$, for all $\eta >0$ there exists $\delta >0$ such that

$$\begin{aligned} \forall \varepsilon \in (0,1),\quad \forall h\in \mathbb {T}^N\quad \left| \int _{\mathbb {R}^N} g\left( h+\frac{x}{\varepsilon }\right) \phi (x)dx-{\mathcal {M}}(g)\int _{\mathbb {R}^N}\phi (x)dx\right| \le \eta . \end{aligned}$$

Proof

Let $g\in C (\mathbb {T}^N)$ be given. We denote for each $h\ge 0$ the quantity

$$\begin{aligned} \omega (h;g)=\sup _{(x,y)\in \mathbb {R}^N,\Vert x-y\Vert \le h}\left| g(x)-g(y)\right| . \end{aligned}$$

Since g is continuous and ${\mathbb {Z}}^N$-periodic, it is uniformly continuous so that

$$\begin{aligned} \lim _{h\rightarrow 0{^{+}}} \omega (g;h)=0. \end{aligned}$$

(53)

To proceed to the proof of the lemma, let us argue by contradiction by assuming that there exists $\phi _0\in L^1(\mathbb {R}^N)$, $\eta _0>0$ and a sequence $\{h_n\}_{n\ge 0}\subset [0,1]^N$ and $\{\varepsilon _n\}_{n\ge 0}\subset (0,\infty )$ tending to 0 as $n\rightarrow \infty$ such that

$$\begin{aligned} \left| \int _{\mathbb {R}^N} g\left( h_n+\frac{x}{\varepsilon _n}\right) \phi _0(x)dx-{\mathcal {M}}(g)\int _{\mathbb {R}^N}\phi _0(x)dx\right| > \eta _0,\quad \forall n\ge 0. \end{aligned}$$

Since $[0,1]^N$ is compact one may assume that $h_n\rightarrow h_\infty \in [0,1]^N$. Now note that one has for each $n\ge 0$:

$$\begin{aligned} \eta _0< & {} \left| \int _{\mathbb {R}^N} g\left( h_n+\frac{x}{\varepsilon _n}\right) \phi _0(x)dx-{\mathcal {M}}(g)\int _{\mathbb {R}^N}\phi _0(x)dx\right| \nonumber \\\le & {} \left| \int _{\mathbb {R}^N} \left[ g\left( h_n+\frac{x}{\varepsilon _n}\right) -g\left( h_\infty +\frac{x}{\varepsilon _n}\right) \right] \phi _0(x)dx\right| \nonumber \\&+ \left| \int _{\mathbb {R}^N} g\left( h_\infty +\frac{x}{\varepsilon _n}\right) \phi _0(x)dx-{\mathcal {M}}(g)\int _{\mathbb {R}^N}\phi _0(x)dx\right| . \end{aligned}$$

(54)

However on the one hand one has

$$\begin{aligned} \left| \int _{\mathbb {R}^N} \left[ g\left( h_n+\frac{x}{\varepsilon _n}\right) -g\left( h_\infty +\frac{x}{\varepsilon _n}\right) \right] \phi _0(x)dx\right| \le \omega \left( \Vert h_n-h_\infty \Vert ;g\right) \Vert \phi _0\Vert _{L^1}. \end{aligned}$$

Since $\Vert h_n-h_\infty \Vert \rightarrow 0$ as $n\rightarrow \infty$ and recalling (53), one obtains that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left| \int _{\mathbb {R}^N} \left[ g\left( h_n+\frac{x}{\varepsilon _n}\right) -g\left( h_\infty +\frac{x}{\varepsilon _n}\right) \right] \phi _0(x)dx\right| =0. \end{aligned}$$

On the other hand, recalling that $\varepsilon _n\rightarrow 0$ as $n\rightarrow \infty$, one gets from (52) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{\mathbb {R}^N} g\left( h_\infty +\frac{x}{\varepsilon _n}\right) \phi _0(x)dx= & {} {\mathcal {M}}\left( g\left( h_\infty +\cdot \right) \right) \int _{\mathbb {R}^N}\phi _0(x)dx\\= & {} {\mathcal {M}}\left( g\right) \int _{\mathbb {R}^N}\phi _0(x)dx. \end{aligned}$$

These two limits contradict (54) and this completes the proof of the lemma. $\square$

As a direct corollary one obtains the following result:

Corollary 6.2

Let $g\in C (\mathbb {T}^N )$ be a given function. Let $\{h_n\}_{n\ge 0}\subset \mathbb {T}^N$ be a given sequence and $\{\varepsilon _n\}_{n\ge 0}\subset (0,\infty )$ be a sequence tending to 0 as $n \rightarrow \infty$ Then one has:

$$\begin{aligned} g\left( h_n+\frac{x}{\varepsilon _n}\right) \rightharpoonup {\mathcal {M}}(g)\quad \text {weakly-} * \text { in } L^\infty (\mathbb {R}^N). \end{aligned}$$

Appendix B: A non-existence result of standing transition waves

In this section we will discuss the statement of Remark 1.11. To that aim we consider a rather specific nonlinear diffusion equation of the form

$$\begin{aligned} \frac{\partial u}{\partial t}=\Delta u+F\left( {\tilde{x}},u\right) , \end{aligned}$$

(55)

posed for $x= (x_1,{\tilde{x}} )\in \mathbb {R}\times \mathbb {R}^{N-1}$ and for some integer $N\ge 2$. In this appendix we derive a sufficient condition ensuring that (55) does not admit any standing transition in the direction $e_0= (1,0_{\mathbb {R}^{N-1}} )\in {\mathbb {S}}^{N-1}$, the direction orthogonal to the heterogeneity. Then this result will be applied to the case of Problem (9) with a periodic row structure to obtain the results stated in Remark 1.11. For that purpose we assume that:

Assumption 7.1

The function $F\equiv F ({\tilde{x}},u):\mathbb {R}^{N-1}\times \mathbb {R}\rightarrow \mathbb {R}$ is continuous, $C^\gamma$ (for some $\gamma \in (0,1)$) in x uniformly with respect to $u \in \mathbb {R}$, of the class $C^1$ in u uniformly with respect to $x\in \mathbb {T}^N$ and $F_u$ is continuous on $\mathbb {T}^N \times \mathbb {R}$. It furthermore satisfies:

• (i) $F({\tilde{x}}, 0 )\equiv F ({\tilde{x}}, 1)\equiv 0$;

• (ii) for any $u\in \{0,1\}$, $\displaystyle \sup _{{\tilde{x}}\in \mathbb {R}^{N-1}}F_u ({\tilde{x}},u )<0$;

• (iii) there exists a sequence $R_n\rightarrow \infty$ such that

  $$\begin{aligned} \lim _{n\rightarrow \infty } \frac{1}{\left| B_{R_n}^{N-1}\right| }\int _{B_{R_n}^{N-1}} W\left( {\tilde{x}},1\right) d{\tilde{x}}\ne 0, \end{aligned}$$

  wherein we have set $W({\tilde{x}},u )=\int _0^u F ({\tilde{x}},s)ds$. Moreover for each $R>0$, $B_R^{N-1}\subset \mathbb {R}^{N-1}$ denotes the ball in $\mathbb {R}^{N-1}$ with the radius $R>0$ and centered at the origin while $|B_{R}^{N-1}\vert$ denotes its measure in $\mathbb {R}^{N-1}$.

Under the above set of assumptions we will prove that the following proposition holds true.

Proposition 7.2

Let Assumption 7.1 be satisfied. Then Problem (55) does not admit any standing transition between $u=0$ and $u=1$ in the direction $e_0$.

Proof

In order to prove the above proposition, let us assume that there exists a standing transition $u:\mathbb {R}^N\rightarrow \mathbb {R}$ connecting $u=0$ and $u=1$ in the direction $e_0$, that is a function $u\equiv u (x_1,{\tilde{x}} )$ satisfying the equation

$$\begin{aligned} \Delta u+F\left( {\tilde{x}}, u\right) =0,\quad \forall x=\left( x_1,{\tilde{x}}\right) \in \mathbb {R}^N, \end{aligned}$$

(56)

as well as the following behaviours when $x_1\rightarrow \pm \infty$:

$$\begin{aligned} \lim _{x_1\rightarrow -\infty } u\left( x_1,{\tilde{x}}\right) =0,\quad \lim _{x_1\rightarrow \infty } u\left( x_1,{\tilde{x}}\right) =1. \end{aligned}$$

(57)

Here the above limits are uniform with respect to ${\tilde{x}}\in \mathbb {R}^{N-1}$.

Now because of Assumption 7.1 (ii) one obtains the following exponential decay with respect to $x_1$: there exist some constants $C>0$ and $\eta >0$ such that for all $x= (x_1,{\tilde{x}})\in \mathbb {R}^N$ one has

$$\begin{aligned}&|u(x)|\le Ce^{\eta x_1},\quad |1-u(x)|\le C e^{-\eta x_1},\nonumber \\&|\partial _{x_1} u(x)|\le C e^{-\eta |x_1|},\quad |\nabla _{{\tilde{x}}} u(x)|\le C. \end{aligned}$$

(58)

We refer for instance to [7, 16] for the derivation of such an exponential decay.

Let us also observe that due to elliptic estimates and the uniform limits in (57) one has:

$$\begin{aligned} \lim _{x_1\rightarrow \pm \infty }\nabla _{{\tilde{x}}} u(x)=0\quad \text {uniformly with respect to} {\tilde{x}}\in \mathbb {R}^{N-1}. \end{aligned}$$

(59)

Let $M>0$ and $R>0$ be given and fixed. Multiplying (56) by $\partial _{x_1} u$ and integrating over the cylinder $(-M,M)\times B_R^{N-1}$ yields

$$\begin{aligned}&\frac{1}{2}\int _{B_R^{N-1}}\left( \partial _{x_1} u(M,{\tilde{x}})\right) ^2 d{\tilde{x}}-\frac{1}{2}\int _{B_R^{N-1}}\left( \partial _{x_1} u(-M,{\tilde{x}})\right) ^2 d{\tilde{x}}\\&\qquad +\int _{-M}^M \int _{\partial B_R^{N-1}}\nabla _{{\tilde{x}}} u(x_1,{\tilde{x}})\cdot {\tilde{\nu }}({\tilde{x}}) \partial _{x_1}u(x_1,{\tilde{x}}) d\sigma \left( {\tilde{x}}\right) dx_1\\&\qquad -\frac{1}{2}\int _{B_R^{N-1}} \left[ |\nabla _{{\tilde{x}}}u(M,{\tilde{x}})|^2-|\nabla _{{\tilde{x}}}u(-M,{\tilde{x}})|^2\right] d{\tilde{x}}\\&\qquad +\int _{B_R^{N-1}} W\left( {\tilde{x}},u(M,{\tilde{x}})\right) d{\tilde{x}}-\int _{B_R^{N-1}} W\left( {\tilde{x}},u(-M,{\tilde{x}})\right) d{\tilde{x}}\\&\quad =0 \end{aligned}$$

In the second line of the above computation, ${\tilde{\nu }}({\tilde{x}})\in \mathbb {R}^{N-1}$ denotes the outward unit vector to $\partial B_R^{N-1}\subset \mathbb {R}^{N-1}$ at ${\tilde{x}}$.

Now using the properties stated in (58) and (59) one can let $M\rightarrow \infty$ in the above formula to obtain that for each $R>0$:

$$\begin{aligned} \int _{-\infty }^\infty \int _{\partial B_R^{N-1}}\nabla _{{\tilde{x}}} u(x_1,{\tilde{x}})\cdot {\tilde{\nu }}({\tilde{x}}) \partial _{x_1}u(x_1,{\tilde{x}}) d\sigma \left( {\tilde{x}}\right) dx_1+\int _{B_R^{N-1}} W\left( {\tilde{x}},1\right) d{\tilde{x}}=0. \end{aligned}$$

Therefore there exists some constant $K>0$ such that for all $R>0$ one has

$$\begin{aligned} \left| \frac{1}{\left| B_R^{N-1}\right| }\int _{B_R^{N-1}} W\left( {\tilde{x}},1\right) d{\tilde{x}}\right| \le K R^{-1}. \end{aligned}$$

This former property contradicts Assumption 7.1 (iii) and this completes the proof of Proposition 7.2. $\square$

We now come back to Problem (9) with $r(x)=r(x_2)$ and $a(x)=a(x_2)$ for all $x= (x_1,x_2 )\in \mathbb {T}^1\times \mathbb {T}^1$. Then the function $f(x,u)=r(x_2)u (u-a(x_2)) (1-u)$ satisfies the conditions of Assumption 7.1 (i) and (ii) and one has

$$\begin{aligned} \lim _{R\rightarrow \infty } \frac{1}{2R}\int _{-R}^R\int _0^1 f(x,u)du={\mathcal {M}}(r) \int _0^1 u\left( u-\overline{\theta }\right) (1-u)du, \end{aligned}$$

with ${\mathcal {M}}(r)=\int _{\mathbb {T}^1} r(x_2)dx_2$ and $\overline{\theta }=\frac{\int _{\mathbb {T}^1} r(x_2)a(x_2)dx_2}{{\mathcal {M}}(r)}$. Hence in that context Assumption 7.1 (iii) is equivalent to $\overline{\theta }\ne \frac{1}{2}$. This condition is satisfied under the assumptions of Theorem 1.9 while under the assumptions of Theorem 1.10 this condition has to be furthermore assumed. This completes the proof of the statements in Remark 1.11.