Appendix
In this appendix, we give the detailed proof of Lemmas 2.2 and 2.5.
Proof of Lemma 2.2
For any \(\epsilon \) with \(0<\epsilon <\min \left\{ \frac{1}{3},r^{(0)}(\infty ),\frac{c_{2,0}^*(\infty )-c^*_1(\infty )-\varepsilon ^{(0)}}{3}\right\} \), let \(\ell \) be a real number such that
$$\begin{aligned} c_{2,0}^*(\ell )=c_{2,0}^*(\infty )-\epsilon . \end{aligned}$$
We choose \(0<\mu _1<\mu _2<\mu _{2,0}^*(\ell )\) such that \(\psi _2(\mu _1)=c^*_1(\infty )+\varepsilon ^{(0)}+\epsilon \) and \(\psi _2(\mu _2)=c_{2,0}^*(\infty )-2\epsilon \). By Lemma 2.1, for any \(\mu \in [\mu _1,\mu _2]\) and sufficiently small \(\beta >0\) and \(\gamma >0\), \(\frac{\beta }{\varphi (\mu ;\sigma (\mu ))}\varphi (\mu ;x-\ell -\psi _2(\mu )t)\) with \(\varphi \) given by (2.7) is a continuous weak lower solution of (2.5).
Since \(\phi _2(x)\ge 0\) and \(\phi _2(x)\not \equiv 0\), \(u^{(0)}_2(t,x,\phi )>0\) for all \((t,x)\in (0,+\infty )\times {\mathbb {R}}\). Choose \(T_0<t_0<T_0+\frac{\sigma (\mu _1)}{\psi _2(\mu _1)}\), and choose sufficiently small \(\beta >0\) and \(\gamma >0\) such that
$$\begin{aligned} \frac{\pi /\gamma +\sigma (\mu _1)-\sigma (\mu _2)}{\psi _2(\mu _2)-\psi _2(\mu _1)}>T_0 \end{aligned}$$
(5.1)
and
$$\begin{aligned} u^{(0)}_2(t_0,x,\phi )\ge \beta ,\ \ \ \forall x\in [\ell +\psi _2(\mu _1)T_0,\ell +4\pi /\gamma +\psi _2(\mu _2)T_0]. \end{aligned}$$
(5.2)
Define
$$\begin{aligned} w(T_0,x)=\left\{ \begin{array}{lll} \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _2(\mu _1)T_0)\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +\psi _2(\mu _1)T_0\le x\le \ell +\sigma (\mu _1)+\psi _2(\mu _1)T_0;\\ \\ \beta \ \ \ \ \text{ if }\ \ell +\sigma (\mu _1)+\psi _2(\mu _1)T_0\le x\le \ell +3\pi /\gamma +\sigma (\mu _2)+\psi _2(\mu _2)T_0;\\ \\ \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma -\psi _2(\mu _2)T_0)\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +3\pi /\gamma +\sigma (\mu _2)+\psi _2(\mu _2)T_0\le x\le \ell +4\pi /\gamma +\psi _2(\mu _2)T_0;\\ \\ 0 \ \ \ \ \ \text{ elsewhere }. \end{array} \right. \end{aligned}$$
It is easily seen that
$$\begin{aligned} w(T_0,x)\ge \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _2(\mu _1)T_0-s),\ \ \forall s\in [0,2\pi /\gamma ] \end{aligned}$$
and
$$\begin{aligned} w(T_0,x)\ge \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma -\psi _2(\mu _2)T_0+s),\ \ \forall s\in [0,2\pi /\gamma ]. \end{aligned}$$
By (5.2) and Lemma 2.1, for any \(t\ge t_0\), we have
$$\begin{aligned}&u^{(0)}_2(t,x,\phi )\nonumber \\&\quad \ge \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _2(\mu _1)(T_0+t-t_0)-s),\ \ \forall s\in [0,2\pi /\gamma ] \end{aligned}$$
(5.3)
and
$$\begin{aligned}&u^{(0)}_2(t,x,\phi )\ge \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma \nonumber \\&\quad -\psi _2(\mu _2)(T_0+t-t_0)+s),\ \ \forall s\in [0,2\pi /\gamma ]. \end{aligned}$$
(5.4)
Inequalities (5.3) and (5.4) imply that for any \(t\ge t_0\),
$$\begin{aligned} u^{(0)}_2(t,x,\phi )\ge \left\{ \begin{array}{lll} \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _2(\mu _1)(T_0+t-t_0))\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +\psi _2(\mu _1)(T_0+t-t_0)\le x\le \ell +\sigma (\mu _1)+\psi _2(\mu _1)(T_0+t-t_0);\\ \\ \beta \ \ \ \ \text{ if }\ \ell +\sigma (\mu _1)+\psi _2(\mu _1)(T_0+t-t_0)\le x\le \ell +2\pi /\gamma \\ \\ \ \ \ \ \ \ \ \ \ +\sigma (\mu _1)+\psi _2(\mu _1)(T_0+t-t_0);\\ \\ 0 \ \ \ \ \ \text{ elsewhere } \end{array} \right. \end{aligned}$$
(5.5)
and
$$\begin{aligned} u^{(0)}_2(t,x,\phi )\ge \left\{ \begin{array}{lll} \beta \ \ \ \ \text{ if }\ \ell +\pi /\gamma +\sigma (\mu _2)+\psi _2(\mu _2)(T_0+t-t_0)\le x\le \ell +3\pi /\gamma \\ \\ \ \ \ \ \ \ \ \ \ +\sigma (\mu _2)+\psi _2(\mu _2)(T_0+t-t_0);\\ \\ \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma -\psi _2(\mu _2)(T_0+t-t_0))\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +3\pi /\gamma +\sigma (\mu _2)+\psi _2(\mu _2)(T_0+t-t_0)\le x\le \ell +4\pi /\gamma \\ \\ \ \ \ \ \ \ \ \ \ +\psi _2(\mu _2)(T_0+t-t_0);\\ \\ 0 \ \ \ \ \ \text{ elsewhere }. \end{array} \right. \end{aligned}$$
(5.6)
Let
$$\begin{aligned} h=\frac{\pi /\gamma +\sigma (\mu _1)-\sigma (\mu _2)}{\psi _2(\mu _2)-\psi _2(\mu _1)}-T_0. \end{aligned}$$
Then (5.1) implies that \(h>0\). Since
$$\begin{aligned}&\ell +2\pi /\gamma +\sigma (\mu _1)+\psi _2(\mu _1)(T_0+t-t_0)\\&\quad \ge \ell +\pi /\gamma +\sigma (\mu _2)+\psi _2(\mu _2)(T_0+t-t_0),\ \ \ \forall t\in [t_0, t_0+h], \end{aligned}$$
inequalities (5.5) and (5.6) imply that
$$\begin{aligned} u^{(0)}_2(t,x,\phi )\ge w(t-t_0^*,x),\ \ \ \forall t\in [t_0, t_0+h], \end{aligned}$$
(5.7)
where \(t_0^*=t_0-T_0\) and
$$\begin{aligned} w(t-t^*_0,x)=\left\{ \begin{array}{lll} \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _2(\mu _1)(t-t_0^*))\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +\psi _2(\mu _1)(t-t_0^*)\le x\le \ell +\sigma (\mu _1)+\psi _2(\mu _1)(t-t_0^*);\\ \\ \beta \ \ \ \ \text{ if }\ \ell +\sigma (\mu _1)+\psi _2(\mu _1)(t-t_0^*)\le x\le \ell +3\pi /\gamma +\sigma (\mu _2)+\psi _2(\mu _2)(t-t_0^*);\\ \\ \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma -\psi _2(\mu _2)(t-t_0^*))\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +3\pi /\gamma +\sigma (\mu _2)+\psi _2(\mu _2)(t-t_0^*)\le x\le \ell +4\pi /\gamma +\psi _2(\mu _2)(t-t_0^*);\\ \\ 0 \ \ \ \ \ \text{ elsewhere }. \end{array} \right. \nonumber \\ \end{aligned}$$
(5.8)
We claim that (5.7) holds true for all \(t\ge t_0\). Assume that (5.7) is valid for \(t\in [t_0, t_0+nh]\) for some positive integer n. Then for any \(s\in [0, 2\pi /\gamma +(\psi _2(\mu _2)-\psi _2(\mu _1))(T_0+nh)]\),
$$\begin{aligned} w(T_0+nh,x)\ge \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _2(\mu _1)(T_0+nh)-s) \end{aligned}$$
and
$$\begin{aligned} w(T_0+nh,x)\ge \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma -\psi _2(\mu _2)(T_0+nh)+s). \end{aligned}$$
Therefore, Lemma 2.1 implies that for any \(t\ge t_0+nh\) and \(s\in [0, 2\pi /\gamma +(\psi _2(\mu _2)-\psi _2(\mu _1))(T_0+nh)]\),
$$\begin{aligned} u^{(0)}_2(t,x,\phi )\ge \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _2(\mu _1)(T_0+nh)-\psi _2(\mu _1)(t-(t_0+nh))-s) \nonumber \\ \end{aligned}$$
(5.9)
and
$$\begin{aligned} u^{(0)}_2(t,x,\phi )\ge & {} \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma \nonumber \\&-\psi _2(\mu _2)(T_0+nh)-\psi _2(\mu _2)(t-(t_0+nh))+s). \end{aligned}$$
(5.10)
Inequalities (5.9) and (5.10) imply that for any \(t\ge t_0+nh\),
$$\begin{aligned} u^{(0)}_2(t,x,\phi )\ge \left\{ \begin{array}{lll} \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _2(\mu _1)(T_0+t-t_0))\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +\psi _2(\mu _1)(T_0+t-t_0)\le x\le \ell +\sigma (\mu _1)+\psi _2(\mu _1)(T_0+t-t_0);\\ \\ \beta \ \ \ \ \text{ if }\ \ell +\sigma (\mu _1)+\psi _2(\mu _1)(T_0+t-t_0)\le x\le \ell +2\pi /\gamma \\ \\ \ \ \ \ \ \ \ \ \ +\sigma (\mu _1)+\psi _2(\mu _1)(t-(t_0+nh))+\psi _2(\mu _2)(T_0+nh);\\ \\ 0 \ \ \ \ \ \text{ elsewhere } \end{array} \right. \nonumber \\ \end{aligned}$$
(5.11)
and
$$\begin{aligned} u^{(0)}_2(t,x,\phi )\ge \left\{ \begin{array}{lll} \beta \ \ \ \ \text{ if }\ \ell +\pi /\gamma +\sigma (\mu _2)+\psi _2(\mu _2)(t-(t_0+nh))+\psi _2(\mu _1)(T_0+nh)\\ \\ \ \ \ \ \ \ \ \ \ \le x\le \ell +3\pi /\gamma +\sigma (\mu _2)+\psi _2(\mu _2)(T_0+t-t_0);\\ \\ \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma -\psi _2(\mu _2)(T_0+t-t_0))\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +3\pi /\gamma +\sigma (\mu _2)+\psi _2(\mu _2)(T_0+t-t_0)\le x\le \ell +4\pi /\gamma \\ \\ \ \ \ \ \ \ \ \ \ +\psi _2(\mu _2)(T_0+t-t_0);\\ \\ 0 \ \ \ \ \ \text{ elsewhere }. \end{array} \right. \end{aligned}$$
(5.12)
Since
for all \(t\in [t_0+nh,t_0+(2n+1)h+2T_0]\), inequalities (5.11) and (5.12) imply that (5.7) holds true for all \(t\in [t_0+nh,t_0+(n+1)h]\). By induction, (5.7) is true for all \(t\ge t_0\).
For the chosen \(\epsilon >0\) and \(\beta >0\), there exists \(L>0\) such that
$$\begin{aligned} \int _{-L}^{L}\frac{1}{\sqrt{\pi }}e^{-x^2}dx\ge 1-\beta \epsilon . \end{aligned}$$
Therefore, for any \(s>0\), we have
$$\begin{aligned} \int _{-L\sqrt{4d_2s}}^{L\sqrt{4d_2s}}\frac{1}{\sqrt{4\pi d_2s}}e^{-\frac{x^2}{4d_2s}}dx\ge 1-\beta \epsilon . \end{aligned}$$
Let \(t_1>t_0\) be sufficiently large. Then, for \(t>t_1\), the solution \(u^{(0)}_2(t,x,\phi )\) of (2.5) satisfies the integral equation
$$\begin{aligned} \begin{aligned} u^{(0)}_2(t,x,\phi )=&\int _{-\infty }^{+\infty }k_2(t-t_1,x-y)u^{(0)}_2(t_1,y,\phi )dy\\&+\int _{t_1}^t\int _{-\infty }^{+\infty }k_2(t-s,x-y)u^{(0)}_2(s,y,\phi )\left[ \rho +R^{(0)}(s,y)-u^{(0)}_2(s,y,\phi )\right] dyds, \end{aligned} \end{aligned}$$
(5.13)
where \(\rho >3r(\infty )-r(-\infty )\) is a real number and
$$\begin{aligned} k_2(t,x)=\frac{1}{\sqrt{4\pi d_2t}}e^{-\rho t-\frac{x^2}{4d_2t}}. \end{aligned}$$
(5.14)
It follows from (5.7) and (5.13) that for any \(t>t_1\),
$$\begin{aligned} \begin{aligned} u^{(0)}_2(t,x,\phi )\ge&\int _{-\infty }^{+\infty }k_2(t-t_1,x-y)w(t_1-t_0^*,y)dy\\&+\int _{t_1}^t\int _{-\infty }^{+\infty }k_2(t-s,x-y)w(s-t_0^*,y)\left[ \rho +R^{(0)}(s,y)-w(s-t_0^*,y)\right] dyds. \end{aligned} \end{aligned}$$
(5.15)
For \(t>t_1\) and x, y satisfying
$$\begin{aligned} \begin{array}{lll} \ell +\sigma (\mu _1)+\psi _2(\mu _1)(t_1-t_0^*)+L\sqrt{4d_2(t-t_1)}\\ \\ \ \ \ \ \ \ \ \ \ \le x\le \ell +\sigma (\mu _2)+\psi _2(\mu _2)(t_1-t_0^*)+3\pi /\gamma -L\sqrt{4d_2(t-t_1)} \end{array} \end{aligned}$$
(5.16)
and
$$\begin{aligned} -L\sqrt{4d_2(t-t_1)}\le y\le L\sqrt{4d_2(t-t_1)}, \end{aligned}$$
(5.17)
we have that
$$\begin{aligned} \ell +\sigma (\mu _1)+\psi _2(\mu _1)(t_1-t_0^*)\le x-y\le \ell +\sigma (\mu _2)+\psi _2(\mu _2)(t_1-t_0^*)+3\pi /\gamma . \qquad \end{aligned}$$
(5.18)
It follows from (5.8) and (5.18) that
$$\begin{aligned} \begin{aligned}&\int _{-\infty }^{+\infty }k_2(t-t_1,x-y)w(t_1-t_0^*,y)dy\\&\quad =\int _{-\infty }^{+\infty }k_2(t-t_1,y)w(t_1-t_0^*,x-y)dy\\&\qquad \ge e^{-\rho (t-t_1)}\int _{-L\sqrt{4d_2(t-t_1)}}^{L\sqrt{4d_2(t-t_1)}} \frac{1}{\sqrt{4\pi d_2(t-t_1)}}e^{-\frac{y^2}{4d_2(t-t_1)}}w(t_1-t_0^*,x-y)dy\\&\quad = \beta e^{-\rho (t-t_1)}\int _{-L\sqrt{4d_2(t-t_1)}}^{L\sqrt{4d_2(t-t_1)}} \frac{1}{\sqrt{4\pi d_2(t-t_1)}}e^{-\frac{y^2}{4d_2(t-t_1)}}dy\\&\quad \ge (1-\beta \epsilon )\beta e^{-\rho (t-t_1)} \end{aligned} \end{aligned}$$
(5.19)
for all x satisfying (5.16). For \(t>t_1\) and x, y satisfying
$$\begin{aligned} \begin{array}{lll} \ell +\sigma (\mu _1)+\psi _2(\mu _1)(s-t_0^*)+L\sqrt{4d_2(t-s)}\\ \\ \ \ \ \ \ \ \ \ \ \le x\le \ell +\sigma (\mu _2)+\psi _2(\mu _2)(s-t_0^*)+3\pi /\gamma -L\sqrt{4d_2(t-s)},\ \ \forall s\in [t_1,t] \end{array} \end{aligned}$$
(5.20)
and
$$\begin{aligned} -L\sqrt{4d_2(t-s)}\le y\le L\sqrt{4d_2(t-s)},\ \ \forall s\in [t_1,t], \end{aligned}$$
(5.21)
we have
$$\begin{aligned}&\ell +\sigma (\mu _1)+\psi _2(\mu _1)(s-t_0^*)\nonumber \\&\quad \le x-y\le \ell +\sigma (\mu _2)+\psi _2(\mu _2)(s-t_0^*)+3\pi /\gamma ,\ \ \forall s\in [t_1,t] \end{aligned}$$
(5.22)
and
$$\begin{aligned} x-y-cs\ge \ell +\sigma (\mu _1)+\psi _2(\mu _1)(s-t_0^*)-cs>\ell +\sigma (\mu _1)+\psi _2(\mu _1)T_0>\ell ,\ \ \forall s\in [t_1,t]. \nonumber \\ \end{aligned}$$
(5.23)
Then it follows from (5.8), (5.22) and (5.23) that
$$\begin{aligned} \begin{aligned}&\int _{t_1}^t\int _{-\infty }^{+\infty }k_2(t-s,x-y)w(s-t_0^*,y)\left[ \rho +R^{(0)}(s,y)-w(s-t_0^*,y)\right] dyds\\&\quad =\int _{t_1}^t\int _{-\infty }^{+\infty }k_2(t-s,y)w(s-t_0^*,x-y)\left[ \rho +R^{(0)}(s,x-y)-w(s-t_0^*,x-y)\right] dyds\\&\quad \ge \int _{t_1}^te^{-\rho (t-s)}\int _{-L\sqrt{4d_2(t-s)}}^{L\sqrt{4d_2(t-s)}} \frac{1}{\sqrt{4\pi d_2(t-s)}}e^{-\frac{y^2}{4d_2(t-s)}}w(s-t_0^*,x-y)\\&\qquad \times \left[ \rho +R^{(0)}(s,x-y)-w(s-t_0^*,x-y)\right] dyds\\&\quad =\beta \int _{t_1}^te^{-\rho (t-s)}\int _{-L\sqrt{4d_2(t-s)}}^{L\sqrt{4d_2(t-s)}} \frac{1}{\sqrt{4\pi d_2(t-s)}}e^{-\frac{y^2}{4d_2(t-s)}}\left[ \rho +R^{(0)}(s,x-y)-\beta \right] dyds\\&\quad \ge \beta (\rho +r^{(0)}(\infty )-\mu _{2,0}^*(\infty )\epsilon -\beta )\int _{t_1}^te^{-\rho (t-s)}\int _{-L\sqrt{4d_2(t-s)}}^{L\sqrt{4d_2(t-s)}} \frac{1}{\sqrt{4\pi d_2(t-s)}}e^{-\frac{y^2}{4d_2(t-s)}}dyds\\&\quad \ge \beta (1-\beta \epsilon )(\rho +r^{(0)}(\infty )-\mu _{2,0}^*(\infty )\epsilon -\beta )\int _{t_1}^te^{-\rho (t-s)}ds \end{aligned} \end{aligned}$$
(5.24)
for all x satisfying (5.20). Here we have used the fact that for x satisfying (5.20) and y satisfying (5.21),
$$\begin{aligned} R^{(0)}(s,x-y)>r^{(0)}(\ell )>r^{(0)}(\infty )-\frac{\epsilon }{2d_2}c_{2,0}^*(\infty )=r^{(0)}(\infty )-\mu _{2,0}^*(\infty )\epsilon . \end{aligned}$$
By (5.15), (5.19) and (5.24), we obtain that
$$\begin{aligned} u^{(0)}_2(t,x,\phi )\ge {\hat{v}}_2^{(1)}(t) \end{aligned}$$
(5.25)
for all \(t>t_1\) and x satisfying (5.16) and (5.20), where
$$\begin{aligned} {\hat{v}}_2^{(1)}(t)= & {} \beta (1-\beta \epsilon )e^{-\rho (t-t_1)} +\beta (1-\beta \epsilon )(\rho +r^{(0)}(\infty )\nonumber \\&-\mu _{2,0}^*(\infty )\epsilon -\beta )\int _{t_1}^te^{-\rho (t-s)}ds. \end{aligned}$$
(5.26)
It then further follows from induction and (5.13) that
$$\begin{aligned} u^{(0)}_2(t,x,\phi )\ge {\hat{v}}_2^{(n)}(t) \end{aligned}$$
(5.27)
for all \(t>t_1\) and x satisfying (5.16) and
$$\begin{aligned} \begin{array}{lll} \ell +\sigma (\mu _1)+\psi _2(\mu _1)(s-t_0^*)+nL\sqrt{4d_2(t-s)}\\ \\ \ \ \ \ \ \ \ \ \ \le x\le \ell +\sigma (\mu _2)+\psi _2(\mu _2)(s-t_0^*)+3\pi /\gamma -nL\sqrt{4d_2(t-s)}, \ \forall s\in [t_1, t], \end{array} \end{aligned}$$
(5.28)
where
$$\begin{aligned} \begin{aligned} {\hat{v}}_2^{(n)}(t)=&\beta (1-\beta \epsilon )e^{-\rho (t-t_1)} +(1-\beta \epsilon )\int _{t_1}^te^{-\rho (t-s)}{\hat{v}}_2^{(n-1)}(s)\\&\times \left[ \rho +r^{(0)}(\infty )-\mu _{2,0}^*(\infty )\epsilon -{\hat{v}}_2^{(n-1)}(s)\right] ds. \end{aligned} \end{aligned}$$
(5.29)
Direct calculations and induction show that
$$\begin{aligned} {\hat{v}}_2^{(n)}(t)={\hat{a}}^{(n)}_2+{\hat{b}}^{(n)}_2(t)e^{-\rho (t-t_1)}, \end{aligned}$$
(5.30)
where
$$\begin{aligned} {\hat{a}}^{(n)}_2= & {} {\hat{a}}^{(n-1)}_2(1-\beta \epsilon )(\rho +r^{(0)}(\infty )-\mu _{2,0}^*(\infty )\epsilon -{\hat{a}}^{(n-1)}_2)/\rho , \end{aligned}$$
(5.31)
$$\begin{aligned} {\hat{a}}^{(1)}_2= & {} \beta (1-\beta \epsilon )(\rho +r^{(0)}(\infty )-\mu _{2,0}^*(\infty )\epsilon -\beta )/\rho \end{aligned}$$
(5.32)
and \({\hat{b}}^{(n)}_2(t)\) is a sum of products of polynomials and exponential functions of the form \(e^{-j\rho (t-t_1)}\) with j being a non-negative integer. Therefore,
$$\begin{aligned} \lim _{t\rightarrow \infty }{\hat{v}}_2^{(n)}(t)={\hat{a}}_2^{(n)} \end{aligned}$$
(5.33)
and \({\hat{a}}_2^{(n)}\le r(\infty )\) for all \(n\ge 1\). Let \({\hat{a}}_2^{(0)}=\beta \). Then for small \(\epsilon \) and \(\beta \), we have
$$\begin{aligned} {\hat{a}}_2^{(1)}-{\hat{a}}_2^{(0)}=(r^{(0)}(\infty )-\mu _{2,0}^*(\infty )\epsilon -\beta -\beta ^2\rho \epsilon )/\rho >0. \end{aligned}$$
(5.34)
It follows from (5.34) and induction that
$$\begin{aligned} \begin{aligned}&{\hat{a}}_2^{(n+1)}-{\hat{a}}_2^{(n)}\\&\quad =\left[ {\hat{a}}^{(n)}_2(\rho +r^{(0)}(\infty )-\mu _{2,0}^*(\infty )\epsilon -{\hat{a}}^{(n)}_2)\right. \\&\qquad \left. -{\hat{a}}^{(n-1)}_2(\rho +r^{(0)}(\infty )-\mu _{2,0}^*(\infty )\epsilon -{\hat{a}}^{(n-1)}_2)\right] \frac{1-\beta \epsilon }{\rho }\\&\quad =({\hat{a}}_2^{(n)}-{\hat{a}}_2^{(n-1)})\left[ \rho +r^{(0)}(\infty )-\mu _{2,0}^*(\infty )\epsilon -{\hat{a}}^{(n-1)}_2-{\hat{a}}^{(n)}_2\right] \frac{1-\beta \delta }{\rho }\\&\quad >0,\ \ \ \forall n\ge 1. \end{aligned} \end{aligned}$$
(5.35)
Thus, \(\left\{ {\hat{a}}_2^{(n)}\right\} _{n=0}^{\infty }\) is increasing and \(\beta <{\hat{a}}_2^{(n)}\le r(\infty )\) for all \(n\ge 1\). So, \(\lim _{n\rightarrow \infty }{\hat{a}}_2^{(n)}\) exists. Let
$$\begin{aligned} \lim _{n\rightarrow \infty }{\hat{a}}_2^{(n)}={\hat{a}}_2^*. \end{aligned}$$
(5.36)
Then \(\beta \le {\hat{a}}_2^*\le r(\infty )\) and by (5.31), we obtain that
$$\begin{aligned} {\hat{a}}^*_2={\hat{a}}^*_2(1-\beta \epsilon )(\rho +r^{(0)}(\infty )-\mu _{2,0}^*(\infty )\epsilon -{\hat{a}}^*_2)/\rho . \end{aligned}$$
(5.37)
Therefore, it follows from (5.37) that
$$\begin{aligned} {\hat{a}}^*_2=r^{(0)}(\infty )-\left( \mu _{2,0}^*(\infty )+\frac{\beta \rho }{1-\beta \epsilon }\right) \epsilon . \end{aligned}$$
(5.38)
Thus, by (5.30), (5.33), (5.36) and (5.38), we obtain that there exist a positive integer N and \(t_2>t_1\) such that
$$\begin{aligned} {\hat{v}}^{(n)}_2(t)>r^{(0)}(\infty )-\left( 1+\mu _{2,0}^*(\infty )+\frac{\beta \rho }{1-\beta \epsilon }\right) \epsilon ,\ \ \forall t>t_2,\ n\ge N. \end{aligned}$$
(5.39)
Clearly, if
$$\begin{aligned} \begin{array}{lll} \ell +\sigma (\mu _1)+\psi _2(\mu _1)(t-t_0^*)+NL\sqrt{4d_2(t-t_1)}\\ \\ \ \ \ \ \ \ \ \ \ \le x\le \ell +\sigma (\mu _2)+\psi _2(\mu _2)(t_1-t_0^*)+3\pi /\gamma -NL\sqrt{4d_2(t-t_1)}, \end{array} \end{aligned}$$
(5.40)
then (5.16) holds and (5.28) with n replaced by N also holds. Choose \(t_1=ml+t_0^*\) and \(t-t_1=l\), where \(m>1\) and \(l>0\) are both sufficiently large. Then we can rewrite (5.40) as
$$\begin{aligned} \begin{array}{lll} \ell +\sigma (\mu _1)+\psi _2(\mu _1)l(m+1)+NL\sqrt{4d_2l}\\ \\ \ \ \ \ \ \ \ \ \ \le x\le \ell +\sigma (\mu _2)+ml\psi _2(\mu _2)+3\pi /\gamma -NL\sqrt{4d_2l}, \end{array} \end{aligned}$$
(5.41)
that is,
$$\begin{aligned} \begin{array}{lll} (t_0^*+l(m+1))\left[ \psi _2(\mu _1)+\frac{\ell +\sigma (\mu _1)}{l(m+1)}+\frac{NL\sqrt{4d_2}}{(m+1)\sqrt{l}}\right] \frac{l(m+1)}{t_0^*+l(m+1)}\\ \\ \ \ \ \ \ \ \ \ \ \le x\le (t_0^*+l(m+1))\left[ \frac{m}{m+1}\psi _2(\mu _2)+\frac{\ell +\sigma (\mu _2)+3\pi /\gamma }{l(m+1)}-\frac{NL\sqrt{4d_2}}{(m+1)\sqrt{l}}\right] \frac{l(m+1)}{t_0^*+l(m+1)}. \end{array} \end{aligned}$$
(5.42)
Now for any given \(\varepsilon \) with \(0<\varepsilon <(c_{2,0}^*(\infty )-c_1^*(\infty )-\varepsilon ^{(0)})/2\), choose \(\epsilon \) sufficiently small such that \(\epsilon <\varepsilon /3\). Then there exist \(l_0\) and \(m_0\) sufficiently large such that for any \(m>m_0,\,\,l>l_0\) and \(t=t_0^*+l(m+1)>t_2\), we have
$$\begin{aligned}&(t_0^*+l(m+1))\left[ \psi _2(\mu _1)+\frac{\ell +\sigma (\mu _1)}{l(m+1)}+\frac{NL\sqrt{4d_2}}{(m+1)\sqrt{l}}\right] \frac{l(m+1)}{t_0^*+l(m+1)}\\&\quad<t(\psi _2(\mu _1)+\epsilon )=t(c_1^*(\infty )+\varepsilon ^{(0)}+\epsilon +\epsilon )\\&\quad <t(c_1^*(\infty )+\varepsilon ^{(0)}+\varepsilon ) \end{aligned}$$
and
$$\begin{aligned}&(t_0^*+l(m+1))\left[ \frac{m}{m+1}\psi _2(\mu _2)+\frac{\ell +\sigma (\mu _2)+3\pi /\gamma }{l(m+1)}-\frac{NL\sqrt{4d_2}}{(m+1)\sqrt{l}}\right] \frac{l(m+1)}{t_0^*+l(m+1)}\\&\quad>t(\psi _2(\mu _2)-\epsilon )=t(c_{2,0}^*(\infty )-2\epsilon -\epsilon )\\&\quad >t(c_{2,0}^*(\infty )-\varepsilon ). \end{aligned}$$
Let \(t_3=t_0^*+l_0(m_0+1)\). If \(t>t_3\), then \(t(c_1^*(\infty )+\varepsilon ^{(0)}+\varepsilon )\le x\le t(c_{2,0}^*(\infty )-\varepsilon )\) implies that (5.40) holds. Thus, by (5.27) and (5.39), we obtain that
$$\begin{aligned}&\lim _{t\rightarrow \infty }\left[ \inf _{t(c_1^*(\infty )+\varepsilon ^{(0)}+\varepsilon )\le x\le t(c_{2,0}^*(\infty )-\varepsilon )}u_2^{(0)}(t,x,\phi )\right] \nonumber \\&\quad \ge r^{(0)}(\infty )-\left( 1+\mu _{2,0}^*(\infty )+\frac{\beta \rho }{1-\beta \epsilon }\right) \epsilon . \end{aligned}$$
(5.43)
Because \(\epsilon \) can be arbitrarily small and (5.43), we have actually shown that
$$\begin{aligned} \lim _{t\rightarrow \infty }\left[ \inf _{t(c_1^*(\infty )+\varepsilon ^{(0)}+\varepsilon )\le x\le t(c_{2,0}^*(\infty )-\varepsilon )}u_2^{(0)}(t,x,\phi )\right] \ge r^{(0)}(\infty ). \end{aligned}$$
(5.44)
Since \(\varepsilon ^{(0)}>0\) is arbitrary, we have actually shown that for every \(\varepsilon \) with \(0<\varepsilon <(c_2^*(\infty )-c_1^*(\infty ))/2\),
$$\begin{aligned} \lim _{t\rightarrow \infty }\left[ \inf _{t(c_1^*(\infty )+\varepsilon )\le x\le t(c_{2}^*(\infty )-\varepsilon )}u_2^{(0)}(t,x,\phi )\right] \ge r(\infty ). \end{aligned}$$
(5.45)
It follows from \(u_2^{(0)}(t,x,\phi )\le r(\infty )\) for all \((t,x)\in [0,+\infty )\times {\mathbb {R}}\) that
$$\begin{aligned} \lim _{t\rightarrow \infty }\left[ \sup _{t(c_1^*(\infty )+\varepsilon )\le x\le t(c_{2}^*(\infty )-\varepsilon )}\left| r(\infty )-u_2^{(0)}(t,x,\phi )\right| \right] =0. \end{aligned}$$
(5.46)
The proof of Lemma 2.2 is completed. \(\square \)
Proof of Lemma 2.5
For any \(\epsilon \) with \(0<\epsilon <\min \left\{ \frac{1}{3},r^{(1)}(\infty ),\frac{c_{1,1}^*(\infty )-c-\varepsilon ^{(1)}}{3}\right\} \), let \(\ell \) be a real number such that
$$\begin{aligned} c_{1,1}^*(\ell )=c_{1,1}^*(\infty )-\epsilon . \end{aligned}$$
We choose \(0<\mu _1<\mu _2<\mu _{1,1}^*(\ell )\) such that \(\psi _1(\mu _1)=c+\varepsilon ^{(1)}+\epsilon \) and \(\psi _1(\mu _2)=c_{1,1}^*(\infty )-2\epsilon \). By Lemma 2.4, for any \(\mu \in [\mu _1,\mu _2]\) and sufficiently small \(\beta >0\) and \(\gamma >0\), \(\frac{\beta }{\varphi (\mu ;\sigma (\mu ))}\varphi (\mu ;x-\ell -\psi _1(\mu )t)\) with \(\varphi \) given by (2.7) is a continuous weak lower solution of (2.34).
Since \(\phi _1(x)\ge 0\) and \(\phi _1(x)\not \equiv 0\), \(u^{(1)}_1(t,x,\phi )>0\) for all \((t,x)\in (0,+\infty )\times {\mathbb {R}}\). Choose \(T_1<t_0<T_1+\frac{\sigma (\mu _1)}{\psi _1(\mu _1)}\), and choose sufficiently small \(\beta >0\) and \(\gamma >0\) such that
$$\begin{aligned} \frac{\pi /\gamma +\sigma (\mu _1)-\sigma (\mu _2)}{\psi _1(\mu _2)-\psi _1(\mu _1)}>T_1 \end{aligned}$$
(5.47)
and
$$\begin{aligned} u^{(1)}_1(t_0,x,\phi )\ge \beta ,\ \ \ \forall x\in [\ell +\psi _1(\mu _1)T_1,\ell +4\pi /\gamma +\psi _1(\mu _2)T_1]. \end{aligned}$$
(5.48)
Define
$$\begin{aligned} w(T_1,x)=\left\{ \begin{array}{lll} \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _1(\mu _1)T_1)\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +\psi _1(\mu _1)T_1\le x\le \ell +\sigma (\mu _1)+\psi _1(\mu _1)T_1;\\ \\ \beta \ \ \ \ \text{ if }\ \ell +\sigma (\mu _1)+\psi _1(\mu _1)T_1\le x\le \ell +3\pi /\gamma +\sigma (\mu _2)+\psi _1(\mu _2)T_1;\\ \\ \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma -\psi _1(\mu _2)T_1)\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +3\pi /\gamma +\sigma (\mu _2)+\psi _1(\mu _2)T_1\le x\le \ell +4\pi /\gamma +\psi _1(\mu _2)T_1;\\ \\ 0 \ \ \ \ \ \text{ elsewhere }. \end{array} \right. \end{aligned}$$
It is easily seen that
$$\begin{aligned} w(T_1,x)\ge \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _1(\mu _1)T_1-s),\ \ \forall s\in [0,2\pi /\gamma ] \end{aligned}$$
and
$$\begin{aligned} w(T_1,x)\ge \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma -\psi _1(\mu _2)T_1+s),\ \ \forall s\in [0,2\pi /\gamma ]. \end{aligned}$$
It follows from (5.48) and Lemma 2.4 that for any \(t\ge t_0\),
$$\begin{aligned}&u^{(1)}_1(t,x,\phi )\nonumber \\&\quad \ge \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _1(\mu _1)(T_1+t-t_0)-s),\ \ \forall s\in [0,2\pi /\gamma ] \quad \qquad \qquad \end{aligned}$$
(5.49)
and
$$\begin{aligned}&u^{(1)}_1(t,x,\phi )\ge \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma \nonumber \\&\quad -\psi _1(\mu _2)(T_1+t-t_0)+s),\ \ \forall s\in [0,2\pi /\gamma ]. \end{aligned}$$
(5.50)
Inequalities (5.49) and (5.50) imply that for any \(t\ge t_0\),
$$\begin{aligned} u^{(1)}_1(t,x,\phi )\ge \left\{ \begin{array}{lll} \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _1(\mu _1)(T_1+t-t_0))\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +\psi _1(\mu _1)(T_1+t-t_0)\le x\le \ell +\sigma (\mu _1)+\psi _1(\mu _1)(T_1+t-t_0);\\ \\ \beta \ \ \ \ \text{ if }\ \ell +\sigma (\mu _1)+\psi _1(\mu _1)(T_1+t-t_0)\le x\le \ell +2\pi /\gamma \\ \\ \ \ \ \ \ \ \ \ \ +\sigma (\mu _1)+\psi _1(\mu _1)(T_1+t-t_0);\\ \\ 0 \ \ \ \ \ \text{ elsewhere } \end{array} \right. \end{aligned}$$
(5.51)
and
$$\begin{aligned} u^{(1)}_1(t,x,\phi )\ge \left\{ \begin{array}{lll} \beta \ \ \ \ \text{ if }\ \ell +\pi /\gamma +\sigma (\mu _2)+\psi _1(\mu _2)(T_1+t-t_0)\le x\le \ell +3\pi /\gamma \\ \ \ \ \ \ \ \ \ \ +\sigma (\mu _2)+\psi _1(\mu _2)(T_1+t-t_0);\\ \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma -\psi _1(\mu _2)(T_1+t-t_0))\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +3\pi /\gamma +\sigma (\mu _2)+\psi _1(\mu _2)(T_1+t-t_0)\le x\le \ell +4\pi /\gamma \\ \ \ \ \ \ \ \ \ \ +\psi _1(\mu _2)(T_1+t-t_0);\\ 0 \ \ \ \ \ \text{ elsewhere }. \end{array} \right. \end{aligned}$$
(5.52)
Let
$$\begin{aligned} h=\frac{\pi /\gamma +\sigma (\mu _1)-\sigma (\mu _2)}{\psi _1(\mu _2)-\psi _1(\mu _1)}-T_1. \end{aligned}$$
Then (5.47) implies that \(h>0\). Since
$$\begin{aligned} \ell +2\pi /\gamma +\sigma (\mu _1)+\psi _1(\mu _1)(T_1+t-t_0)\ge \ell +\pi /\gamma +\sigma (\mu _2)+\psi _1(\mu _2)(T_1+t-t_0) \end{aligned}$$
for all \(t\in [t_0, t_0+h]\), inequalities (5.51) and (5.52) imply that
$$\begin{aligned} u^{(1)}_1(t,x,\phi )\ge w(t-t_0^*,x) \end{aligned}$$
(5.53)
for all \(t\in [t_0, t_0+h]\), where \(t_0^*=t_0-T_1\) and
$$\begin{aligned} w(t-t^*_0,x)=\left\{ \begin{array}{lll} \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _1(\mu _1)(t-t_0^*))\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +\psi _1(\mu _1)(t-t_0^*)\le x\le \ell +\sigma (\mu _1)+\psi _1(\mu _1)(t-t_0^*);\\ \\ \beta \ \ \ \ \text{ if }\ \ell +\sigma (\mu _1)+\psi _1(\mu _1)(t-t_0^*)\le x\le \ell +3\pi /\gamma +\sigma (\mu _2)+\psi _1(\mu _2)(t-t_0^*);\\ \\ \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma -\psi _1(\mu _2)(t-t_0^*))\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +3\pi /\gamma +\sigma (\mu _2)+\psi _1(\mu _2)(t-t_0^*)\le x\le \ell +4\pi /\gamma +\psi _1(\mu _2)(t-t_0^*);\\ \\ 0 \ \ \ \ \ \text{ elsewhere }. \end{array} \right. \nonumber \\ \end{aligned}$$
(5.54)
We claim that (5.53) holds true for all \(t\ge t_0\). Assume that (5.53) is valid for \(t\in [t_0, t_0+nh]\) for some positive integer n. Then for any \(s\in [0, 2\pi /\gamma +(\psi _1(\mu _2)-\psi _1(\mu _1))(T_1+nh)]\),
$$\begin{aligned} w(T_1+nh,x)\ge \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _1(\mu _1)(T_1+nh)-s) \end{aligned}$$
and
$$\begin{aligned} w(T_1+nh,x)\ge \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma -\psi _1(\mu _2)(T_1+nh)+s). \end{aligned}$$
Therefore, Lemma 2.4 implies that
$$\begin{aligned} u^{(1)}_1(t,x,\phi )\ge \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _1(\mu _1)(T_1+nh)-\psi _1(\mu _1)(t-(t_0+nh))-s) \nonumber \\ \end{aligned}$$
(5.55)
and
$$\begin{aligned}&u^{(1)}_1(t,x,\phi )\ge \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma \nonumber \\&\quad -\psi _1(\mu _2)(T_1+nh)-\psi _1(\mu _2)(t-(t_0+nh))+s) \end{aligned}$$
(5.56)
for all \(t\ge t_0+nh\) and \(s\in [0, 2\pi /\gamma +(\psi _1(\mu _2)-\psi _1(\mu _1))(T_1+nh)]\). Inequalities (5.55) and (5.56) imply that for \(t\ge t_0+nh\),
$$\begin{aligned} u^{(1)}_1(t,x,\phi )\ge \left\{ \begin{array}{lll} \frac{\beta }{\varphi (\mu _1; \sigma (\mu _1))}\varphi (\mu _1; x-\ell -\psi _1(\mu _1)(T_1+t-t_0))\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +\psi _1(\mu _1)(T_1+t-t_0)\le x\le \ell +\sigma (\mu _1)+\psi _1(\mu _1)(T_1+t-t_0);\\ \\ \beta \ \ \ \ \text{ if }\ \ell +\sigma (\mu _1)+\psi _1(\mu _1)(T_1+t-t_0)\le x\le \ell +2\pi /\gamma \\ \\ \ \ \ \ \ \ \ \ \ +\sigma (\mu _1)+\psi _1(\mu _1)(t-(t_0+nh))+\psi _1(\mu _2)(T_1+nh);\\ \\ 0 \ \ \ \ \ \text{ elsewhere } \end{array} \right. \end{aligned}$$
(5.57)
and
$$\begin{aligned} u^{(1)}_1(t,x,\phi )\ge \left\{ \begin{array}{lll} \beta \ \ \ \ \text{ if }\ \ell +\pi /\gamma +\sigma (\mu _2)+\psi _1(\mu _2)(t-(t_0+nh))+\psi _1(\mu _1)(T_1+nh)\\ \\ \ \ \ \ \ \ \ \ \ \le x\le \ell +3\pi /\gamma +\sigma (\mu _2)+\psi _1(\mu _2)(T_1+t-t_0);\\ \\ \frac{\beta }{\varphi (\mu _2; \sigma (\mu _2))}\varphi (\mu _2; x-\ell -3\pi /\gamma -\psi _1(\mu _2)(T_1+t-t_0))\\ \\ \ \ \ \ \ \ \text{ if }\ \ell +3\pi /\gamma +\sigma (\mu _2)+\psi _1(\mu _2)(T_1+t-t_0)\le x\le \ell +4\pi /\gamma \\ \\ \ \ \ \ \ \ \ \ \ +\psi _1(\mu _2)(T_1+t-t_0);\\ \\ 0 \ \ \ \ \ \text{ elsewhere }. \end{array} \right. \end{aligned}$$
(5.58)
Since
$$\begin{aligned} \begin{aligned}&\ell +2\pi /\gamma +\sigma (\mu _1)+\psi _1(\mu _1)(t-(t_0+nh))+\psi _1(\mu _2)(T_1+nh)\\&\ge \ell +\pi /\gamma +\sigma (\mu _2)+\psi _1(\mu _2)(t-(t_0+nh))+\psi _1(\mu _1)(T_1+nh) \end{aligned} \end{aligned}$$
for all \(t\in [t_0+nh,t_0+(2n+1)h+2T_1]\), inequalities (5.57) and (5.58) imply that (5.53) holds true for all \(t\in [t_0+nh,t_0+(n+1)h]\). By induction, (5.53) is true for all \(t\ge t_0\).
For the chosen \(\epsilon >0\) and \(\beta >0\), there exists \(L>0\) such that
$$\begin{aligned} \int _{-L}^{L}\frac{1}{\sqrt{\pi }}e^{-x^2}dx\ge 1-\beta \epsilon . \end{aligned}$$
Therefore, for any \(s>0\), we have
$$\begin{aligned} \int _{-L\sqrt{4d_1s}}^{L\sqrt{4d_1s}}\frac{1}{\sqrt{4\pi d_1s}}e^{-\frac{x^2}{4d_1s}}dx\ge 1-\beta \epsilon . \end{aligned}$$
Let \(t_1>t_0\) be sufficiently large. Then, for \(t>t_1\), the solution \(u^{(1)}_1(t,x,\phi )\) of (2.34) satisfies the integral equation
$$\begin{aligned} u^{(1)}_1(t,x,\phi )= & {} \int _{-\infty }^{+\infty }k_1(t-t_1,x-y)u^{(1)}_1(t_1,y,\phi )dy\nonumber \\&+\int _{t_1}^t\int _{-\infty }^{+\infty }k_1(t-s,x-y)u^{(1)}_1(s,y,\phi )\nonumber \\&\left[ \rho +R^{(1)}(s,y)-u^{(1)}_1(s,y,\phi )\right] dyds, \end{aligned}$$
(5.59)
where \(\rho >3r(\infty )-r(-\infty )\) is a real number and
$$\begin{aligned} k_1(t,x)=\frac{1}{\sqrt{4\pi d_1t}}e^{-\rho t-\frac{x^2}{4d_1t}}. \end{aligned}$$
(5.60)
It follows from (5.53) and (5.59) that for any \(t>t_1\),
$$\begin{aligned} \begin{aligned} u^{(1)}_1(t,x,\phi )\ge&\int _{-\infty }^{+\infty }k_1(t-t_1,x-y)w(t_1-t_0^*,y)dy\\&+\int _{t_1}^t\int _{-\infty }^{+\infty }k_1(t-s,x-y)w(s-t_0^*,y)\\&\quad \left[ \rho +R^{(1)}(s,y)-w(s-t_0^*,y)\right] dyds. \end{aligned} \end{aligned}$$
(5.61)
For \(t>t_1\) and x, y satisfying
$$\begin{aligned} \begin{array}{lll} \ell +\sigma (\mu _1)+\psi _1(\mu _1)(t_1-t_0^*)+L\sqrt{4d_1(t-t_1)}\\ \\ \ \ \ \ \ \ \ \ \ \le x\le \ell +\sigma (\mu _2)+\psi _1(\mu _2)(t_1-t_0^*)+3\pi /\gamma -L\sqrt{4d_1(t-t_1)} \end{array} \end{aligned}$$
(5.62)
and
$$\begin{aligned} -L\sqrt{4d_1(t-t_1)}\le y\le L\sqrt{4d_1(t-t_1)}, \end{aligned}$$
(5.63)
we have that
$$\begin{aligned} \ell +\sigma (\mu _1)+\psi _1(\mu _1)(t_1-t_0^*)\le x-y\le \ell +\sigma (\mu _2)+\psi _1(\mu _2)(t_1-t_0^*)+3\pi /\gamma . \end{aligned}$$
(5.64)
It follows from (5.54) and (5.64) that
$$\begin{aligned} \begin{aligned}&\int _{-\infty }^{+\infty }k_1(t-t_1,x-y)w(t_1-t_0^*,y)dy\\&\quad =\int _{-\infty }^{+\infty }k_1(t-t_1,y)w(t_1-t_0^*,x-y)dy\\&\quad \ge e^{-\rho (t-t_1)}\int _{-L\sqrt{4d_1(t-t_1)}}^{L\sqrt{4d_1(t-t_1)}} \frac{1}{\sqrt{4\pi d_1(t-t_1)}}e^{-\frac{y^2}{4d_1(t-t_1)}}w(t_1-t_0^*,x-y)dy\\&\quad = \beta e^{-\rho (t-t_1)}\int _{-L\sqrt{4d_1(t-t_1)}}^{L\sqrt{4d_1(t-t_1)}} \frac{1}{\sqrt{4\pi d_1(t-t_1)}}e^{-\frac{y^2}{4d_1(t-t_1)}}dy\\&\quad \ge (1-\beta \epsilon )\beta e^{-\rho (t-t_1)} \end{aligned} \end{aligned}$$
(5.65)
for all x satisfying (5.62). For \(t>t_1\) and x, y satisfying
$$\begin{aligned} \begin{array}{lll} \ell +\sigma (\mu _1)+\psi _1(\mu _1)(s-t_0^*)+L\sqrt{4d_1(t-s)}\\ \\ \ \ \ \ \ \ \ \ \ \le x\le \ell +\sigma (\mu _2)+\psi _1(\mu _2)(s-t_0^*)+3\pi /\gamma -L\sqrt{4d_1(t-s)},\ \ \ \forall s\in [t_1,t] \end{array} \end{aligned}$$
(5.66)
and
$$\begin{aligned} -L\sqrt{4d_1(t-s)} \le y\le L\sqrt{4d_1(t-s)},\ \ \ \forall s\in [t_1,t], \end{aligned}$$
(5.67)
we have
$$\begin{aligned}&\ell +\sigma (\mu _1)+\psi _1(\mu _1)(s-t_0^*)\nonumber \\&\quad \le x-y\le \ell +\sigma (\mu _2)+\psi _1(\mu _2)(s-t_0^*)+3\pi /\gamma ,\ \ \ \forall s\in [t_1,t] \end{aligned}$$
(5.68)
and
$$\begin{aligned}&x-y-cs\ge \ell +\sigma (\mu _1)+\psi _1(\mu _1)(s-t_0^*)-cs>\ell \nonumber \\&\quad +\sigma (\mu _1)+\psi _1(\mu _1)T_1>\ell ,\ \ \ \forall s\in [t_1,t]. \end{aligned}$$
(5.69)
Then it follows from (5.54), (5.68) and (5.69) that
$$\begin{aligned} \begin{aligned}&\int _{t_1}^t\int _{-\infty }^{+\infty }k_1(t-s,x-y)w(s-t_0^*,y)\left[ \rho +R^{(1)}(s,y)-w(s-t_0^*,y)\right] dyds\\&\quad =\int _{t_1}^t\int _{-\infty }^{+\infty }k_1(t-s,y)w(s-t_0^*,x-y)\left[ \rho +R^{(1)}(s,x-y)-w(s-t_0^*,x-y)\right] dyds\\&\quad \ge \int _{t_1}^te^{-\rho (t-s)}\int _{-L\sqrt{4d_1(t-s)}}^{L\sqrt{4d_1(t-s)}} \frac{1}{\sqrt{4\pi d_1(t-s)}}e^{-\frac{y^2}{4d_1(t-s)}}w(s-t_0^*,x-y)\\&\qquad \times \left[ \rho +R^{(1)}(s,x-y)-w(s-t_0^*,x-y)\right] dyds\\&\quad = \beta \int _{t_1}^te^{-\rho (t-s)}\int _{-L\sqrt{4d_1(t-s)}}^{L\sqrt{4d_1(t-s)}} \frac{1}{\sqrt{4\pi d_1(t-s)}}e^{-\frac{y^2}{4d_1(t-s)}}\left[ \rho +R^{(1)}(s,x-y)-\beta \right] dyds\\&\quad \ge \beta (\rho +r^{(1)}(\infty )-\mu _{1,1}^*(\infty )\epsilon -\beta )\int _{t_1}^te^{-\rho (t-s)}\int _{-L\sqrt{4d_1(t-s)}}^{L\sqrt{4d_1(t-s)}} \frac{1}{\sqrt{4\pi d_1(t-s)}}e^{-\frac{y^2}{4d_1(t-s)}}dyds\\&\quad \ge \beta (1-\beta \epsilon )(\rho +r^{(1)}(\infty )-\mu _{1,1}^*(\infty )\epsilon -\beta )\int _{t_1}^te^{-\rho (t-s)}ds \end{aligned} \end{aligned}$$
(5.70)
for all x satisfying (5.66). Here we have used the fact that for x satisfying (5.66) and y satisfying (5.67),
$$\begin{aligned} R^{(1)}(s,x-y)>r^{(1)}(\ell )>r^{(1)}(\infty )-\frac{\epsilon }{2d_1}c_{1,1}^*(\infty )=r^{(1)}(\infty )-\mu _{1,1}^*(\infty )\epsilon . \end{aligned}$$
By (5.61), (5.65) and (5.70), we obtain that
$$\begin{aligned} u^{(1)}_1(t,x,\phi )\ge {\hat{v}}_1^{(1)}(t) \end{aligned}$$
(5.71)
for all \(t>t_1\) and x satisfying (5.62) and (5.66), where
$$\begin{aligned}&{\hat{v}}_1^{(1)}(t)=\beta (1-\beta \epsilon )e^{-\rho (t-t_1)} +\beta (1-\beta \epsilon )(\rho +r^{(1)}(\infty )\nonumber \\&\quad -\mu _{1,1}^*(\infty )\epsilon -\beta )\int _{t_1}^te^{-\rho (t-s)}ds. \end{aligned}$$
(5.72)
It then further follows from induction and (5.59) that
$$\begin{aligned} u^{(1)}_1(t,x,\phi )\ge {\hat{v}}_1^{(n)}(t) \end{aligned}$$
(5.73)
for all \(t>t_1\) and x satisfying (5.62) and
$$\begin{aligned} \begin{array}{lll} \ell +\sigma (\mu _1)+\psi _1(\mu _1)(s-t_0^*)+nL\sqrt{4d_1(t-s)}\\ \\ \ \ \ \ \ \ \ \ \ \le x\le \ell +\sigma (\mu _2)+\psi _1(\mu _2)(s-t_0^*)+3\pi /\gamma -nL\sqrt{4d_1(t-s)}, \ \forall s\in [t_1, t], \end{array} \end{aligned}$$
(5.74)
where
$$\begin{aligned} \begin{aligned} {\hat{v}}_1^{(n)}(t)=&\beta (1-\beta \epsilon )e^{-\rho (t-t_1)} +(1-\beta \epsilon )\int _{t_1}^te^{-\rho (t-s)}{\hat{v}}_1^{(n-1)}(s)\\&\times \left[ \rho +r^{(1)}(\infty )-\mu _{1,1}^*(\infty )\epsilon -{\hat{v}}_1^{(n-1)}(s)\right] ds. \end{aligned} \end{aligned}$$
(5.75)
Direct calculations and induction show that
$$\begin{aligned} {\hat{v}}_1^{(n)}(t)={\hat{a}}^{(n)}_1+{\hat{b}}^{(n)}_1(t)e^{-\rho (t-t_1)}, \end{aligned}$$
(5.76)
where
$$\begin{aligned} {\hat{a}}^{(n)}_1= & {} {\hat{a}}^{(n-1)}_1(1-\beta \epsilon )(\rho +r^{(1)}(\infty )-\mu _{1,1}^*(\infty )\epsilon -{\hat{a}}^{(n-1)}_1)/\rho , \end{aligned}$$
(5.77)
$$\begin{aligned} {\hat{a}}^{(1)}_1= & {} \beta (1-\beta \epsilon )(\rho +r^{(1)}(\infty )-\mu _{1,1}^*(\infty )\epsilon -\beta )/\rho \end{aligned}$$
(5.78)
and \({\hat{b}}^{(n)}_1(t)\) is a sum of products of polynomials and exponential functions of the form \(e^{-j\rho (t-t_1)}\) with j being a non-negative integer. Therefore,
$$\begin{aligned} \lim _{t\rightarrow \infty }{\hat{v}}_1^{(n)}(t)={\hat{a}}_1^{(n)} \end{aligned}$$
(5.79)
and \({\hat{a}}_1^{(n)}\le r(\infty )\) for all \(n\ge 1\). Let \({\hat{a}}_1^{(0)}=\beta \). Then for small \(\epsilon \) and \(\beta \), we have
$$\begin{aligned} {\hat{a}}_1^{(1)}-{\hat{a}}_1^{(0)}=(r^{(1)}(\infty )-\mu _{1,1}^*(\infty )\epsilon -\beta -\beta ^2\rho \epsilon )/\rho >0. \end{aligned}$$
(5.80)
It follows from (5.80) and induction that
$$\begin{aligned} \begin{aligned}&{\hat{a}}_1^{(n+1)}-{\hat{a}}_1^{(n)}\\&\quad =\left[ {\hat{a}}^{(n)}_1(\rho +r^{(1)}(\infty )-\mu _{1,1}^*(\infty )\epsilon -{\hat{a}}^{(n)}_1) -{\hat{a}}^{(n-1)}_1(\rho +r^{(1)}(\infty )\right. \\&\qquad \left. -\mu _{1,1}^*(\infty )\epsilon -{\hat{a}}^{(n-1)}_1)\right] \frac{1-\beta \epsilon }{\rho }\\&\quad =({\hat{a}}_1^{(n)}-{\hat{a}}_1^{(n-1)})\left[ \rho +r^{(1)}(\infty )-\mu _{1,1}^*(\infty )\epsilon -{\hat{a}}^{(n-1)}_1-{\hat{a}}^{(n)}_1\right] \frac{1-\beta \delta }{\rho }\\&\quad >0,\ \ \ \forall n\ge 1. \end{aligned} \end{aligned}$$
(5.81)
Thus, \(\left\{ {\hat{a}}_1^{(n)}\right\} _{n=0}^{\infty }\) is increasing and \(\beta <{\hat{a}}_1^{(n)}\le r(\infty )\) for \(n\ge 1\). So, \(\lim _{n\rightarrow \infty }{\hat{a}}_1^{(n)}\) exists. Let
$$\begin{aligned} \lim _{n\rightarrow \infty }{\hat{a}}_1^{(n)}={\hat{a}}_1^*. \end{aligned}$$
(5.82)
Then \(\beta \le {\hat{a}}_1^*\le r(\infty )\) and by (5.77), we obtain that
$$\begin{aligned} {\hat{a}}^*_1={\hat{a}}^*_1(1-\beta \epsilon )(\rho +r^{(1)}(\infty )-\mu _{1,1}^*(\infty )\epsilon -{\hat{a}}^*_1)/\rho . \end{aligned}$$
(5.83)
Therefore, it follows from (5.83) that
$$\begin{aligned} {\hat{a}}^*_1=r^{(1)}(\infty )-\left( \mu _{1,1}^*(\infty )+\frac{\beta \rho }{1-\beta \epsilon }\right) \epsilon . \end{aligned}$$
(5.84)
Thus, by (5.76), (5.79), (5.82) and (5.84), we obtain that there exist a positive integer N and \(t_2>t_1\) such that
$$\begin{aligned} {\hat{v}}^{(n)}_1(t)>r^{(1)}(\infty )-\left( 1+\mu _{1,1}^*(\infty )+\frac{\beta \rho }{1-\beta \epsilon }\right) \epsilon ,\ \ \ \forall t>t_2,\ n\ge N. \end{aligned}$$
(5.85)
Clearly, if
$$\begin{aligned} \begin{array}{lll} \ell +\sigma (\mu _1)+\psi _1(\mu _1)(t-t_0^*)+NL\sqrt{4d_1(t-t_1)}\\ \\ \ \ \ \ \ \ \ \ \ \le x\le \ell +\sigma (\mu _2)+\psi _1(\mu _2)(t_1-t_0^*)+3\pi /\gamma -NL\sqrt{4d_1(t-t_1)}, \end{array} \end{aligned}$$
(5.86)
then (5.62) holds and (5.74) with n replaced by N also holds. Choose \(t_1=ml+t_0^*\) and \(t-t_1=l\), where \(m>1\) and \(l>0\) are both sufficiently large. Then we can rewrite (5.86) as
$$\begin{aligned} \begin{array}{lll} \ell +\sigma (\mu _1)+\psi _1(\mu _1)l(m+1)+NL\sqrt{4d_1l}\\ \\ \ \ \ \ \ \ \ \ \ \le x\le \ell +\sigma (\mu _2)+ml\psi _1(\mu _2)+3\pi /\gamma -NL\sqrt{4d_1l}, \end{array} \end{aligned}$$
(5.87)
that is,
$$\begin{aligned} \begin{array}{lll} (t_0^*+l(m+1))\left[ \psi _1(\mu _1)+\frac{\ell +\sigma (\mu _1)}{l(m+1)}+\frac{NL\sqrt{4d_1}}{(m+1)\sqrt{l}}\right] \frac{l(m+1)}{t_0^*+l(m+1)}\\ \\ \ \ \ \ \ \ \ \ \ \le x\le (t_0^*+l(m+1))\left[ \frac{m}{m+1}\psi _1(\mu _2)+\frac{\ell +\sigma (\mu _2)+3\pi /\gamma }{l(m+1)}-\frac{NL\sqrt{4d_1}}{(m+1)\sqrt{l}}\right] \frac{l(m+1)}{t_0^*+l(m+1)}. \end{array} \end{aligned}$$
(5.88)
Now for any given \(\varepsilon \) with \(0<\varepsilon <(c_{1,1}^*(\infty )-c-\varepsilon ^{(1)})/2\), choose \(\epsilon \) sufficiently small such that \(\epsilon <\varepsilon /3\). Then there exist \(l_0\) and \(m_0\) sufficiently large such that for \(m>m_0,\,\,l>l_0\) and \(t=t_0^*+l(m+1)>t_2\),
$$\begin{aligned}&(t_0^*+l(m+1))\left[ \psi _1(\mu _1)+\frac{\ell +\sigma (\mu _1)}{l(m+1)}+\frac{NL\sqrt{4d_1}}{(m+1)\sqrt{l}}\right] \frac{l(m+1)}{t_0^*+l(m+1)}\\&\quad<t(\psi _1(\mu _1)+\epsilon )=t(c+\varepsilon ^{(1)}+\epsilon +\epsilon )\\&\quad <t(c+\varepsilon ^{(1)}+\varepsilon ) \end{aligned}$$
and
$$\begin{aligned}&(t_0^*+l(m+1))\left[ \frac{m}{m+1}\psi _1(\mu _2)+\frac{\ell +\sigma (\mu _2)+3\pi /\gamma }{l(m+1)}-\frac{NL\sqrt{4d_1}}{(m+1)\sqrt{l}}\right] \frac{l(m+1)}{t_0^*+l(m+1)}\\&\quad>t(\psi _1(\mu _2)-\epsilon )=t(c_{1,1}^*(\infty )-2\epsilon -\epsilon )\\&\quad >t(c_{1,1}^*(\infty )-\varepsilon ). \end{aligned}$$
Let \(t_3=t_0^*+l_0(m_0+1)\). If \(t>t_3\), then \(t(c+\varepsilon ^{(1)}+\varepsilon )\le x\le t(c_{1,1}^*(\infty )-\varepsilon )\) implies that (5.86) holds. Thus, by (5.73) and (5.85), we obtain that
$$\begin{aligned}&\lim _{t\rightarrow \infty }\left[ \inf _{t(c+\varepsilon ^{(1)}+\varepsilon )\le x\le t(c_{1,1}^*(\infty )-\varepsilon )}u_1^{(1)}(t,x,\phi )\right] \nonumber \\&\quad \ge r^{(1)}(\infty )-\left( 1+\mu _{1,1}^*(\infty )+\frac{\beta \rho }{1-\beta \epsilon }\right) \epsilon . \end{aligned}$$
(5.89)
Because \(\epsilon \) can be arbitrarily small and (5.89), we have actually shown that
$$\begin{aligned} \lim _{t\rightarrow \infty }\left[ \inf _{t(c+\varepsilon ^{(1)}+\varepsilon )\le x\le t(c_{1,1}^*(\infty )-\varepsilon )}u_1^{(1)}(t,x,\phi )\right] \ge r^{(1)}(\infty ). \end{aligned}$$
(5.90)
Since \(\varepsilon ^{(1)}>0\) is arbitrary, we have actually shown that for every \(\varepsilon \) with \(0<\varepsilon <(c_1^*(\infty )-c)/2\),
$$\begin{aligned} \lim _{t\rightarrow \infty }\left[ \inf _{t(c+\varepsilon )\le x\le t(c_{1}^*(\infty )-\varepsilon )}u_1^{(1)}(t,x,\phi )\right] \ge r(\infty ). \end{aligned}$$
(5.91)
It follows from \(u_1^{(1)}(t,x,\phi )\le r(\infty )\) for all \((t,x)\in [0,+\infty )\times {\mathbb {R}}\) that
$$\begin{aligned} \lim _{t\rightarrow \infty }\left[ \sup _{t(c+\varepsilon )\le x\le t(c_{1}^*(\infty )-\varepsilon )}\left| r(\infty )-u_1^{(1)}(t,x,\phi )\right| \right] =0. \end{aligned}$$
(5.92)
The proof of Lemma 2.5 is completed. \(\square \)
