Propagation Phenomena for Nonlocal Dispersal Equations in Exterior Domains

Abstract

This paper is concerned with the spatial propagation of bistable nonlocal dispersal equations in exterior domains. We first obtain the existence and uniqueness of an entire solution which behaves like a planar traveling wave front for large negative time. Then, when the entire solution comes to the interior domain, the profile of the front will be disturbed. However, the disturbance is local in space for finite time, which means the disturbance disappears as its location is far away from the interior domain. Furthermore, we prove that the solution can gradually recover its planar wave profile uniformly in space and continue to propagate in the same direction for large positive time provided that the interior domain is compact and convex. Our work generalizes the local (Laplace) diffusion results obtained by Berestycki et al. (2009) to the nonlocal dispersal setting by using new known Liouville results and Lipschitz continuity of entire solutions due to Li et al. (2010).

Appendix

1.1 Proof of Proposition 3.2

In this subsection we intend to show the results of Proposition 3.2. For convenience we define the operator ${\mathcal {L}}$ as follows

$$\begin{aligned} {\mathcal {L}}\omega (x,t)=\omega _t(x,t)-\int _\Omega J(x-y)[\omega (y,t)-\omega (x,t)]dy-f(\omega (x,t)). \end{aligned}$$

We further show that $W^-(x,t)$ is a sub-solution. A straightforward computation shows that

$$\begin{aligned}&{\mathcal {L}}W^-(x,t)\\&\quad =\left\{ \begin{aligned}&-\int _\Omega J(x-y)W^-(y,t)dy ,\qquad x_1<0, \\&(c-{\dot{\xi }}(t))[\phi '(x_1+ct-\xi (t))-\phi '(-x_1+ct-\xi (t))]-\int _\Omega J(x-y)[W^-(y,t)\\&\quad -W^-(x,t)]dy-f(\phi (x_1+ct-\xi (t))-\phi (-x_1+ct-\xi (t))),\qquad x_1\ge 0. \end{aligned}\right. \end{aligned}$$

For $x_1<0$, since $J(x)\ge 0$ and $W^-\ge 0$, we have

$$\begin{aligned} {\mathcal {L}}W^-=-\int _\Omega J(x-y)W^-(y,t)dy\le 0. \end{aligned}$$

For $x_1\ge 0$, in view of that

$$\begin{aligned}&\int _\Omega J(x-y)[W^-(y,t)-W^-(x,t)]dy\\&\quad =\int _{{\mathbb {R}}^N}J(x-y)[W^-(y,t)-W^-(x,t)]dy -\int _K J(x-y)[W^-(y,t)-W^-(x,t)]dy \\&\quad =\int _{{\mathbb {R}}^N\cap \{y_1>0\}} J(x-y)[W^-(y,t)-W^-(x,t)]dy\\&\qquad +\int _{{\mathbb {R}}^N\cap \{y_1<0\}} J(x-y)[W^-(y,t)-W^-(x,t)]dy\\&\qquad -\int _K J(x-y)[W^-(y,t)-W^-(x,t)]dy\\&\quad \ge \int _{{\mathbb {R}}^N}J(x-y)[(\phi (y_1+ct-\xi (t)) -\phi (-y_1+ct-\xi (t)))-(\phi (x_1+ct-\xi (t))\\&\qquad -\phi (-x_1+ct-\xi (t)))]dy-\int _K J(x-y)[W^-(y,t)-W^-(x,t)]dy, \end{aligned}$$

we have

$$\begin{aligned} \begin{aligned} {\mathcal {L}}W^-\le&-{\dot{\xi }}(t)[\phi '(z_+(t))-\phi '(z_-(t))] \\&+f(\phi (z_+(t)))-f(\phi (z_-(t)))-f(\phi (z_+(t))-\phi (z_-(t)))\\&+\int _KJ(x-y)[W^-(y,t)-W^-(x,t)]dy, \end{aligned} \end{aligned}$$

where $z_+(t)=x_1+ct-\xi (t),~z_-(t)=-x_1+ct-\xi (t)$. Recall that $K\subset \{x\in {\mathbb {R}}^N\mid x_1\le 0\}$, it follows that

$$\begin{aligned} W^-(y,t)=0 ~\text {for all}~y\in K, \end{aligned}$$

which implies that

$$\begin{aligned} \begin{aligned} {\mathcal {L}}W^-(x,t)\le&-{\dot{\xi }}(t)[\phi '(z_+(t))-\phi '(z_-(t))] \\&+f(\phi (z_+(t)))-f(\phi (z_-(t)))-f(\phi (z_+(t))-\phi (z_-(t))). \end{aligned} \end{aligned}$$

Now we go further to show ${\mathcal {L}}W^-\le 0$ in two subcases.

Case A: $0<x_1<-ct+\xi (t)$.

In this case the following lemma holds.

Lemma 6.1

Suppose that (F) holds. Let $(\phi ,c)$ be the unique solution of (1.4) and $\phi ''(\xi )\ge 0$ for $\xi \le 0$. Then there exists $k_3>0$ such that

$$\begin{aligned} \phi '(\xi _1)-\phi '(\xi _2)\ge k_3[\phi (\xi _1)-\phi (\xi _2)] \end{aligned}$$

(6.1)

for $\xi _2<\xi _1<0$.

Proof

It follows from (1.6) that there exists some ${\mathfrak {c}}>0$ such that

$$\begin{aligned} \frac{\phi '(\xi _2)}{\phi '(\xi _1)}\le {\mathfrak {c}}e^{\lambda (\xi _2-\xi _1)}. \end{aligned}$$

Then one can choose $M'>\frac{\ln 2{\mathfrak {c}}}{\lambda }$ and $(\xi _1,\xi _2)\in {\mathbb {R}}^2$ with $\xi _1-M'<\xi _2<\xi _1<0$ such that

$$\begin{aligned} \frac{\phi '(\xi _2)}{\phi '(\xi _1)}\le {\mathfrak {c}}e^{-\lambda M'}\le \frac{1}{2}. \end{aligned}$$

If $\xi _1-M'<\xi _2<\xi _1<0$, then we have

$$\begin{aligned} \phi '(\xi _1)-\phi '(\xi _2)=\phi ''(\theta ^1)(\xi _1-\xi _2),~\phi (\xi _1)-\phi (\xi _2)=\phi '(\theta ^2)(\xi _1-\xi _2) \end{aligned}$$

for some $(\theta ^1,\theta ^2)\in [\xi _2,\xi _1]^2$ with $|\theta ^1-\theta ^2|<M'.$ This and the facts $\phi ''(\xi )\ge 0$ for $\xi \le 0$ and $\phi '(\psi )>0$ for all $\psi \in {\mathbb {R}}$ imply that (6.1) holds true for $\xi _1-M'<\xi _2<\xi _1<0$.

When $\xi _2+M'<\xi _1<0$, it follows that

$$\begin{aligned} \phi '(\xi _2)\le \frac{1}{2}\phi '(\xi _1). \end{aligned}$$

This and the inequalities in (1.5) and (1.6) yield

$$\begin{aligned} \phi '(\xi _1)-\phi '(\xi _2)\ge \frac{1}{2}\phi '(\xi _1)\ge k_3\phi (\xi _1)\ge k_3[\phi (\xi _1)-\phi (\xi _2)]. \end{aligned}$$

Thus we finish the proof. $\square$

Now we are ready to show ${\mathcal {L}}W^-(x,t)\le 0$. By Lemma 6.1, we have

$$\begin{aligned} {\mathcal {L}}W^-(x,t)\le&-{\dot{\xi }}(t)(\phi '(z_+(t))-\phi '(z_-(t)))+L_f\phi (z_-(t)) (\phi (z_+(t))-\phi (z_-(t)))\\ \le&\left[ -Mk_3e^{\lambda _0(ct+\xi (t))}+L_f\phi (z_-(t))\right] [\phi (z_+(t))-\phi (z_-(t))]\\ \le&\left[ L_f\beta _0e^{\lambda (-x_1+ct-\xi (t))}-Mk_3e^{\lambda _0(ct+\xi (t))}\right] [\phi (z_+(t))-\phi (z_-(t))]\\ \le&e^{\lambda _0(ct+\xi (t))}\left[ L_f\beta _0e^{(\lambda -\lambda _0)(ct+\xi (t))-2\lambda \xi (t)} -Mk_3\right] [\phi (z_+(t))-\phi (z_-(t))]\\ \le&(L_f\beta _0-Mk_3)[\phi (z_+(t))-\phi (z_-(t))]\\ \le&0. \end{aligned}$$

The last inequality holds provided that $M\ge \frac{L_f\beta _0}{k_3}$.

Case B: $x_1\ge -ct+\xi (t)$.

A direct calculation gives that

$$\begin{aligned} \begin{aligned} {\mathcal {L}}W^-(x,t)\le&-Me^{\lambda _0(ct+\xi (t))}[\phi '(z_+(t))-\phi '(z_-(t))] +L_f\phi (z_-(t))[\phi (z_+(t))-\phi (z_-(t))]\\ \le&L_f\phi (z_-(t))-Me^{\lambda _0(ct-\xi (t))+2\lambda _0\xi (t)}[\phi '(z_+(t))-\phi '(z_-(t))]\\ \le&e^{-\lambda x_1+\lambda _0(ct-\xi (t))+2\lambda _0\xi (t)}\bigg [L_f\beta _0e^{(\lambda -\lambda _0)(ct-\xi (t)) -2\lambda _0\xi (t)}\\&-M\left( \gamma _1e^{(\lambda -\mu )x_1-\mu (ct-\xi (t))} -\delta _0e^{\lambda (ct-\xi (t))}\right) \bigg ]\\ \le&e^{-\lambda x_1+\lambda _0(ct-\xi (t))+2\lambda _0\xi (t)}\left[ L_f\beta _0 -M\left( \gamma _1e^{(\lambda -\mu )x_1-\mu (ct-\xi (t))} -\delta _0e^{\lambda (ct-\xi (t))}\right) \right] . \end{aligned} \end{aligned}$$

If $\lambda \ge \mu$, then

$$\begin{aligned} \begin{aligned} {\mathcal {L}}W^-(x,t)\le&e^{-\lambda x_1+\lambda _0(ct-\xi (t))+2\lambda _0\xi (t)}\left[ L_f\beta _0 -M\left( \gamma _1e^{-\mu (ct-\xi (t))} -\delta _0e^{\lambda (ct-\xi (t))}\right) \right] \\ \le&0 \end{aligned} \end{aligned}$$

for $ct-\xi (t)\ll -1$ and $M>1$ is sufficiently large.

When $\lambda <\mu$, which means $|f'(1)-f'(0)|>0$, there holds

$$\begin{aligned}&f(\phi (z_+(t))))-f(\phi (z_-(t)))-f(\phi (z_+(t)))-\phi (z_-(t))))\\&\quad =f'(\phi (z_+(t)))\phi (z_-(t)))-o(\phi ^2(z_-(t))))-f'(\phi (z_-(t)))) \phi (z_-(t)))+o(\phi ^2(z_-(t))))\\&\quad \le -k_4\phi (z_-(t)) \end{aligned}$$

for $x_1+ct-\xi (t)>L_2>0$ with $L_2$ being large enough, where $0<k_4<\frac{1}{2}|f'(1)-f'(0)|$. The inequality above follows from that $f'(\phi (z_+(t)))\rightarrow f'(1)$ and $f'(\phi (z_-(t)))\rightarrow f'(0)$ as $L_2\rightarrow +\infty$. Then

$$\begin{aligned} \begin{aligned} {\mathcal {L}}W^-(x,t)\le&Me^{\lambda _0(ct+\xi (t))}\phi '(z_-(t))-k_4\phi (z_-(t))\\ \le&Me^{\lambda _0(ct+\xi (t))}\delta _0e^{\lambda (-x_1+ct-\xi (t))}-k_4\alpha _0e^{\lambda (-x_1+ct-\xi (t))}\\ \le&e^{\lambda (-x_1+ct-\xi (t))}\left( Me^{\lambda _0(ct+\xi (t))}-k_4\alpha _0\right) \\ \le&0, \end{aligned} \end{aligned}$$

provided that $ct+\xi (t)\ll -1$.

In addition, for $0<x_1+ct-\xi (t)<L_2$, there holds

$$\begin{aligned} \begin{aligned} {\mathcal {L}}W^-(x,t)\le&e^{-\lambda x_1+\lambda _0(ct-\xi (t))+2\lambda _0\xi (t)}\bigg [L_f\beta _0e^{(\lambda -\lambda _0)(ct-\xi (t)) -2\lambda _0\xi (t)}\\&-M\left( \gamma _1e^{(\lambda -\mu )x_1-\mu (ct-\xi (t))} -\delta _0e^{\lambda (ct-\xi (t))}\right) \bigg ]\\ \le&e^{-\lambda x_1+\lambda _0(ct-\xi (t))+2\lambda _0\xi (t)}\Big [L_f\beta _0 -M\left( \gamma _1e^{(\lambda -\mu )L_2}e^{-\lambda (ct-\xi (t))} -\delta _0e^{\lambda (ct-\xi (t))}\right) \Big ]. \end{aligned} \end{aligned}$$

Since $ct-\xi (t)\ll -1$ and $M\gg 1$, we have ${\mathcal {L}}W^-(x,t)\le 0$.

Next, we show the $W^+(x,t)$ is a super-solution. A straightforward computation shows that

$$\begin{aligned} {\mathcal {L}}W^+(x,t)=\left\{ \begin{aligned}&2(c+{\dot{\xi }}(t))\phi '(x_1+ct)-f(2\phi (ct+\xi (t)))\\&-\int _\Omega J(x-y)[W^+(y,t) -W^+(x,t)]dy,~x_1<0, \\&(c+{\dot{\xi }}(t))[\phi '(x_1+ct+\xi (t))+\phi '(-x_1+ct+\xi (t))]\\&\quad -\int _\Omega J(x-y)[W^+(y,t)-W^+(x,t)]dy-f(\phi (x_1+ct+\xi (t))\\&\quad +\phi (-x_1+ct+\xi (t))),~ x_1>0. \end{aligned}\right. \end{aligned}$$

When $x_1\ge 0$, denote

$$\begin{aligned} \Gamma ^{+}=\{x\in {\mathbb {R}}^N\mid y_1>0\},~\Gamma ^-=\{x\in {\mathbb {R}}^N\mid y_1\le 0\}. \end{aligned}$$

In view of that $K\subset {\mathbb {R}}^N\setminus$supp$(J)\cap \{x\in {\mathbb {R}}^N:~x_1\le 0\}$, one gets

$$\begin{aligned}&\int _{\Omega }J(x-y)[W^+(y,t)-W^+(x,t)]dy\\&\quad =\int _{\Omega \cap \Gamma ^+}J(x-y)[(\phi (y_1+ct+\xi (t))+\phi (-y_1+ct+\xi (t)))\\&\qquad -(\phi (x_1+ct+\xi (t)+\phi (-x_1+ct+\xi (t))]dy\\&\qquad +\int _{\Omega \cap \Gamma ^-}J(x-y)[2\phi (ct+\xi (t)) -(\phi (x_1+ct+\xi (t))+\phi (-x_1+ct+\xi (t)))]dy\\&\quad =\int _{{\mathbb {R}}^N}J(x-y)[(\phi (y_1+ct+\xi (t))+\phi (-y_1+ct+\xi (t))) -(\phi (x_1+ct+\xi (t))\\&\qquad +\phi (-x_1+ct+\xi (t)))]dy+\int _{\Omega \cap \Gamma ^-}J(x-y)[2\phi (ct+\xi (t)) -(\phi (y_1+ct+\xi (t))\\&\qquad +\phi (-y_1+ct+\xi (t)))]dy\\&\quad =c(\phi '(x_1+ct+\xi (t))+\phi '(-x_1+ct+\xi (t)))-f(\phi (x_1+ct+\xi (t)))\\&\qquad -f(\phi (-x_1+ct+\xi (t)))\\&\qquad +\int _{\Omega \cap \Gamma ^-}J(x-y)[2\phi (ct+\xi (t)) -(\phi (y_1+ct+\xi (t))+\phi (-y_1+ct+\xi (t)))]. \end{aligned}$$

Observe that, if $x_1>|ct+\xi (t)|>L$, where L is the diameter of the compact support of J, then the integral item of the last equality is equal to 0. Therefore, we obtain

$$\begin{aligned} {\mathcal {L}}W^+(x,t)=&{\dot{\xi }}[\phi '(x_1+ct+\xi (t))+\phi '(-x_1+xt+\xi (t))]+f(\phi (x_1+ct+\xi (t)))\\&+f(\phi (-x_1+ct+\xi (t)))-f(\phi (x_1+ct+\xi (t))+\phi (-x_1+ct+\xi (t)))\\ \ge&{\dot{\xi }}\phi '(x_1+ct+\xi (t))-L_f\phi (x_1+ct+\xi (t))\phi (-x_1+ct+\xi (t))\\ \ge&e^{\lambda _0(ct+\xi (t))}\left( M\gamma _1e^{-\mu (x_1+ct+\xi (t))}-L_f\alpha _0e^{-\lambda x_1} e^{(\lambda -\lambda _0)(ct+\xi (t))}\right) . \end{aligned}$$

If $\mu \le \lambda$, by choosing $M\gamma _1\ge L_f\alpha _0$, it is obvious that ${\mathcal {L}}W^+(x,t)\ge 0$ with $x_1>|ct+\xi (t)|$ being sufficiently large.

For $\mu >\lambda$, we have $f'(1)<f'(0)$. Consider the case $x_1+ct+\xi (t)\ge L_0\gg 1$. Then $\phi (x_1+ct+\xi (t))\approx 1$ while $\phi (-x_1+ct+\xi (t))\approx 0$. Furthermore,

$$\begin{aligned}&f(\phi (x_1+ct+\xi (t)))+f(\phi (-x_1+ct+\xi (t)))\\&\qquad -f(\phi (x_1+ct+\xi (t))+\phi (-x_1+ct+\xi (t)))\\&\quad \ge \frac{1}{2}(f'(0)-f'(1))\phi (-x_1+ct+\xi (t))\\&\quad \ge 0, \end{aligned}$$

which implies that ${\mathcal {L}}W^+(x,t)\ge 0$. For the other case $x_1+ct+\xi (t)\le L_0$, we know

$$\begin{aligned} {\mathcal {L}}W^+(x,t)\ge&e^{\lambda _0(ct+\xi (t))}\left( M\gamma _1e^{-\mu L_0}-L_f\alpha _0e^{-\lambda x_1} e^{(\lambda -\lambda _0)(ct+\xi (t))}\right) . \end{aligned}$$

Since $\lambda _0<\lambda$, we obtain ${\mathcal {L}}W^+(x,t)\ge 0$ holds provided that $M\ge \frac{L_f\alpha _0}{\gamma _1}e^{\mu L_0}$.

For the case $0<x_1<|ct+\xi (t)|$, one can see that

$$\begin{aligned}&\int _{\Omega \cap \Gamma ^-}J(x-y)[2\phi (ct+\xi (t)) -(\phi (y_1+ct+\xi (t))+\phi (-y_1+ct+\xi (t)))]dy \\&\quad =\int _{\Omega \cap \{y_1<ct+\xi (t)\}}J(x-y)[2\phi (ct+\xi (t)) -(\phi (y_1+ct+\xi (t))+\phi (-y_1+ct+\xi (t)))]dy\\&\qquad +\int _{\Omega \cap \{ct+\xi (t)<y_1<0\}}J(x-y)[2\phi (ct+\xi (t)) -(\phi (y_1+ct+\xi (t))+\phi (-y_1+ct+\xi (t)))]dy\\&\quad :=I_1+I_2. \end{aligned}$$

Since $\phi (ct+\xi (t))\le \frac{\theta }{2}$ for $ct+\xi (t)\ll -1$, $\phi (0)\le \theta ,$ and $\phi '>0$, we get that $\phi (-y_1+ct+\xi (t))>\theta \ge 2\phi (ct+\xi (t))$ for $y_1<ct+\xi (t)$. It follows that $I_1\le 0$. We know that

$$\begin{aligned} \begin{aligned} I_2\le&\int _{\Omega \cap \{ct+\xi (t)<y_1<0\}}J(x-y)C_{\phi }e^{\lambda (ct+\xi (t))} \left( 2-\left( e^{\lambda y_1}+e^{-\lambda y_1}\right) \right) dy\\&+K_\phi e^{(k_{\phi }+\lambda )(ct+\xi (t))}\int _{\Omega \cap \{ct+\xi (t)<y_1<0\}}J(x-y) \left( 2+e^{\lambda y_1}+e^{-\lambda y_1}\right) dy\\ \le&C_0e^{(k_{\phi }+\lambda )(ct+\xi (t))}. \end{aligned} \end{aligned}$$

The first inequality is follows from that there exist two numbers $K_\phi >0$ and $k_\phi >0$ such that $\left| \phi (x_1)-C_{\phi }e^{\lambda x_1}\right| \le K_{\phi }e^{(k_{\phi }+\lambda )x_1}$ for $x_1\le 0$ which is easy to obtain by (1.5). Then we have

$$\begin{aligned} {\mathcal {L}}W^+\ge&Me^{\lambda _0(ct+\xi (t))}(\phi '(x_1+ct+\xi (t))+\phi '(-x_1+ct+\xi (t))) +f(\phi (x_1+ct+\xi (t)))\\&+f(\phi (-x_1+ct+\xi (t))) -f(\phi (x_1+ct+\xi (t))+\phi (-x_1+ct+\xi (t)))\\&-C_0e^{(k_{\phi }+\lambda )(ct+\xi (t))}\\ \ge&Me^{\lambda _0(ct+\xi (t))}(\phi '(x_1+ct+\xi (t))+\phi '(-x_1+ct+\xi (t)))\\&-L_f\phi (x_1+ct+\xi (t))\phi (-x_1+ct+\xi (t)) -C_0e^{(k_{\phi }+\lambda )(ct+\xi (t))}\\ \ge&e^{(\lambda _0+\lambda )(ct+\xi (t))}\left[ 2M\gamma _0-L_f\beta _0e^{(\lambda -\lambda _0)(ct+\xi (t))} -C_0e^{(k_{\phi }-\lambda _0)(ct+\xi (t))}\right] . \end{aligned}$$

This gives that ${\mathcal {L}}W^+\ge 0$, provided $2M\alpha _0>L_f\beta _0+C_0$ and $\lambda _0<\min \{k_{\phi },\lambda \}$.

For $x_1<0$, we just deal with the case $-L<x_1<0$ because that for $x_1\le -L$,

$$\begin{aligned}&\int _\Omega J(x-y)[W^+(y,t)-W^+(x,t)]dy\\&\quad =\int _{\Gamma ^+}J(x-y)[\phi (y_1+ct+\xi (t))+\phi (y_1+ct+\xi (t))-2\phi (ct+\xi (t))]dy=0. \end{aligned}$$

Since $\phi ''(x)\ge 0$ for $x\le 0$, we have that

$$\begin{aligned}&\int _\Omega J(x-y)[W^+(y,t)-W^+(x,t)]dy\\&\quad \le c\phi '(ct+\xi (t))-f(\phi (ct+\xi (t)))+\int _{-\infty }^{x_1}J_1(y_1)[\phi (y_1-x_1+ct+\xi (t))\\&\qquad +\phi (x_1-y_1+ct+\xi (t))-2\phi (ct+\xi (t))]dy_1\\&\qquad -\int _{{\mathbb {R}}}J_1(y_1)[\phi (ct+\xi (t)-y_1)-\phi (ct+\xi (t))]dy_1\\&\quad =c\phi '(ct+\xi (t))-f(\phi (ct+\xi (t)))+\int _{-\infty }^{0}J_1(y_1)[\phi (y_1-x_1+ct+\xi (t))\\&\qquad +\phi (x_1-y_1+ct+\xi (t))-2\phi (ct+\xi (t))]dy_1\\&\qquad +\int _{x_1}^0J_1(y_1)[2\phi (ct+\xi (t))-\phi (y_1-x_1+ct+\xi (t))-\phi (x_1-y_1+ct+\xi (t))]\\&\qquad -\int _{-\infty }^0J_1(y_1)[\phi (ct+\xi (t)-y_1)+\phi (ct+\xi (t)+y_1)-2\phi (ct+\xi (t))]dy_1\\&\quad \le c\phi '(ct+\xi (t))-f(\phi (ct+\xi (t)))+C_0e^{(k_{\phi }+\lambda )(ct+\xi (t))}\\&\qquad +\int _{-\infty }^{0}J_1(y_1)[\phi (y_1-x_1+ct+\xi (t))\\&\qquad +\phi (x_1-y_1+ct+\xi (t))-\phi (ct+\xi (t)-y_1)-\phi (ct+\xi (t)+y_1)]dy_1. \end{aligned}$$

Observe that, if $x_1<0$ then $\left| \phi (x_1)-C_{\phi }e^{\lambda x_1}\right| \le K_{\phi }e^{(k_{\phi }+\lambda )x_1}$. Thus there is $C_0>0$ such that

$$\begin{aligned}&\int _{-\infty }^{0}J_1(y_1)[\phi (y_1-x_1+ct+\xi (t))+\phi (x_1-y_1+ct+\xi (t))\\&\qquad -\phi (ct+\xi (t)-y_1)-\phi (ct+\xi (t)+y_1)]dy_1\\&\quad \le C_{\phi }e^{\lambda (ct+\xi (t))}\int _{-\infty }^{0}J_1(y_1)\left[ \left( e^{\lambda (x_1-y_1)} +e^{\lambda (y_1-x_1)}\right) -\left( e^{\lambda y_1}+e^{-\lambda y_1}\right) \right] dy_1\\&\qquad +2K_{\phi }e^{(k_{\phi }+\lambda )(ct+\xi (t))}\int _{-\infty }^{0}J_1(y_1)\bigg [\left( e^{(k_{\phi }+\lambda )(x_1-y_1)} +e^{(k_{\phi }+\lambda )(y_1-x_1)}\right) \\&\qquad -\left( e^{(k_{\phi }+\lambda )y_1}+e^{-(k_{\phi }+\lambda )y_1}\right) \bigg ]dy_1\\&\quad \le C_0e^{(k_{\phi }+\lambda )(ct+\xi (t))}. \end{aligned}$$

The last inequality above holds true, since $x_1<0$ and $f(\upsilon )=\upsilon +\frac{1}{\upsilon }$ is monotonically increasing in $\upsilon \in (1,\infty )$. Then it follows that for $M\ge \frac{C_0}{\gamma _0}$,

$$\begin{aligned} {\mathcal {L}}W^+\ge&2{\dot{\xi }}(t)\phi '(ct+\xi (t))+f(\phi (ct+\xi (t)))-f(2\phi (ct+\xi (t))) -C_0e^{(k_{\phi }+\lambda )(ct+\xi (t))}\\ \ge&2M\gamma _0e^{(\lambda _0+\lambda )(ct+\xi (t))}-2C_0e^{(k_{\phi }+\lambda )(ct+\xi (t))}\\ =&e^{(k_{\phi }+\lambda )(ct+\xi (t))}(2M\gamma _0-2C_0)\\ \ge&0. \end{aligned}$$

The second inequality follows from that $f'(s)<0$ in $[\phi (ct+\xi (t)),2\phi (ct+\xi (t))]$ for $ct+\xi (t)\ll -1$. The proof of Proposition 3.2 has been finished.

1.2 Proof of Lemma 4.2

We know that

$$\begin{aligned} {\mathcal {M}}{\underline{u}}:=&{\underline{u}}_t(x,t)-\int _{{\mathbb {R}}^N}J(x-y)[{\underline{u}}(y,t) -{\underline{u}}(x,t)]dy-f({\underline{u}}(x,t))\\ =&(c-2\epsilon \Vert f'\Vert \delta ^{-1}e^{-\omega (t-t_0)})\phi '+\epsilon \omega e^{-\omega (t-t_0)}\\&-\int _{{\mathbb {R}}^N}J(x-y)[\phi (\xi _-(y,t)) -\phi (\xi _-(x,t))]dy\\&-f(\phi (\xi _-(x,t))-\epsilon e^{-\omega (t-t_0)})\\ =&-2\epsilon \Vert f'\Vert \delta ^{-1}e^{-\omega (t-t_0)}\phi '+\epsilon \omega e^{-\omega (t-t_0)}+f(\phi (\xi _-(x,t)))\\&-f\left( \phi (\xi _-(x,t))-\epsilon e^{-\omega (t-t_0)}\right) . \end{aligned}$$

When $\xi _-(x,t)\in [-A, A]$, there holds $\phi '(\xi _-(x,t))\ge \delta$. Therefore,

$$\begin{aligned} {\mathcal {M}}{\underline{u}}\le \epsilon e^{-\omega (t-t_0)}(-2\Vert f'\Vert +\omega +\Vert f'\Vert )\le 0. \end{aligned}$$

For $|\xi _-(x,t)|\ge A$, we have

$$\begin{aligned} \phi (\xi _-(x,t)),{\underline{u}}(x,t)\in [-\infty ,\eta ]\cup [1-\eta ,+\infty ]. \end{aligned}$$

Then $f'(s)\le -\omega$ for $s\in [\phi (\xi _-(xt))-\epsilon e^{-\omega (t-t_0)},\phi (\xi _-(xt))]$. Hence,

$$\begin{aligned} {\mathcal {M}}{\underline{u}}\le \epsilon \omega e^{-\omega (t-t_0)}-\omega \epsilon e^{-\omega (t-t_0)}=0. \end{aligned}$$

For $t_0\le -T$, one get

$$\begin{aligned} {\underline{u}}(x,t_0)=\phi (x_1+ct)-\epsilon \le u(x,t_0). \end{aligned}$$

Until now, we have show the function ${\underline{u}}$ is a sub-solution to (4.2). Similarly one can show ${\overline{u}}$ is a super-solution to (4.2).