Propagation Phenomena for Man–Environment Epidemic Model with Nonlocal Dispersals

Abstract

In this paper, we focus on the propagation phenomena of a bistable man–environment epidemic model with nonlocal dispersals, where there exists a positive feedback interaction between the concentration of infectious agent and infectious human population. The monostable and bistable traveling wave solutions and three-wave entire solutions are studied. First, by applying and developing the known results for monostable case, we give a summary of the existence and asymptotic behavior of all monostable traveling wave solutions in two different monostable intervals and further find some relationship between them. The existence of bistable traveling wave solutions is obtained by introducing the results about monotone semiflows with weak compactness. Second, we give twelve types of three-wave entire solutions, which contain all possibilities of three-wave entire solutions originating from three traveling wave solutions with different nonzero wave speeds, by constructing new auxiliary functions and super- and sub-solutions for every type. We also show that these entire solutions are globally Lipschitz continuous with respect to spatial variable. In addition, the nonexistence result of entire solutions originating from more than four traveling wave solutions is obtained.

Appendix A. Proofs of Some Lemmas in Section 3.2

1.1 A.1 Proof of Lemma 3.8

We denote

$$\begin{aligned}&\epsilon _1:=\min \Big \{\frac{\phi _{11}(0)}{u_1^*},\frac{\phi _{12}(0)}{v_1^*}\Big \},\quad \epsilon _2:=\max \Big \{\frac{\phi _{11}(0)}{u_2^*},\frac{\phi _{12}(0)}{v_2^*}\Big \},\\&\epsilon _3:=\min \Big \{\frac{\phi _{21}(0)}{u_1^*},\frac{\phi _{22}(0)}{v_1^*}\Big \},\quad \epsilon _4:=\max \Big \{\frac{\phi _{21}(0)}{u_2^*},\frac{\phi _{22}(0)}{v_2^*}\Big \},\\&\epsilon _5:=\min \Big \{\frac{\phi _{31}(0)}{u_1^*},\frac{\phi _{32}(0)}{v_1^*}\Big \},\quad \epsilon _6:=\max \Big \{\frac{\phi _{31}(0)}{u_2^*},\frac{\phi _{32}(0)}{v_2^*}\Big \}. \end{aligned}$$

It follows that $\epsilon _1$, $\epsilon _3>1$, $\epsilon _2$, $\epsilon _4$, $\epsilon _5$, $\epsilon _6\in (0,1)$ and

$$\begin{aligned} E_1\ll \epsilon _1 E_1\le \varvec{\phi }_1(0)\le \epsilon _2 E_2\ll E_2,\\ E_0\ll \epsilon _3 E_1\le \varvec{\phi }_2(0)\le \epsilon _4 E_2\ll E_2,\\ E_0\ll \epsilon _5 E_1\le \varvec{\phi }_3(0)\le \epsilon _6 E_2\ll E_2. \end{aligned}$$

In what follows, we give the upper and lower bounds of $\varvec{\phi }_1(\xi -q_1)$, $\varvec{\phi }_2(\xi +q_2)$, and $\varvec{\phi }_3(\xi +q_2)$ in four cases. The calculations are not complicated and we omit them. First, when $\xi \le q_1(t)$, it holds that

$$\begin{aligned} E_1\le \varvec{\phi }_1(\xi -q_1)\le \epsilon _2 E_2, ~\epsilon _3 E_1\le \varvec{\phi }_2(\xi +q_2)\le E_2, ~E_0 \le \varvec{\phi }_3(\xi +q_2)\le \epsilon _6 E_2. \end{aligned}$$

Second, when $q_1(t)\le \xi \le -q_1(t)$, we have

$$\begin{aligned} \epsilon _1 E_1\le \varvec{\phi }_1(\xi -q_1)\le E_2, ~\epsilon _3 E_1\le \varvec{\phi }_2(\xi +q_2)\le E_2, ~E_0 \le \varvec{\phi }_3(\xi +q_2)\le \epsilon _6 E_2. \end{aligned}$$

Third, when $-q_1(t)\le \xi \le -q_2(t)$, it follows that

$$\begin{aligned} \epsilon _1 E_1\le \varvec{\phi }_1(\xi -q_1)\le E_2,~ E_0\le \varvec{\phi }_2(\xi +q_2)\le \epsilon _4 E_2,~ E_0 \le \varvec{\phi }_3(\xi +q_2)\le \epsilon _6 E_2. \end{aligned}$$

Fourth, when $\xi \ge -q_2(t)$, we can get

$$\begin{aligned} \epsilon _1 E_1\le \varvec{\phi }_1(\xi -q_1)\le E_2, ~E_0\le \varvec{\phi }_2(\xi +q_2)\le \epsilon _4 E_2, ~\epsilon _5 E_1\le \varvec{\phi }_3(\xi +q_2)\le E_2. \end{aligned}$$

Some calculations imply that

$$\begin{aligned} \begin{aligned} 0&<u_1^*u_2^*\min \{\epsilon _3(1-\epsilon _6)(u_2^*-u_1^*),(\epsilon _1-1)(1-\epsilon _4)u_2^*\}\\&\le u_2^*[\phi _{11}(\xi -q_1)-u_1^*][u_2^*-\phi _{21}(\xi +q_1)]\\&\quad +(u_2^*-u_1^*)\phi _{21}(\xi +q_1)[u_2^*-\phi _{31}(\xi +q_2)]\\&\le 2{u_2^*}^2(u_2^*-u_1^*) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} 0&<v_1^*v_2^*\min \{\epsilon _3(1-\epsilon _6)(v_2^*-v_1^*),(\epsilon _1-1)(1-\epsilon _4)v_2^*\}\\&\le v_2^*\left[ \phi _{12}(\xi -q_1)-v_1^*\right] [v_2^*-\phi _{22}(\xi +q_1)]\\&\quad +(v_2^*-v_1^*)\phi _{22}(\xi +q_1)[v_2^*-\phi _{32}(\xi +q_2)]\\&\le 2{v_2^*}^2(v_2^*-v_1^*) \end{aligned} \end{aligned}$$

for any $\xi \in {\mathbb {R}}$ and $q_2$, $q_1$ satisfying (3.17). Then, the second and third inequalities in Lemma 3.8 can be easily obtained. For the first inequality, we get from (3.12) and $\varvec{\phi }_2'<0$ that

$$\begin{aligned} (\phi _{21}(\xi +q_1),\phi _{22}(\xi +q_1))\ge (\phi _{21}(0),\phi _{22}(0))\ge \left( \frac{u_1^*+u_2^*}{2},\frac{v_1^*+v_2^*}{2}\right) ~~\text {when}~\xi \le q_1(t). \end{aligned}$$

It follows that

$$\begin{aligned} \begin{aligned}&u_2^*[\phi _{31}(\xi +q_2)-u_1^*][u_2^*-\phi _{21}(\xi +q_1)]+\phi _{21}(\xi +q_1)[u_2^*-\phi _{31}(\xi +q_2)](u_2^*-u_1^*)\\&\quad \ge u_2^*[\phi _{31}(\xi +q_2)-u_1^*]\left( u_2^*-\frac{u_1^*+u_2^*}{2}\right) +\frac{u_1^*+u_2^*}{2}[u_2^*-\phi _{31}(\xi +q_2)](u_2^*-u_1^*)\\&\quad \ge \frac{u_2^*}{2}(u_2^*-u_1^*)^2, \end{aligned} \end{aligned}$$

and then

$$\begin{aligned} \begin{aligned} P^{x}(\xi ,t)&= \phi _{21}(u_2^*-\phi _{31})\frac{(u_2^*-u_1^*)[u_2^*(\phi _{31}-u_1^*)(u_2^*-\phi _{21})+\phi _{21}(u_2^*-\phi _{31})(u_2^*-u_1^*)]}{[u_2^*(\phi _{11}-u_1^*)(u_2^*-\phi _{21})+(u_2^*-u_1^*)\phi _{21}(u_2^*-\phi _{31})]^2}\\&\ge \frac{\epsilon _3(1-\epsilon _6)u_1^*u_2^*(u_2^*-u_1^*)\frac{u_2^*}{2}(u_2^*-u_1^*)^2}{[2{u_2^*}^2(u_2^*-u_1^*)]^2}=\frac{\epsilon _3(1-\epsilon _6)u_1^*(u_2^*-u_1^*)}{8{u_2^*}^2}. \end{aligned} \end{aligned}$$

It completes the proof.

1.2 A.2 Proof of Lemma 3.9

When $\xi \le -q_1(t)$, we have that

$$\begin{aligned} \phi _{31}(\xi +q_2)\le \phi _{31}(0)\le u_1^*\le \phi _{11}(\xi -q_1),\quad ~\text {and}~P^{y}(\xi ,t)\ge 0. \end{aligned}$$

By $P^{z},\phi _{11}',\phi _{31}'\ge 0$ and $\phi _{21}'\le 0$, we have that when $\xi \le q_1(t)$,

$$\begin{aligned} A_1(\xi ,t)=P^{x}(\xi ,t)\phi _{11}'-P^{y}(\xi ,t)\phi _{21}'+P^{z}(\xi ,t)\phi _{31}'\ge P^x(\xi ,t)\phi _{11}'\ge \frac{1}{2}P^x(\xi ,t)|\phi _{11}'|, \end{aligned}$$

and when $\xi \le -q_1(t)$,

$$\begin{aligned} A_1(\xi ,t)\ge \frac{1}{2}\left[ P^{x}(\xi ,t)|\phi _{11}'|+P^{y}(\xi ,t)|\phi _{21}'|\right] . \end{aligned}$$

When $-q_1(t)\le \xi \le -q_2(t)$, we have that $P^{y}$, $P^{z}\ge 0$. Then, we compute that

$$\begin{aligned}&A_1(\xi ,t)-\frac{1}{2}\left[ P^{y}(\xi ,t)|\phi _{21}'(\xi +q_1)|+P^{z}(\xi ,t)|\phi _{31}'(\xi +q_2)|\right] \\&\quad =P^x(\xi ,t)\phi _{11}'+\frac{1}{2}P^y(\xi ,t)|\phi _{21}'|+\frac{1}{2}P^z(\xi ,t)\phi _{31}'\ge P^x(\xi ,t)\phi _{11}'+\frac{1}{2}P^y(\xi ,t)|\phi _{21}'|\\&\quad \ge \frac{\phi _{21}(u_2^*-\phi _{31})(u_2^*-u_1^*)[u_2^*(\phi _{31}-u_1^*)(u_2^*-\phi _{21})+\phi _{21}(u_2^*-\phi _{31})(u_2^*-u_1^*)]}{[u_2^*(\phi _{11}-u_1^*)(u_2^*-\phi _{21})+(u_2^*-u_1^*)\phi _{21}(u_2^*-\phi _{31})]^2}\phi _{11}'\\&\qquad +\frac{\rho \phi _{21}(\phi _{11}-u_1^*)(u_2^*-\phi _{31}){u_2^*}^2(u_2^*-u_1^*)(\phi _{11}-\phi _{31})}{2[u_2^*(\phi _{11}-u_1^*)(u_2^*-\phi _{21})+(u_2^*-u_1^*)\phi _{21}(u_2^*-\phi _{31})]^2}\\&\quad \ge \frac{\phi _{21}(u_2^*-u_1^*)u_1^*{u_2^*}^3}{[2u_2^*(u_2^*-u_1^*)]^2}\left[ -C_0e^{-2\eta _2\delta }+\frac{\rho }{2}(\varepsilon _1-1)(1-\varepsilon _6)(\phi _{11}(0)-\phi _{31}(0))\right] \ge 0 \end{aligned}$$

for $\delta$ sufficiently large. When $\xi \ge -q_2(t)$, we have that $P^z\ge 0$. Similarly, we obtain that

$$\begin{aligned}\begin{aligned}&A_1(\xi ,t)-\frac{1}{2}P^z(\xi ,t)|\phi _{31}'(\xi +q_2)|=P^x(\xi ,t)\phi _{11}'+P^y(\xi ,t)|\phi _{21}'|\\&\quad +\frac{1}{2}P^z(\xi ,t)\phi _{31}' \ge \frac{u_2^*-\phi _{31}}{[2u_2^*(u_2^*-u_1^*)]^2}\left\{ -(u_2^*-u_1^*)u_1^*{u_2^*}^3C_0e^{-2\eta _2\delta }\right. \\&\left. \quad -(u_2^*-u_1^*)^3{u_2^*}^2C_0e^{-\eta _2\delta }+\frac{\rho }{2}[(\varepsilon _1-1)u_1^*(1-\varepsilon _4){u_2^*}^2]^2\right\} \ge 0 \end{aligned}\end{aligned}$$

for $\delta$ sufficiently large. It completes the proof.

1.3 A.3 Proof of Lemma 3.10

For given $(\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-$ and $s\in {\mathbb {R}}$, when there is no confusion, we simply write

$$\begin{aligned} {\hat{\phi }}_{11}(\theta ):= & {} \phi _{11}(\xi -q_1(t)-\theta s),~{\hat{\phi }}_{21}(\theta ) :=\phi _{21}(\xi +q_1(t)-\theta s),~\text {and}~{\hat{\phi }}_{31}(\theta )\\:= & {} \phi _{31}(\xi +q_2(t)-\theta s), \end{aligned}$$

where $\theta \in [0,1]$. Then, $H_1(\xi ,t)$ can be represented as follows:

$$\begin{aligned} \begin{aligned} H_1(\xi ,t)=&\int _{{\mathbb {R}}}J_1(s)[P({\hat{\phi }}_{11}(1),{\hat{\phi }}_{21}(1),{\hat{\phi }}_{31}(1))-P({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0))]{\text {d}}s\\&\quad -P_{x}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0))\int _{{\mathbb {R}}}J_1(s)[{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)]{\text {d}}s\\&\quad -P_{y}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0))\int _{{\mathbb {R}}}J_1(s)[{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)]{\text {d}}s\\&\quad -P_{z}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0))\int _{{\mathbb {R}}}J_1(s)[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)]{\text {d}}s. \end{aligned}\end{aligned}$$

We denote

$$\begin{aligned} \begin{aligned}&{\mathcal {I}}_1=P_{x}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0)) [{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)],\\&{\mathcal {I}}_2=P_{y}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0))[{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)],\\&{\mathcal {I}}_3=P_{z}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0))[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)]. \end{aligned} \end{aligned}$$

By mean value theorem, there exist $\theta _1$, $\theta _2$, and $\theta _3$ in (0, 1) such that

$$\begin{aligned} P({\hat{\phi }}_{11}(1),{\hat{\phi }}_{21}(1),{\hat{\phi }}_{31}(1))-P({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0)) ={\mathcal {I}}_4+{\mathcal {I}}_5+{\mathcal {I}}_6, \end{aligned}$$

where

$$\begin{aligned}\begin{aligned}&{\mathcal {I}}_4=P_{x}(\theta _1{\hat{\phi }}_{11}(1)+(1-\theta _1){\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(1),{\hat{\phi }}_{31}(1))[{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)],\\&{\mathcal {I}}_5=P_{y}({\hat{\phi }}_{11}(0),\theta _2{\hat{\phi }}_{21}(1)+(1-\theta _2){\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(1))[{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)],\\&{\mathcal {I}}_6=P_{z}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),\theta _3{\hat{\phi }}_{31}(1)+(1-\theta _3){\hat{\phi }}_{31}(0))[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)]. \end{aligned}\end{aligned}$$

Then, we can get that

$$\begin{aligned} H_1(\xi ,t)=\int _{{\mathbb {R}}}J_1(s)[{\mathcal {I}}_4+{\mathcal {I}}_5+{\mathcal {I}}_6-{\mathcal {I}}_1-{\mathcal {I}}_2-{\mathcal {I}}_3]{\text {d}}s. \end{aligned}$$

Also by mean value theorem, there exist $\theta _4$, $\theta _5$, and $\theta _6$ in (0, 1) such that

$$\begin{aligned}\begin{aligned}&{\mathcal {I}}_4-{\mathcal {I}}_1=P_{xx}(\theta _4{\hat{\phi }}_{11}(1)+(1-\theta _4){\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(1),{\hat{\phi }}_{31}(1))\theta _1[{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)]^2\\&\quad +P_{xy}({\hat{\phi }}_{11}(0),\theta _5{\hat{\phi }}_{21}(1)+(1-\theta _5){\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(1))[{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)][{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)]\\&\quad +P_{xz}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),\theta _6{\hat{\phi }}_{31}(1)+(1-\theta _6){\hat{\phi }}_{31}(0))[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)][{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)]\\&\quad =\theta _1G_1(\xi ,t,s;\theta _4)+G_2(\xi ,t,s;\theta _5)+G_3(\xi ,t,s;\theta _6), \end{aligned}\end{aligned}$$

where

$$\begin{aligned}\begin{aligned} G_1(\xi ,t,s;\theta _4)&= P_{xx}(\theta _4{\hat{\phi }}_{11}(1)+(1-\theta _4){\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(1),{\hat{\phi }}_{31}(1))[{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)]^2,\\ G_2(\xi ,t,s;\theta _5)&= P_{xy}({\hat{\phi }}_{11}(0),\theta _5{\hat{\phi }}_{21}(1)+(1-\theta _5){\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(1))\\&\quad [{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)][{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)],\\ G_3(\xi ,t,s;\theta _6)&= P_{xz}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),\theta _6{\hat{\phi }}_{31}(1)+(1-\theta _6){\hat{\phi }}_{31}(0))[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)]\\&\quad [{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)]. \end{aligned}\end{aligned}$$

There exist $\theta _7$ and $\theta _8$ in (0, 1) such that

$$\begin{aligned}\begin{aligned} {\mathcal {I}}_5-{\mathcal {I}}_2&=P_{yy}({\hat{\phi }}_{11}(0),\theta _7{\hat{\phi }}_{21}(1)+(1-\theta _7){\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(1))\theta _2[{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)]^2\\&\quad -P_{yz}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),\theta _8{\hat{\phi }}_{31}(1)+(1-\theta _8){\hat{\phi }}_{31}(0))\\&\quad [{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)][{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)]\\&=\theta _2G_4(\xi ,t,s;\theta _7)+G_5(\xi ,t,s;\theta _8), \end{aligned}\end{aligned}$$

where

$$\begin{aligned}\begin{aligned} G_4(\xi ,t,s;\theta _7)&= P_{yy}({\hat{\phi }}_{11}(0),\theta _7{\hat{\phi }}_{21}(1)+(1-\theta _7){\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(1))[{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)]^2,\\ G_5(\xi ,t,s;\theta _8)&= P_{yz}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),\theta _6{\hat{\phi }}_{31}(1)+(1-\theta _6){\hat{\phi }}_{31}(0))[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)]\\&\quad [{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)]. \end{aligned}\end{aligned}$$

There is a constant $\theta _9\in (0,1)$ such that

$$\begin{aligned} \begin{aligned} {\mathcal {I}}_6-{\mathcal {I}}_3&=P_{zz}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),\theta _9{\hat{\phi }}_{31}(1)+(1-\theta _9){\hat{\phi }}_{31}(0))\theta _3[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)]^2\\&=\theta _3G_6(\xi ,t,s;\theta _9), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} G_6(\xi ,t,s;\theta _9):=P_{zz}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),\theta _9{\hat{\phi }}_{31}(1)+(1-\theta _9){\hat{\phi }}_{31}(0))[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)]^2. \end{aligned}$$

Based on the above formulas, we can get that

$$\begin{aligned} \begin{aligned} H_1(\xi ,t)&=\int _{{\mathbb {R}}}J_1(s)[\theta _1G_1(\xi ,t,s;\theta _4)+G_2(\xi ,t,s;\theta _5)+G_3(\xi ,t,s;\theta _6)\\&\quad +\theta _2G_4(\xi ,t,s;\theta _7)+G_5(\xi ,t,s;\theta _8)+\theta _3G_6(\xi ,t,s;\theta _9)]{\text {d}}s. \end{aligned} \end{aligned}$$

(3.31)

In what follows, we provide the detailed estimations only for $|G_1(\xi ,t,s;\theta _4)/ A_1(\xi ,t)|$, and for $|G_i(\xi ,t,s;\theta _{i+3})/ A_1(\xi ,t)|$ with $i=2,\ldots ,6$, the methods are similar. We get from the mean value theorem that

$$\begin{aligned} |{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)|= & {} |{\phi }_{11}(\xi -q_1-s)-{\phi }_{11}(\xi -q_1)| =|{\phi }_{11}'(\xi -q_1-\theta _{10}s)s|\\= & {} |{\hat{\phi }}_{11}'(\theta _{10})s|, \end{aligned}$$

where $\theta _{10}$ is some constant in [0, 1]. In addition, for the $C_3$ given by Lemma 3.6, there is a sufficiently large constant $C_4$ such that

$$\begin{aligned} C_3|{\phi }_{11}(\xi -q_1-s)-{\phi }_{11}(\xi -q_1)|<C_4\quad \text {for any}~(\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-,s\in {\mathbb {R}}. \end{aligned}$$

Now, we consider the following six cases. Recall that the constant m, which appears below, is defined in assumption (J4).

Case 1: When $\xi \le q_1(t)$, based on Lemmas 3.6, 3.7, 3.8, and 3.9 (i), we have that for $s\in \text {supp}(J_1)$, there exists a positive constant ${\tilde{M}}_1$ such that

$$\begin{aligned} \bigg |\frac{G_1(\xi ,t,s;\theta _4)}{A_1(\xi ,t)}\bigg |\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }} _{11}(1)-{\hat{\phi }}_{11}(0)|}{P^{x}(\xi ,t)|\phi _{11}'(\xi -q_1)|}\\\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}'(\theta _{10})s|}{\mu _1|\phi _{11}'(\xi -q_1)|} \le \frac{2C_4m}{\mu _1}|\phi _{21}'(\xi +q_1-s)|\\&\cdot \frac{|u_2^*-\phi _{21}(\xi +q_1-s)|}{|\phi _{21}'(\xi +q_1-s)|}\cdot \frac{|\phi _{11}'(\xi -q_1-\theta _{10}s)|}{|\phi _{11}(\xi -q_1-\theta _{10}s)-u_1^*|}\\&\cdot \frac{|\phi _{11}(\xi -q_1-\theta _{10}s)-u_1^*|}{|\phi _{11}(\xi -q_1)-u_1^*|}\cdot \frac{|\phi _{11}(\xi -q_1)-u_1^*|}{|\phi _{11}'(\xi -q_1)|}\\\le & {} \frac{2C_4m}{\mu _1} C_0e^{\eta _1(\xi +q_1-s)}\frac{1}{C_1} C_2C\frac{1}{C_1}\le {\tilde{M}}_1e^{\eta _1q_1}. \end{aligned}$$

Case 2: When $q_1(t)\le \xi \le 0$, based on Lemmas 3.6, 3.7, 3.8, and 3.9 (ii), we have that for $s\in \text {supp}(J_1)$, there exists a positive constant ${\tilde{M}}_2$ such that

$$\begin{aligned} \bigg |\frac{G_1(\xi ,t,s;\theta _4)}{A_1(\xi ,t)}\bigg |\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)|}{P^x(\xi ,t)|\phi _{11}'(\xi -q_1)|+P^y(\xi ,t)|\phi _{21}'(\xi +q_1)|}\\\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}'(\theta _{10})s|}{\mu _1|\phi _{11}'(\xi -q_1)|}\le \frac{2C_4m}{\mu _1}|\phi _{21}'(\xi +q_1-s)|\\&\cdot \frac{|u_2^*-\phi _{21}(\xi +q_1-s)|}{|\phi _{21}'(\xi +q_1-s)|}\cdot \frac{|\phi _{11}'(\xi -q_1-\theta _{10}s)|}{|u_2^*-\phi _{11}(\xi -q_1-\theta _{10}s)|}\\&\cdot \frac{|u_2^*-\phi _{11}(\xi -q_1-\theta _{10}s)|}{|u_2^*-\phi _{11} (\xi -q_1)|}\cdot \frac{|u_2^*-\phi _{11}(\xi -q_1)|}{|\phi _{11}'(\xi -q_1)|}\\\le & {} \frac{2C_4m}{\mu _1}C_0e^{\eta _1(\xi +q_1-s)}\frac{1}{C_1} C_2C\frac{1}{C_1}\le {\tilde{M}}_2e^{\eta _1q_1}. \end{aligned}$$

Case 3: When $0\le \xi \le -q_1(t)$, based on Lemmas 3.6, 3.7,  3.8, and 3.9 (ii), we have that for $s\in \text {supp}(J_1)$, there exists a positive constant ${\tilde{M}}_3$ such that

$$\begin{aligned} \bigg |\frac{G_1(\xi ,t,s;\theta _4)}{A_1(\xi ,t)}\bigg |\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)|}{P^x(\xi ,t)|\phi _{11}'(\xi -q_1)|+P^y(\xi ,t)|\phi _{21}'(\xi +q_1)|}\\\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}'(\theta _{10})s|}{\mu _2|\phi _{21}'(\xi +q_1)|}\le \frac{2C_4m}{\mu _2}|\phi _{11}'(\xi -q_1-\theta _{10}s)|\\&\cdot \frac{|u_2^*-\phi _{21}(\xi +q_1-s)|}{|u_2^*-\phi _{21}(\xi +q_1)|}\cdot \frac{|u_2^*-\phi _{21}(\xi +q_1)|}{|\phi _{21}'(\xi +q_1)|}\\\le & {} \frac{2C_4m}{\mu _2}C_0e^{-\eta _2(\xi -q_1-\theta _{10}s)}C\frac{1}{C_1}\le {\tilde{M}}_3e^{\eta _2q_1}. \end{aligned}$$

Case 4: When $-q_1(t)v\le \xi \le (-q_1(t)-q_2(t))/2$, based on Lemma 3.6, 3.7,  3.8, and 3.9(iii), we have that for $s\in \text {supp}(J_1)$, there exists a positive constant ${\tilde{M}}_4$ such that

$$\begin{aligned} \bigg |\frac{G_1(\xi ,t,s;\theta _4)}{A_1(\xi ,t)}\bigg |\le & {} \frac{2C_4|{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)|}{P^y(\xi ,t)|\phi _{21}'(\xi +q_1)|+P^z(\xi ,t)|\phi _{31}'(\xi +q_2)|}\\\le & {} \frac{2C_4|{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}'(\theta _{10})s|}{\mu _2|\phi _{21}'(\xi +q_1)|} \le \frac{2C_4m}{\mu _2}|\phi _{11}'(\xi -q_1-\theta _{10}s)|\\&\cdot \frac{|\phi _{21}(\xi +q_1-s)|}{|\phi _{21}(\xi +q_1)|}\cdot \frac{|\phi _{21}(\xi +q_1)|}{|\phi _{21}'(\xi +q_1)|}\\\le & {} \frac{2C_4m}{\mu _2}C_0e^{-\eta _2(\xi -q_1-\theta _{10}s)}C\frac{1}{C_1}\le {\tilde{M}}_4e^{\eta _2q_1}. \end{aligned}$$

Case 5: When $(-q_1(t)-q_2(t))/2\le \xi \le -q_2(t)$, based on Lemmas 3.6, 3.7,  3.8, and 3.9 (iii), we have that for $s\in \text {supp}(J_1)$, there exists a positive constant ${\tilde{M}}_5$ such that

$$\begin{aligned} \bigg |\frac{G_1(\xi ,t,s;\theta _4)}{A_1(\xi ,t)}\bigg |\le & {} \frac{2C_4|{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)|}{P^y(\xi ,t)|\phi _{21}'(\xi +q_1)|+P^z(\xi ,t)|\phi _{31}'(\xi +q_2)|}\\\le & {} \frac{2C_4|{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}'(\theta _{10})s|}{\mu _2|\phi _{21}'(\xi +q_1)|}\le \frac{2C_4m}{\mu _2}|\phi _{11}'(\xi -q_1-\theta _{10}s)|\\&\cdot \frac{|\phi _{21}(\xi +q_1-s)|}{|\phi _{21}(\xi +q_1)|}\cdot \frac{|\phi _{21}(\xi +q_1)|}{|\phi _{21}'(\xi +q_1)|}\\\le & {} \frac{2C_4m}{\mu _2}C_0e^{-\eta _2(\xi -q_1-\theta _{10}s)}C\frac{1}{C_1}\le {\tilde{M}}_5e^{\eta _2q_1}. \end{aligned}$$

Case 6: When $\xi \ge -q_2(t)$, based on Lemmas 3.6, 3.7,  3.8, and 3.9 (iv), we have that for $s\in \text {supp}(J_1)$, there exists a positive constant ${\tilde{M}}_6$ such that

$$\begin{aligned} \bigg |\frac{G_1(\xi ,t,s;\theta _4)}{A_1(\xi ,t)}\bigg |\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{31}(1)|\cdot |{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)|}{P^z(\xi ,t)|\phi _{31}'(\xi +q_2)|}\\\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{31}(1)|\cdot |{\hat{\phi }}_{11}'(\theta _{10})s|}{\mu _3|\phi _{31}'(\xi +q_2)|}\le \frac{2C_4m}{\mu _3}|\phi _{11}'(\xi -q_1-\theta _{10}s)|\\&\cdot \frac{|u_2^*-\phi _{31}(\xi +q_2-s)|}{|u_2^*-\phi _{31}(\xi +q_2)|}\cdot \frac{|u_2^*-\phi _{31}(\xi +q_2)|}{|\phi _{31}'(\xi +q_2)|}\\\le & {} \frac{2C_4m}{\mu _3}C_0e^{-\eta _2(\xi -q_1-\theta _{10}s)}C\frac{1}{C_1}\le {\tilde{M}}_6e^{\eta _2q_1}. \end{aligned}$$

With the above estimates, we can get (3.19) from (3.31) and $\int _{{\mathbb {R}}}J_1(x) dx=1$, immediately. It completes the proof.