Entire Solutions with Annihilating Fronts to a Nonlocal Dispersal Equation with Bistable Nonlinearity and Spatio-Temporal Delay

Abstract

This paper deals with the entire solutions to a nonlocal dispersal bistable equation with spatio-temporal delay. Assuming that the equation has a traveling wave front with non-zero wave speed, we establish the existence of entire solutions with annihilating-fronts by using the comparison principle combined with explicit constructions of sub- and supersolutions. These entire solutions constitute a two-dimensional manifold and the traveling wave fronts belong to the boundary of the manifold. We also prove the uniqueness, Liapunov stability and continuous dependence on the shift parameters of the entire solutions.

1 Introduction

In recent years, in order to investigate the interaction effects of spatial diffusion and time delay on the evolutionary behavior of biological systems, the study of reaction-diffusion equations with spatio-temporal delay (or nonlocal delay) has drawn great attention. We refer the readers to the survey papers of Gourley et al. [10] and Ruan [23] for more results and references. For example, the following equation is a typical and important model describing the evolution of matured population of a single species (see [1, 2, 24]):

$$\begin{aligned} \frac{\partial u}{\partial t}=D \frac{\partial ^2 u}{\partial x^2}-d u(x,t)+ \int _0^\tau \int _{-\infty }^{+ \infty }G(x-y,s)b(u(y,t-s))dyds, \end{aligned}$$

(1.1)

where $u(x,t)$ denotes the density of the adult population at location $x\in \mathbb {R}$ and time $t\ge 0$; $D>0$ and $d>0$ are the diffusion rate and death rate of the adult population, respectively; $b(\cdot )$ is related to the birth function. The kernel function $G(x-y, s)$ represents the probability of the population which have been born at location $y$ and time $t-s$, and become mature at location $x$ and time $t$.

The basic assumption for the model (1.1) is that the internal interaction of the species is random and local, i.e. any individual moves randomly between the adjacent spatial locations. However, in realistic world, the movements and interactions of many species in ecology and biology can occur between the non-adjacent spatial locations, see e.g. Lee et al. [14] and Murray [22]. Taking this fact into account, the authors of [34] recently introduced a nonlocal dispersal equation for a structured population with spatio-temporal delay. The governing equation is

$$\begin{aligned} \frac{\partial u}{\partial t}=D \big [(J*u)(x,t)-u(x,t)\big ]-d u + \int _0^\tau \int _{-\infty }^{+ \infty }G(x-y,s)b(u(y,t-s))dyds, \end{aligned}$$

(1.2)

where $(J*u)(x,t)-u(x,t)$ means the “nonlocal dispersal operator” and $(J*u)(x,t)$ is a “spatial convolution operator” defined by

$$\begin{aligned} (J*u)(x,t):=\int _{-\infty }^{+ \infty }J(x-y) u(y,t) dy. \end{aligned}$$

(1.3)

In biological and epidemiological models, the existence of traveling wave solutions is an important issue due to their significant applications. Many mathematical results related to traveling wave solutions have been established in the past decades. For example, the traveling wave solutions of reaction-diffusion equations with spatial-temporal delay and nonlocal dispersal equations have been widely studied in the literature [4, 5, 7, 9, 24, 26]. On the other hand, from the viewpoint of dynamical systems, it is significant to understand the dynamical structure of the global attractor (or the maximal invariant set) which consists of entire solutions, i.e. solutions defined for all time variable $t\in \mathbb {R}$. It is clear that the traveling wave solution is a special type of entire solutions.

Although the traveling wave solutions constitute important parts of the global attractor, the structure of global attractor could be quite complicated. Recently, many types of front-like entire solutions have been observed for various evolution equations by mixing the traveling wave solutions and some spatially independent solutions, see [11–13, 15–17, 20, 21, 25, 27–33]. For examples, Hamel and Nadirashvili [12] established three-, four- and five-dimensional manifolds of entire solutions for the Fisher-KPP equation. In [13], Hamel and Nadirashvili further obtained an infinite-dimensional manifold of entire solutions for the Fisher-KPP equation in high-dimensional spaces. Different from those entire solutions obtained in [12, 13], Morita and Ninomiya [21] further constructed other types of entire solutions for some bistable reaction-diffusion equations. As mentioned in [21], we see that such entire solutions also play important roles in some other areas, such as, transient dynamics and distinct history of two solutions, etc..

For Eq. (1.2), Wu and Ruan [34] recently established the existence and qualitative properties of entire solutions under the monostable assumption of birth functions. However, for the case of bistable birth functions, the study for entire solutions of (1.2) other than traveling wave solutions still remains open. Therefore, the purpose of this paper is to study the entire solutions of (1.2) with bistable birth functions. To this end, we make the following assumptions for the kernel functions $J(\cdot )$, $G(\cdot )$ and the birth function $b(\cdot )$.

$\mathrm (G)\$ :

$J(-x)=J(x)\ge 0,$ $G(x,t)=G(-x,t)\ge 0$, $\forall x\in \mathbb {R}$, $t\in [0,\tau ]$,

$$\begin{aligned} \int _{-\infty }^{+ \infty }J(y)dy=\int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)dyds=1\ \mathrm{(normalization)}, \end{aligned}$$

and for any $c,\lambda \ge 0$,

$$\begin{aligned} \displaystyle \int _{-\infty }^{+ \infty }e^{-\lambda y}J(y)dy<+\infty \ \text { and } \int _0^\tau \int _{-\infty }^{+ \infty }e^{-\lambda (y+cs)}G(y,s)dyds<+\infty . \end{aligned}$$

$\mathrm (B1)$ :

There exists some $K>0$ such that $b(\cdot )\in C^2 ( [0,K] ,\mathbb {R})$, $d>\max \{b'(0),b'(K)\}$, $b(0)=dK-b(K)=0$, and $b'(u)\ge 0$ for $u\in [0,K]$.

It is well-known that a solution $u(x,t)$ of (1.2) is called a traveling wave solution connecting $0$ and $K$ with speed $c$, if $u(x,t)=\phi (x+ct)$, $x,t\in \mathbb {R}$, for some function $\phi (\cdot )\in C^1(\mathbb {R})$ (called wave profile) such that $\phi (-\infty )=0$ and $\phi (+\infty )=K$. Following the above definition, we see that $(c, \phi )$ satisfies the following equation

$$\begin{aligned} c\phi '(\xi )=D \big [(J*\phi )(\xi )-\phi (\xi )\big ]-d\phi (\xi )+ \int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)b\big (\phi (\xi -y-cs)\big )dyds, \end{aligned}$$

(1.4)

where

$$\begin{aligned} (J*\phi )(\xi )=\int _{-\infty }^{+\infty }J(y)\phi \big (\xi -y\big )dy. \end{aligned}$$

Under the basic assumptions $\mathrm{(G)}$ and $\mathrm{(B1)}$, the following condition ensures the existence of traveling wave fronts of (1.2) connecting $0$ and $K$

$\mathrm (B2)$ :

There exists an $a\in (0,K)$ such that $d<b'(a)$, $b(u)<du$ for $u\in (0,a)$ and $b(u)>du$ for $u\in (a,K)$.

In fact, under the assumptions $\mathrm{(G)},$ $\mathrm{(B1)}$–$\mathrm{(B2)}$ and applying the abstract theory established by Chen [4] and Fang and Zhao [8], one can show that (1.4) has a monotone solution $U(x+ct)$ (called a traveling wave front of (1.2)) connecting $0$ and $K$ with wave speed $c$. Clearly, $U(-x+ct)$ is also a traveling wave front of (1.2) connecting $0$ and $K$. Moreover, it could be verified that $b(u)=pu^2e^{-\alpha u}$ with $p>0$ and $\alpha >0$ satisfies the assumptions (B1)–(B2) for a wide range of the parameters $p$ and $\alpha$. Such specific birth function has been widely used in mathematical biology literature, see e.g. Ma and Zou [18] and Wang et al. [29].

Throughout this paper, we always assume that $\mathrm (G)$ and $\mathrm (B1)$ hold and (1.2) has a traveling wave front $U(x+ct)$ connecting $0$ and $K$ with speed $c\ne 0$. Using the traveling wave fronts $U(x+ct+\theta _1)$ and $U(-x+ct+\theta _2)$, where $\theta _1,\theta _2$ are the shift parameters, we first construct a pair of sub- and supersolutions of (1.2) (see Definition 2.1). Then we establish the existence of entire solutions of (1.2) by using the comparison principle combining with the sub- and supersolutions. According to our constructions, one can see that entire solutions behave as two traveling wave fronts approaching each other from both sides of the $x$-axis as $t\rightarrow -\infty$ and annihilating as time increases. We call such entire solutions as “annihilating-front” entire solutions. In addition, based on the construction of different pairs of sub- and supersolutions via the derived entire solutions (see Lemmas 4.2 and 4.3), we prove the uniqueness, Liapunov stability and continuous dependence on the shift parameters $\theta _1,\theta _2$ of the entire solutions. Here we point out that the assumption $\mathrm{(B2)}$ will not be needed in studying the problems on the entire solutions.

For convenience, hereinafter we denote $\Phi _{\theta }^\pm (x,t):=U(\pm x+ct+\theta )$ for any $\theta \in \mathbb {R}$. Our main results are stated as follows.

Theorem 1.1

Assume that $\mathrm (G)$ and $\mathrm (B1)$ hold and (1.2) has a traveling wave front $U(x+ct)$ connecting $0$ and $K$ with speed $c>0$. Then for any $\theta _1,\theta _2\in \mathbb {R}$, there exists a unique entire solution $\Phi _{\theta _1,\theta _2}(x,t)$ of (1.2) which satisfies

$$\begin{aligned} \lim _{t\rightarrow -\infty }\Big \{\sup _{x\le 0}\big |\Phi _{\theta _1,\theta _2}(x,t)-\Phi _{\theta _1}^-(x,t)\big | +\sup _{x\ge 0}\big |\Phi _{\theta _1,\theta _2}(x,t)-\Phi _{\theta _2}^+(x,t)\big |\Big \}=0. \end{aligned}$$

(1.5)

Furthermore, the following statements hold:

$\mathrm (1)$ :

${\partial _t}\Phi _{\theta _1,\theta _2}(x,t)>0$ and $0<\Phi _{\theta _1,\theta _2}(x,t)<K$ for all $(x,t)\in \mathbb {R}^2$.

$\mathrm (2)$ :

$\lim \limits _{t\rightarrow +\infty }\sup \limits _{x\in \mathbb {R}}|\Phi _{ \theta _1,\theta _2}(x,t)- K|=0$, $\lim \limits _{t\rightarrow -\infty }\sup \limits _{|x|\le N_0}\Phi _{ \theta _1,\theta _2}(x,t)=0$ for any $N_0\in \mathbb {R}_+$, and $\lim \limits _{|x|\rightarrow +\infty }\sup \limits _{t\ge t_0}|\Phi _{ \theta _1,\theta _2}(x,t)- K|=0$ for any $t_0\in \mathbb {R}$.

$\mathrm (3)$ :

For any $(x,t)\in \mathbb {R}^2$, $\Phi _{\theta _1,\theta _2}(x,t)$ converges to $\left\{ \begin{array}{ll}\Phi _{\theta _1}^-(x,t)\ \text{ as } \theta _2\rightarrow -\infty ,\\ \Phi _{\theta _2}^+(x,t)\ \text{ as } \theta _1\rightarrow -\infty .\end{array}\right.$

$\mathrm (4)$ :

For any $\theta _1^*,\theta _2^*\in \mathbb {R}$, there exists $(x_0,t_0)\in \mathbb {R}^2$ depending on $\theta _1,\theta _2,\theta _1^*,\theta _2^*$ such that $\Phi _{\theta _1^*,\theta _2^*}(\cdot ,\cdot )=\Phi _{\theta _1,\theta _2}(\cdot +x_0,\cdot +t_0)$ on $\mathbb {R}^2$.

$\mathrm (5)$ :

For any $(x,t)\in \mathbb {R}^2$, $\Phi _{ \theta _1,\theta _2}(x,t)$ is increasing with respect to $(\theta _1,\theta _2)\in \mathbb {R}^2$.

$\mathrm (6)$ :

$\Phi _{\theta _1,\theta _2}(x,t)$ depends continuously on $(\theta _1,\theta _2)\in \mathbb {R}^2$.

$\mathrm (7)$ :

The entire solution $\Phi _{\theta _1,\theta _2}(x,t)$ is Liapunov stable in the following sense:

$\forall \epsilon >0$, $\exists$ $\bar{\delta }>0$ such that $\forall \varphi \in \mathcal {C}_{[{ 0},{ {K}}]}$ (see (2.2) for the definition) satisfying

$$\begin{aligned} \sup _{x\in \mathbb {R}}\left\| \varphi (x,\cdot )-\Phi _{\theta _1,\theta _2} (x+x_0,\cdot +t_0)\right\| _{L^\infty [-\tau ,0]}<\bar{\delta }, \end{aligned}$$

the solution $u(x,t;\varphi )$ of (1.2) with initial value $\varphi$ satisfies

$$\begin{aligned} \left| u(x,t;\varphi ) - \Phi _{\theta _1,\theta _2}(x+x_0,t+t_0)\right| <\epsilon \end{aligned}$$

for any $x\in \mathbb {R}$ and $t\ge 0$, where $x_0,t_0\in \mathbb {R}$ are two constants.

Following the same discussions in Hamel and Nadirashvili [12], we see that the entire functions $\Phi _{\theta _1,\theta _2}(x,t)$ established by Theorem 1.1 constitute a two-dimensional manifold $\mathcal {M}_2$. In addition, (1.2) possesses two one-dimensional manifolds $\mathcal {M}_1^-$ and $\mathcal {M}_1^+$ of entire solutions of traveling wave type, namely $\Phi _{\theta _1}^-(x,t)$ and $\Phi _{\theta _2}^+(x,t)$ respectively. Then, from (3) of Theorem 1.1, we know that $\mathcal {M}_1^-$ (or $\mathcal {M}_1^+$) belongs to the boundary of $\mathcal {M}_2$ by taking the limit $\theta _2\rightarrow -\infty$ (or $\theta _1\rightarrow -\infty$).

Similar to Theorem 1.1, when $c<0$, we can obtain the following results.

Theorem 1.2

Assume that $\mathrm (G)$ and $\mathrm (B1)$ hold and (1.2) has a traveling wave front $U(x+ct)$ connecting $0$ and $K$ with speed $c<0$. Then for any $\theta _1,\theta _2\in \mathbb {R}$, there exists a unique entire solution $\tilde{\Phi }_{\theta _1,\theta _2}(x,t)$ of (1.2) which satisfies

$$\begin{aligned} \lim _{t\rightarrow -\infty }\Big \{\sup _{x\le 0}\big |\tilde{\Phi }_{\theta _1,\theta _2}(x,t)-\Phi _{\theta _1}^+(x,t)\big | +\sup _{x\ge 0}\big |\tilde{\Phi }_{\theta _1,\theta _2}(x,t)-\Phi _{\theta _2}^-(x,t)\big |\Big \}=0. \end{aligned}$$

(1.6)

Moreover, the assertions $\mathrm (4)$–$\mathrm (7)$ in Theorem 1.1 and the following statements hold:

$\mathrm (1)'$ :

${\partial _t}\tilde{\Phi }_{ \theta _1,\theta _2}(x,t)<0$ and $0<\tilde{\Phi }_{ \theta _1,\theta _2}(x,t)<K$ for all $(x,t)\in \mathbb {R}^2$.

$\mathrm (2)'$ :

$\lim \limits _{t\rightarrow +\infty }\sup \limits _{x\in \mathbb {R}}\tilde{\Phi }_{ \theta _1,\theta _2}(x,t)=0$, $\lim \limits _{t\rightarrow -\infty }\sup \limits _{|x|\le N_1}|\tilde{\Phi }_{ \theta _1,\theta _2}(x,t)-K|=0$ for any $N_1\in \mathbb {R}_+$ and $\lim \limits _{|x|\rightarrow +\infty }\sup \limits _{t\ge t_1}\tilde{\Phi }_{ \theta _1,\theta _2}(x,t) =0$ for any $t_1\in \mathbb {R}$.

$\mathrm (3)'$ :

For any $(x,t)\in \mathbb {R}^2$, $\tilde{\Phi }_{\theta _1,\theta _2}(x,t)$ converges to $\left\{ \begin{array}{ll}\Phi _{\theta _1}^+(x,t)\ \text{ as } \theta _2\rightarrow +\infty ,\\ \Phi _{\theta _2}^-(x,t)\ \text{ as } \theta _1\rightarrow +\infty .\end{array}\right.$

Remark 1.3

• (1) Here we note that Theorem 1.2 is a consequence of Theorem 1.1. In fact, let us denote $\tilde{c}:=-c>0$ and $\tilde{U}(x+\tilde{c}t):=K- U (-(x+\tilde{c}t))= K- U (-x+ct).$ Then, $\tilde{U}(-\infty ) =0$, $\tilde{U}(+\infty ) =K$, and $\tilde{U}(x+\tilde{c}t)$ is an increasing traveling wave solution of the following equation

  $$\begin{aligned} \frac{\partial v}{\partial t}=D [(J*v)(x,t)-v(x,t)]-d v + \int _0^\tau \int _{-\infty }^{+ \infty }G(x-y,s)\tilde{b}(v(y,t-s))dyds, \end{aligned}$$

  (1.7)

  where $\tilde{b}(v):=b(K)-b(K-v)$. Clearly, $\tilde{b}(\cdot )$ satisfies the condition $\mathrm (B1)$. Then it follows from Theorem 1.1 that there exists an entire solution $W (x,t)$ of (1.7) such that

  $$\begin{aligned} \lim _{t\rightarrow -\infty }\{\sup _{x\le 0}|W(x,t)-\tilde{U}(-x+\tilde{c}t-\theta _1)| +\sup _{x\ge 0}|W(x,t)-\tilde{U}(x+\tilde{c}t-\theta _2)|\}=0. \end{aligned}$$

  (1.8)

  Denote $\tilde{\Phi }_{\theta _1,\theta _2}(x,t):=K-W(x,t)$. Since $\Phi _{\theta _1}^+(x,t)=K-\tilde{U}(-x+\tilde{c}t-\theta _1)$ and $\Phi _{\theta _2}^-(x,t)=K-\tilde{U}(x+\tilde{c}t-\theta _2)$, according to (1.8), we see that $\tilde{\Phi }_{\theta _1,\theta _2}(x,t)$ is an entire solution of (1.2) which satisfies the statement of Theorem 1.2. Therefore, in the following of this work, we only prove Theorem 1.1.

• (2) We prove the main results under the assumption that (1.2) has a bistable traveling wave front with non-zero wave speed. Due to the non-zero wave speed, we can establish the entire solutions by constructing an appropriate pair of sub- and supersolution of (1.2) (see Lemmas 3.2 and 3.3). However, when the wave speed is zero, there occurs the propagation failure or pinning phenomenon for the wave front of (1.2). This fact causes the construction of sub- and supersolutions becoming very difficult. We will consider this problem in future research. Moreover, it is also an interesting and important problem to consider the sign of wave speed of the bistable traveling wave front of (1.2).

The rest of the paper is organized as follows. In Sect. 2, we first establish the existence and comparison principle for solutions of the Cauchy problem of (1.2). Then we investigate the asymptotic behavior of the traveling wave fronts at $\pm \infty$. Sect. 3 is devoted to the construction of a pair of sub- and supersolution of (1.2). Using the sub- and supersolutions and comparison principle, we first prove the existence and qualitative properties of entire solutions in Sect. 4. Based on the construction of different pairs of sub- and supersolutions via the derived entire solutions, the uniqueness, Liapunov stability and continuous dependence on the shift parameters of the entire solutions are then proved.

2 Preliminaries

We first establish the existence and comparison principle for solutions of the Cauchy problem of (1.2). Then we investigate the asymptotic behavior of the traveling wave fronts at $\pm \infty$. It could be seen that the asymptotic decay rates of the traveling wave fronts play an important role in the constructions of sub- and supersolutions of (1.2) (see Lemma 3.2).

$$\begin{aligned} \text {Let us define } \hat{b}(\cdot ):[0,2K]\rightarrow \mathbb {R} \text { by } \hat{b}(u):=\Bigg \{ \begin{array}{lll} b(u),&{} \quad u\in [0,K], \\ b(K)+b'(K)(u-K),&{} \quad u\in [K,2K]. \end{array} \end{aligned}$$

Obviously, $\hat{b}(u)$ is an extension of $b(u)$, $\hat{b}'(u)\ge 0$ and

$$\begin{aligned} \big |\hat{b}'(u_1)-\hat{b}'(u_2)\big |\le \max \limits _{u\in [0,K]}\big |b''(u)\big ||u_1-u_2|, \quad \hbox { for } u_1,u_2\in [0,2K]. \end{aligned}$$

For the sake of convenience, we still denote $\hat{b}(u)$ by $b(u)$ in the remainder of this paper.

2.1 Cauchy problem and comparison principle

Let $X$ be the Banach space of all bounded and uniformly continuous functions from $\mathbb {R}$ into $\mathbb {R}$ with the supremum norm $\Vert \cdot \Vert _X$ and $\mathcal {C }= C([-\tau ,0],X)$ be the Banach space of continuous functions from $[-\tau ,0]$ into $X$ with the supremum norm. Then we denote the following spaces:

$$\begin{aligned} X_{[ { 0}, K]}:=&\big \{ \varphi \in X: \varphi (x)\in [ { 0}, K], x\in \mathbb {R} \big \}, \end{aligned}$$

(2.1)

$$\begin{aligned} \mathcal {C }_{[ { 0}, K]}:=&\big \{ \varphi \in \mathcal {C }: \varphi (x,s)\in [ { 0}, K], x\in \mathbb {R},s\in [-\tau ,0] \big \}. \end{aligned}$$

(2.2)

As usual, we identify an element $\varphi \in \mathcal {C}$ as a function from $\mathbb {R}\times [-\tau ,0]$ into $\mathbb {R}$ defined by $\varphi (x,s)=\varphi (s)(x)$. For any continuous function $u(\cdot ):[-\tau ,\ell )\rightarrow X$, $\ell >0$, we define $u^t\in \mathcal {C}$, $t\in [0,\ell )$ by $u^t(s)=u(t+s)$, $s\in [-\tau ,0]$. Then $t \rightarrow u^t(\cdot )$ is a continuous function from $[0, \ell )$ to $\mathcal {C}$. Define $F[\cdot ] :\mathcal {C}_{[{ 0},{ {K}}]}\rightarrow X$ by

$$\begin{aligned} F[\varphi ](x) := \big (J*\varphi \big )(x,0) +\int _0^\tau \int _{-\infty }^{+ \infty }G(x-y,s)b\big (\varphi (y,-s)\big )dyds. \end{aligned}$$

It is easy to see that $F[\cdot ]:\mathcal {C}_{[{ 0},{ {K}}]}\rightarrow X$ is globally Lipschitz continuous and $T(t):= e^{-(D +d)t}$ is a linear semigroup on $X$. Then the definitions of super- and subsolutions of (1.2) are given as follows.

Definition 2.1

A continuous function $u(\cdot ):[-\tau ,\ell )\rightarrow X_{[{ 0},{ {K}}]}$, $\ell >0$, is called a supersolution (or a subsolution) of (1.2) on $[0,\ell )$ if

$$\begin{aligned} u(t)\ge (or \le )\ T(t-s)u(s)+\int _s^tT(t-r)F[u^r]dr \end{aligned}$$

for any $0\le s<t<\ell$.

Applying the theory of abstract functional differential equations [19], Corollary5], we have the following result (see also [34]).

Lemma 2.2

Assume that $\mathrm (G)$ and $\mathrm (B1)$ hold and $\varphi (\cdot )$ is the Cauchy data of Eq. (1.2). We have the following results.

(1):

For any $\varphi \in \mathcal {C}_{[{ 0},{ {K}}]}$, (1.2) has a classical and unique solution $u(x,t;\varphi )$ satisfying ${ 0}\le u(x,t;\varphi )\le { {K}}$ for $(x,t)\in \mathbb {R}\times (0,\infty )$.

(2):

Let $u^-(x,t)$ and $u^+(x,t)$ be a pair of sub- and supersolutions of (1.2) on $\mathbb {R}\times [-\tau ,\infty )$ such that $u^-(x,s)\le u^+(x,s)$ for $(x,s)\in \mathbb {R}\times [-\tau ,0]$, then ${ 0}\le u^-(x,t)\le u^+(x,t)\le { {K}}$ for $(x,t)\in \mathbb {R}\times [0,\infty )$.

2.2 Asymptotic behavior of traveling wave fronts

By elementary computations, the characteristic functions of the profile equation (1.4) with respect to the equilibria $0,K$ can be represented by

$$\begin{aligned}&\triangle _1(\lambda ):=c\lambda -D\big [\mathcal {J}(\lambda )-1\big ]+d -b'(0) \mathcal {G}(\lambda ), \\&\triangle _2(\lambda ):=c\lambda -D\big [\mathcal {J}(\lambda )-1\big ]+d -b'(K) \mathcal {G}(\lambda ), \end{aligned}$$

respectively, where

$$\begin{aligned} \mathcal {J}(\lambda ):=\int _{-\infty }^{+\infty }e^{-\lambda y}J(y) dy\ \text { and }\ \mathcal {G}(\lambda ):= \int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)e^{-\lambda (y+cs)}dyds. \end{aligned}$$

Since $d>\max \{b'(0),b'(K)\}$, one can easily obtain the following result.

Lemma 2.3

Assume $\mathrm (G)$ and $\mathrm (B1)$. The equation $\triangle _j(\lambda )=0$ ($j=1,2$) has two real roots $\lambda _{j1}:=\lambda _{j1} (c)< 0$ and $\lambda _{j2}:=\lambda _{j2}(c) > 0$ such that $\triangle _j(\lambda )>0$ if $\lambda \in (\lambda _{j1},\lambda _{j2})$, and $\triangle _j(\lambda )<0$ if $\lambda \in \mathbb {R}\setminus [\lambda _{j1},\lambda _{j2}].$

To establish the asymptotic behavior of traveling wave fronts at $\pm \infty$, we first recall the following Ikehara’s Theorem, see e.g. [3, 6].

Theorem 2.4

Let $u(\xi )$ be a positive decreasing function and $F(\Lambda ) :=\int _0^{+\infty }e^{-\Lambda \xi }u(\xi )d\xi$. If $F(\Lambda )$ can be written as $F(\Lambda )= H (\Lambda )(\Lambda +\Lambda _0)^{-(k+1)}$, where $k>-1$, $\Lambda _0>0$ are two constants and $H(\Lambda )$ is analytic in the strip $-\Lambda _0 \le \mathrm{Re} \Lambda <0$, then

$$\begin{aligned} \lim _{\xi \rightarrow +\infty }u(\xi )e^{\Lambda _0 \xi }/\xi ^k = {H(-\Lambda _0)}/{\Gamma (\Lambda _0+1)}. \end{aligned}$$

Here $\Gamma (\cdot )$ means the gamma-function.

For convenience, we denote

$$\begin{aligned} \big (G\star \phi \big )(\xi )&:= \int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)\phi \big (\xi -y-cs\big )dyds,\ \forall \phi \in C\big (\mathbb {R},[0,2K]\big ). \end{aligned}$$

(2.3)

Since $G(x,t)=G(-x,t)\ge 0$, $\forall x\in \mathbb {R}$, $t\in [0,\tau ]$, it is clear that

$$\begin{aligned} \big (G\star \phi \big )(\xi ) = \int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)\phi \big (\xi +y-cs\big )dyds. \end{aligned}$$

(2.4)

By Lemma 2.3 and Theorem 2.4, we have the following results.

Lemma 2.5

Assume $\mathrm (G)$ and $\mathrm (B1)$. Let $U(x +ct)$ be a traveling wave front of (1.2) connecting 0 and $K$ with $c\in \mathbb {R}$. Then,

$$\begin{aligned}&\lim _{\xi \rightarrow -\infty }U(\xi )e^{-\lambda _{12}\xi }=a_1 ,\ \lim _{\xi \rightarrow -\infty }U'(\xi )e^{-\lambda _{12}\xi }=a_1\lambda _{12} , \end{aligned}$$

(2.5)

$$\begin{aligned}&\lim _{\xi \rightarrow -\infty } \big (G\star U^m\big )^\frac{1}{m}(\xi )e^{-\lambda _{12}\xi }= a_1 \mathcal {G}^\frac{1}{m}\big (m\lambda _{12}\big ),\ m=1,2, \end{aligned}$$

(2.6)

$$\begin{aligned}&\lim _{\xi \rightarrow +\infty }\big (K-U(\xi )\big )e^{-\lambda _{21}\xi }=b_1 ,\ \lim _{\xi \rightarrow +\infty }U'(\xi )e^{-\lambda _{21}\xi }=-b_1\lambda _{21} ,\end{aligned}$$

(2.7)

$$\begin{aligned}&\lim _{\xi \rightarrow +\infty }\big [K- (G\star U^m)^\frac{1}{m}(\xi ) \big ]e^{-\lambda _{21}\xi }= b_1 \mathcal {G}^\frac{1}{m}\big (m\lambda _{21}\big ) ,\ m=1,2, \end{aligned}$$

(2.8)

where $a_1$ and $b_1$ are positive constants.

Proof

The proof is similar to those of [27], Theorem 3.5] and [3], Theorem 1]. For the sake of completeness and reader’s convenience, we sketch the outline for assertions (2.5) and (2.6) in the following three steps. Note that the other assertions can be considered by the same way.

• Step 1. We show that $U(\xi )$ is integrable on $(-\infty ,\xi ']$ for some $\xi '\in \mathbb {R}$.

• Step 2. We show that $U(\xi )=O(e^{\gamma \xi })$ as $\xi \rightarrow -\infty$ for some $\gamma >0$. We obtain this assertion by showing that there exists a $\gamma >0$ such that $V(\xi )=O(e^{\gamma \xi })$ as $\xi \rightarrow -\infty$, where $V(\xi ):=\int _{-\infty }^{\xi } U(s)ds$.

• Step 3. For $0<\text {Re}\lambda <\gamma$, let us define a two-sided Laplace transformation of $U$ by

  $$\begin{aligned} F(\lambda ) :=\int _{-\infty }^{+\infty }U(\xi )e^{-\lambda \xi }d\xi . \end{aligned}$$

The first part of assertion (2.5) follows from Lemma 2.3, Theorem 2.4 and a property of Laplace transformation. In addition, it follows that (2.6) and the second part of (2.5) hold. The proof is complete.$\square$

3 Construction of sub- and supersolutions

According to Remark 1.3, we may assume $c>0$ in the following of this work. By Lemma 2.5, we know that there exist positive constants $k,L,\eta ,\mu$ such that

$$\begin{aligned}&ke^{\lambda _{12}\xi }\le U(\xi )\le Le^{\lambda _{12} \xi }, ~\mu U(\xi )\le U'(\xi ), {\ \xi \le 0}, \end{aligned}$$

(3.1)

$$\begin{aligned}&ke^{\lambda _{12} \xi }\le \big (G\star U^m\big )^\frac{1}{m}(\xi )\le Le^{\lambda _{12} \xi }, ~\mu \big (G\star U^m\big )^\frac{1}{m}(\xi )\le U'(\xi ), {\ \xi \le 0},\ m=1,2, \end{aligned}$$

(3.2)

$$\begin{aligned}&\mu \eta e^{\lambda _{21} \xi }\le \mu \big (K- U(\xi )\big ), ~\mu \big [K- (G\star U^m)^\frac{1}{m}(\xi )\big ]\le U'(\xi ), {\ \xi \ge 0},\ m=1,2. \end{aligned}$$

(3.3)

In order to construct appropriate sub- and superolutions of (1.2), we give the following definitions and then introduce two important functions $p_1(t)$ and $p_2(t)$.

Definition 3.1

• (1) Let $k,L,\eta$ and $\mu$ be the constants stated in (3.1)–(3.3), we denote

  $$\begin{aligned} L_1:= & {} \max \limits _{u\in [0,2K]}b'(u),\ L_2:=\max \limits _{u\in [0,K]}|b''(u)|,\\\ N:= & {} \max \{\displaystyle {\mu ^{-1}k^{-1}L_2 L^2},\ \displaystyle {\mu ^{-1}\eta ^{-1}L_2LK }\}. \end{aligned}$$

• (2) For any $\rho _1\in (-\infty ,0]$, we denote the function

  $$\begin{aligned} \omega (\rho _1):=\rho _1-\frac{1}{\lambda _{12}}\ln \left( 1+\frac{N}{c}e^{\lambda _{12}\rho _1}\right) \quad \text{ and }\quad \bar{\omega }:=\omega (0)<0. \end{aligned}$$

  (3.4)

Since $\omega (\rho _1)$ is increasing in $\rho _1\in (-\infty ,0]$, we may denote its inverse function by $\rho _1=\rho _1(\omega ): (-\infty ,\bar{\omega }]\rightarrow (-\infty ,0]$. Then, for any $(\omega ,\tilde{\omega })\in (-\infty ,\bar{\omega }]^2$, we further define

$$\begin{aligned} \rho _2(\omega ,\tilde{\omega })&:= \tilde{\omega }+ \frac{1}{\lambda _{12}}\ln \left( 1+\frac{N}{c}e^{ \lambda _{12}\rho _1(\omega )}\right) ,\\ \tilde{p}_1(t;\omega )&:=\rho _1(\omega )+ct-\frac{1}{\lambda _{12} }\ln \left\{ 1+\frac{N}{c}e^{\lambda _{12} \rho _1(\omega )}(1-e^{c\lambda _{12} t})\right\} ,\quad \text{ for } t\le 0,\\ \tilde{p}_2(t;\omega ,\tilde{\omega })&:=\rho _2(\omega ,\tilde{\omega })+c t-\frac{1}{\lambda _{12} }\ln \left\{ 1+\frac{N}{c}e^{\lambda _{12} \rho _1(\omega )}(1-e^{c\lambda _{12} t})\right\} ,\quad \text{ for } t\le 0. \end{aligned}$$

Elementary computations show that $\tilde{p}_1(t;\omega )$ and $\tilde{p}_2(t;\omega ,\tilde{\omega })$ satisfy the problems:

$$\begin{aligned} \left\{ \begin{array}{lll} \tilde{p}_1'(t;\omega )=c+Ne^{\lambda _{12} \tilde{p}_1(t;\omega )},\\ \tilde{p}_1(0;\omega )=\rho _1(\omega ), \ \end{array} \right. \ \text{ and } \ \left\{ \begin{array}{lll} \tilde{p}_2'(t;\omega ,\tilde{\omega })=c+Ne^{\lambda _{12} \tilde{p}_1(t;\omega )}, \\ \tilde{p}_2(0;\omega ,\tilde{\omega })=\rho _2(\omega ,\tilde{\omega }). \end{array} \right. \end{aligned}$$

(3.5)

Obviously, we have $\tilde{\omega }-\omega =\rho _2(\omega ,\tilde{\omega })-\rho _1(\omega )$,

$$\begin{aligned} \tilde{p}_2(t;\omega ,\tilde{\omega })-\tilde{p}_1(t;\omega )&= \rho _2(\omega ,\tilde{\omega })-\rho _1(\omega )= \tilde{\omega }-\omega ,\end{aligned}$$

(3.6)

$$\begin{aligned} \tilde{p}_1(t;\omega )- ct-\omega&=\tilde{p}_2(t;\omega ,\tilde{\omega })- c t- \tilde{\omega }=-\frac{1}{\lambda _{12}}\ln \left( 1-\frac{r}{1+r}e^{c\lambda _{12} t}\right) , \end{aligned}$$

(3.7)

for $t\le 0,$ where $r:={N}{c}^{-1}e^{\lambda _{12} \rho _1(\omega )}$.

Moreover, given any $(\omega _1,\omega _2)\in (-\infty ,\bar{\omega }]^2$, we set

$$\begin{aligned} \text{ or } \begin{array}{ll} &{}p_1(t)=p_1(t;\omega _1,\omega _2):=\tilde{p}_1(t;\omega _1),\ p_2(t)=p_2(t;\omega _1,\omega _2)\\ &{}\quad :=\tilde{p}_2(t;\omega _1,\omega _2),\ \text{ if } \omega _2\le \omega _1;\\ &{} p_1(t)=p_1(t;\omega _1,\omega _2):=\tilde{p}_2(t;\omega _2,\omega _1),\ p_2(t)=p_2(t;\omega _1,\omega _2)\\ &{}\quad :=\tilde{p}_1(t;\omega _2),\ \text{ if } \omega _1\le \omega _2. \end{array} \end{aligned}$$

Then, $p_2(t)\le p_1(t)\le 0$ when $\omega _2\le \omega _1$; and $p_1(t)\le p_2(t)\le 0$ when $\omega _1\le \omega _2$. By (3.7), there exists a positive constant $R_0$, independent of $\omega _1$ and $\omega _2$, such that

$$\begin{aligned} 0<p_1(t)-ct-\omega _1=p_2(t)-c t-\omega _2\le R_0e^{c\lambda _{12} t},\quad \text { for }t\le 0. \end{aligned}$$

Using $p_1(t)$ and $p_2(t)$, we are ready to establish the supersolution of (1.2). For simplicity, we denote

$$\begin{aligned} (G\star v)(x,t):= \int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)v(x-y,t-s)dyds,\quad \forall v\in C\big (\mathbb {R}^2,[0,2K]\big ). \end{aligned}$$

Lemma 3.2

For any $(\omega _1,\omega _2)\in (-\infty ,\bar{\omega }]^2$, there exists a $T<0$ such that the function $\overline{u}(x,t)$ defined by

$$\begin{aligned} \overline{u}(x,t)=U(x+p_1(t))+U(-x+p_2(t)) \end{aligned}$$

is a supersolution of (1.2) on $\mathbb {R}\times (-\infty ,T)$.

Proof

We only consider the case $\omega _1\le \omega _2$, since the other case can be discussed in the same way. In this case, $p_1(t)\le p_2(t)$ and $p_i'(t)=c+Ne^{\lambda _{12} p_2(t)}$, $i=1,2,$ for $t\le 0$. By direct computations, we have

$$\begin{aligned} \mathcal {F}(\overline{u})(x,t):=&\,\overline{u}_t-D\big [(J*\overline{u})(x,t)-\overline{u}(x,t)\big ]+d \overline{u}- \big (G\star b(\overline{u})\big )(x,t)\\ =\,&p'_1(t)U'(x+p_1)+p'_2(t)U'(-x+p_2)\\&-D\big [(J*U)(x+p_1)+(J*U)(-x+p_2)-U(x+p_1)-U(-x+p_2)\big ]\\&+d\big [U(x+p_1)+U(-x+p_2)\big ]-\big (G\star b(\overline{u})\big )(x,t)\\ =\,&\big (p'_1(t)-c\big )U'(x+p_1)+\big (p'_2(t)-c\big )U'(-x+p_2)-H(x,t)\\ =\,&\big [U'(x+p_1)+U'(-x+p_2)\big ]\big [Ne^{\lambda _{12} p_2(t)}-R(x,t)\big ], \end{aligned}$$

where

$$\begin{aligned} \text{ and }\quad \begin{array}{ll} R(x,t):=\displaystyle {H(x,t)}/\big [{U'(x+p_1)+U'(-x+p_2)}\big ]\\ H(x,t):= \big (G\star b(\overline{u})\big )(x,t)-\big (G\star b(U)\big )(x+p_1(t))-\big (G\star b(U)\big )(-x+p_2(t)). \end{array} \end{aligned}$$

Clearly, $p_i(t-s)\le p_i(t)-cs$ for $s\in [0, \tau ], i=1,2$. Note that $|b'(u_1)-b'(u_2)|\le L_2 |u_1-u_2|$ for $u_1,u_2\in [0,2K]$. For any $v_1,v_2\in [0,K]$, we have

$$\begin{aligned} \big |b(v_1+v_2)-b(v_1)-b(v_2)\big |=\Big |\int _0^1 v_2\big [b'(v_1+\theta v_2)- b'( \theta v_2)\big ] d\theta \Big | \le L_2v_1v_2. \end{aligned}$$

Then, using Cauchy–Schwarz inequality and (2.3) and (2.4), we obtain

$$\begin{aligned} H(x,t)=&\int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)\big [b\big (U(x-y+p_1(t-s))+U(-x{+y}+p_2(t-s))) \nonumber \\&\qquad -b(U(x+p_1(t)-y-cs))-b(U(-x+p_2(t){+y}-cs)\big )\big ]dyds \nonumber \\ \le&\int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)\big [b\big (U(x+p_1(t)-y-cs)+U(-x+p_2(t){+y}-cs)\big ) \nonumber \\&\qquad -b(U(x+p_1(t)-y-cs))-b(U(-x+p_2(t){+y}-cs))\big ]dyds \nonumber \\ \le&L_2\int _0^\tau \int _{-\infty }^{+ \infty }G(y,s) U(x+p_1(t)-y-cs)U(-x+p_2(t){+y}-cs)dyds \nonumber \\ \le&L_2 \big (G\star U^2\big )^\frac{1}{2}(x+p_1(t))\big (G\star U^2\big )^\frac{1}{2}(-x+p_2(t)). \end{aligned}$$

(3.8)

Now we estimate $R(x,t)$ by dividing $\mathbb {R}$ into the following 3 regions:

$$\begin{aligned} (1)\,p_2(t)\le x\le -p_1(t) \quad (2)\,x\ge -p_1(t),\quad (3)\,x\le p_2(t). \end{aligned}$$

(1) By (3.1), (3.2) and (3.8), we have $H(x,t)\le L_2 L^2e^{\lambda _{12} (p_1+p_2)}$ and

$$\begin{aligned} U'(x+p_1)+U'(-x+p_2)&\ge \mu \big [ U(x+p_1)+U(-x+p_2)\big ]\\&\ge \mu k \big [e^{\lambda _{12} (x+p_1)}+e^{\lambda _{12} (-x+p_2)}\big ]\ge 2 \mu k e^{\lambda _{12} p_1}. \end{aligned}$$

Hence, it follows that

$$\begin{aligned} R(x,t)\le {2^{-1}\mu ^{-1} k^{-1}L_2 L^2}e^{\lambda _{12} p_2}. \end{aligned}$$

(3.9)

(2) In this case, we further consider two sub-cases:

$$\begin{aligned} (2\text {-}1) b'(0)\le b'(K) \hbox { and } (2\text {-}2) b'(K)< b'(0). \end{aligned}$$

(2-1) Let us denote

$$\begin{aligned} \triangle _3(\lambda ):=\triangle _2(-\lambda )=-c\lambda -D\big [ \mathcal {J}(\lambda ) -1\big ]+d -b'(K) \mathcal {G}(-\lambda ) . \end{aligned}$$

It is clear that $\triangle _3(\lambda )$ has exactly one positive zero $-\lambda _{21}$. Moreover,

$$\begin{aligned} \triangle _1(\lambda )-\triangle _3(\lambda )=2c\lambda + b'(K) \mathcal {G}(-\lambda )-b'(0) \mathcal {G}(\lambda )\ge 0,\ \forall \lambda \ge 0. \end{aligned}$$

By the properties of $\triangle _1(\lambda )$ and $\triangle _3(\lambda )$, we see that $\lambda _{12}\ge -\lambda _{21}$. Then it follows from (3.2), (3.3) and (3.8) that

$$\begin{aligned} R(x,t)&\le {L_2K\big (G\star U^2\big )^\frac{1}{2}(-x+p_2)}/{U'(x+p_1) } \le \mu ^{-1}\eta ^{-1}L_2LK e^{\lambda _{12} (-x+p_2)}e^{-\lambda _{21} (x+p_1)}\nonumber \\&= {\mu ^{-1}\eta ^{-1}L_2LK e^{\lambda _{12} p_2}}{ e^{ - (\lambda _{12}+\lambda _{21})x} e^{-\lambda _{21} p_1}}\le {\mu ^{-1}\eta ^{-1}L_2LK }e^{\lambda _{12} p_2}. \end{aligned}$$

(3.10)

   (2-2) A direct computation shows that

$$\begin{aligned} H(x,t)\le&\int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)\big [b\big (U(x+p_1(t)-y-cs)+U(-x+p_2(t)+y-cs)\big ) \nonumber \\&\qquad -b(U(x+p_1(t)-y-cs))-b(U(-x+p_2(t){+y}-cs))\big ]dyds \nonumber \\ =&\int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)\big [b'\big (U(x+p_1(t)-y-cs)+\theta _1U(-x+p_2(t){+y}-cs)\big ) \nonumber \\&\qquad -b'(\theta _2 U(-x+p_2(t){+y}-cs))\big ]U(-x+p_2(t){+y}-cs)dyds, \end{aligned}$$

(3.11)

where $\theta _1,\theta _2\in (0,1).$ Let $\epsilon :=b'(0)-b'(K)>0.$ Noting that $b'(u)=b'(K)$ for $u\in [K,2K]$, there exists a $\delta >0$ such that

$$\begin{aligned} b'(u)<b'(0)-{\epsilon }/{2}, \quad \text { for all }u\in [K-{\delta }/{2},2K]. \end{aligned}$$

(3.12)

By Lemma 2.5, we have

$$\begin{aligned} \lim \limits _{\xi \rightarrow -\infty }{U'(\xi )}/{U(\xi )}=\lambda _{12}\quad \text { and } \quad \lim \limits _{\xi \rightarrow +\infty }{U'(\xi )}/{U(\xi )}=0. \end{aligned}$$

Then, we can choose $\beta >0$ such that

$$\begin{aligned} \frac{d}{d\xi }\Big [U(\xi )e^{-\beta \xi }\Big ]=e^{-\beta \xi }U(\xi ) \left[ \frac{U'(\xi )}{U(\xi )}-\beta \right] \le 0 , \quad \forall \xi \in \mathbb {R}, \end{aligned}$$

that is, $U(\xi )e^{-\beta \xi }$ is decreasing in $\mathbb {R}$. Noting that

$$\begin{aligned} \lim _{\xi \rightarrow -\infty }{U(\xi ) }/{(G\star U)(\xi )}={1}/{ \mathcal {G}(\lambda _{12})}\ \ \text { and }\ \ p_i(-\infty )=-\infty ,\ i=1,2, \end{aligned}$$

then there exists a $T <0$ such that

$$\begin{aligned} U(-x+p_2(t))\le \frac{2}{ \mathcal {G}(\lambda _{12})} (G\star U)(-x+p_2(t)),\ \text{ for } x\ge -p_1(t) \text{ and } t\le T. \end{aligned}$$

By assumption $\mathrm (G)$, we can choose $B>c\tau$ such that

$$\begin{aligned} \frac{2L _1}{ \mathcal {G}(\lambda _{12})}\left\{ \int _0^\tau \int _{-\infty }^{-B} + \int _0^\tau \int _{B}^{+\infty }e^{\beta (y-cs)} \right\} G(y,s)dyds \le \frac{\epsilon }{2}. \end{aligned}$$

Thus, for $x\ge -p_1(t)$ and $t\le T$, we have

$$\begin{aligned}&\left\{ \int _0^\tau \int _{-\infty }^{-B}\!+\int _0^\tau \int _{B}^{+\infty } \right\} G(y,s) b'\big (U(x+p_1(t)-y-cs)\!+ \theta _1U(-x+p_2(t)+y-cs)\big ) \nonumber \\&\qquad \qquad \qquad \qquad \qquad \ \times U(-x+p_2(t)+y-cs)dyds \nonumber \\&\quad \le L_1\left\{ \int _0^\tau \int _{-\infty }^{-B}+\int _0^\tau \int _{B}^{+\infty }\right\} G(y,s)U(-x+p_2(t)+y-cs)dyds \nonumber \\&\quad \le L_1\left\{ \int _0^\tau \int _{-\infty }^{-B} + \int _0^\tau \int _{B}^{+\infty } e^{\beta (y-cs)}\right\} G(y,s)dydsU(-x+p_2(t)) \nonumber \\&\quad \le \frac{ 2L_1}{ \mathcal {G}(\lambda _{12})} \left\{ \int _0^\tau \int _{-\infty }^{-B}+ \int _0^\tau \int _{B}^{+\infty } e^{\beta (y-cs)} \right\} G(y,s)dyds(G\star U)(-x+p_2(t)) \nonumber \\&\quad \le \frac{\epsilon }{2}\big (G\star U\big )(-x+p_2(t)). \end{aligned}$$

(3.13)

Since $U(+\infty )=K$, we may assume $U(\xi )> K-{\delta }/{2} \text { for }\xi \ge -B-c\tau$ by translations if necessary. If $x\ge -p_1(t)$ and $t\le T$, then it follows from (3.12) and (3.13) that

$$\begin{aligned}&\int _0^\tau \int _{-\infty }^{+ \infty }G(y,s) b'\big (U(x+p_1(t)-y-cs) \nonumber \\&\qquad \qquad \qquad + \theta _1U(-x+p_2(t)+y-cs)\big ) U(-x+p_2(t)+y-cs)dyds \nonumber \\&\quad = \left\{ \int _0^\tau \int _{-\infty }^{-B}+\int _0^\tau \int _{B}^{+\infty }+\int _0^\tau \int _{-B}^{B}\right\} G(y,s) b'\big (U(x+p_1(t)-y-cs) \nonumber \\&\qquad \qquad \qquad + \theta _1U(-x+p_2(t)+y-cs)\big ) U(-x+p_2(t)+y-cs)dyds \nonumber \\&\quad \le \frac{\epsilon }{2}\big (G\star U\big )(-x+p_2(t)) +\left( b'(0)-\frac{\epsilon }{2}\right) \big (G\star U\big )(-x+p_2(t)) \nonumber \\&\quad =b'(0)\big (G\star U\big )(-x+p_2(t)) . \end{aligned}$$

(3.14)

Using (3.11) and (3.14), for $x\ge -p_1(t)$ and $t\le T$, we conclude that

$$\begin{aligned} H(x,t)&\le \int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)\big [b'(0) -b'(\theta _2 U(-x+p_2(t)+y-cs))\big ] \nonumber \\&\qquad \quad \ \qquad \times U(-x+p_2(t)+y-cs)dyds \nonumber \\&\le L_2\int _0^\tau \int _{-\infty }^{+ \infty }G(y,s) U^2(-x+p_2(t)+y-cs)dyds \nonumber \\&= L_2 \big (G\star U^2\big )(-x+p_2(t)) . \end{aligned}$$

(3.15)

Therefore, by (3.1), (3.2) and (3.15), we have

$$\begin{aligned} R(x,t)&\le \frac{L_2\big (G\star U^2\big )(-x+p_2(t))}{U'(-x+p_2) } \le \frac{L_2 L^2 e^{2\lambda _{12} (-x+p_2)}}{ \mu k e^{\lambda _{12} (-x+p_2)}} \le {\mu ^{-1} k^{-1}L_2L^2 }e^{\lambda _{12} p_2}. \end{aligned}$$

(3.16)

(3) Similar to the discussion of case (2), we can also derive

$$\begin{aligned} R(x,t) \le \max \big \{ \mu ^{-1} k^{-1} L_2 L^2 ,\ \mu ^{-1}\eta ^{-1}L_2LK \big \}e^{\lambda _{12} p_2}. \end{aligned}$$

(3.17)

Thus, combining (3.9), (3.10), (3.16) and (3.17), we have $\mathcal {F}(\overline{u})(x,t)\ge 0$, that is $\overline{u}(x,t)$ is a supersolution of (1.2) on $\mathbb {R}\times (-\infty ,T)$. This completes the proof. $\square$

Moreover, we have the following subsolution of (1.2).

Lemma 3.3

For any $(\omega _1,\omega _2)\in (-\infty ,\bar{\omega }]^2$, the function $\underline{u}(x,t)$ defined by

$$\begin{aligned} \underline{u}(x,t):=\max \big \{\Phi _{\omega _1}^+(x,t),\Phi _{\omega _2}^-(x,t)\big \} \end{aligned}$$

is a subsolution of (1.2) on $\mathbb {R}\times (-\infty ,+\infty )$.

Proof

The proof is obvious. We omit it here. $\square$

4 Proof of the main result

Based on the construction of sub- and supersolutions of (1.2), we first prove the assertions of Theorem 1.1 for the case $(\theta _1,\theta _2)=(\omega _1,\omega _2)\in (-\infty ,\bar{\omega }]^2$. Then, we improve the results to any $(\theta _1,\theta _2)\in \mathbb {R}^2$.

4.1 Entire solutions for $(\theta _1,\theta _2)=(\omega _1,\omega _2)\in (-\infty ,\bar{\omega }]^2$

As mentioned in the Introduction, we always assume that $\mathrm (G)$ and $\mathrm (B1)$ hold and (1.2) has a traveling wave front $U(x+ct)$ with speed $c>0$.

4.1.1 Existence of entire solutions

Theorem 4.1

For any $(\omega _1,\omega _2)\in (-\infty ,\bar{\omega }]^2$, there exists an entire solution $\Phi _{\omega _1,\omega _2}(x,t)$ of (1.2) satisfying the following statements.

$\mathrm (1)$ :

$\partial _t\Phi _{\omega _1,\omega _2}(x,t)>0$ and $0<\Phi _{\omega _1,\omega _2}(x,t)<K$ for all $(x,t)\in \mathbb {R}^2$.

$\mathrm (2)$ :

For any $(x,t)\in \mathbb {R}^2$, $\Phi _{\omega _1,\omega _2}(x,t)\ \text{ is } \text{ increasing } \text{ with } \text{ respect } \text{ to } (\omega _1,\omega _2)$.

$\mathrm (3)$ :

$\lim \limits _{t\rightarrow -\infty }\{\sup \limits _{x\le 0}|\Phi _{\omega _1,\omega _2}(x,t)-\Phi _{\omega _1}^-(x,t)| +\sup \limits _{x\ge 0}|\Phi _{\omega _1,\omega _2}(x,t)-\Phi _{\omega _2}^+(x,t)|\}=0,$

$\lim \limits _{t\rightarrow +\infty }\sup \limits _{x\in \mathbb {R}}|\Phi _{\omega _1,\omega _2}(x,t)- K|=0, \ \lim \limits _{t\rightarrow -\infty }\sup \limits _{|x|\le N_0}\Phi _{\omega _1,\omega _2}(x,t)=0,\ \forall \ N_0\in \mathbb {R},$

$\lim \limits _{|x|\rightarrow +\infty }\sup \limits _{t\ge t_0}|\Phi _{\omega _1,\omega _2}(x,t)- K|=0,\ \forall \ t_0\in \mathbb {R}.$

$\mathrm (4)$ :

For any $\omega _1^*,\omega _2^*\in (-\infty ,\bar{\omega }]$, there exists $(x_0,t_0)\in \mathbb {R}^2$ depending on $\omega _1,$ $\omega _2,$ $\omega _1^*,$ $\omega _2^*$ such that $\Phi _{\omega _1^*,\omega _2^*}(\cdot ,\cdot )=\Phi _{\omega _1,\omega _2}(\cdot +x_0,\cdot +t_0)\ \text{ on } \mathbb {R}^2.$

Proof

For any $(\omega _1,\omega _2)\in (-\infty ,\bar{\omega }]^2$ and $n\in (-T,+\infty )\cap \mathbb {N}$, let $\Phi ^n(x,t)$ be the unique solution of the following initial value problem:

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} \Phi ^n_t \!=\! D [J* \Phi ^n\!-\! \Phi ^n]\!-\!d \Phi ^n \!+\! (G\star b(\Phi ^n))(x,t), &{}{} \quad \text { for } x\in \mathbb {R}, \ t\!>\! -n;\\ \Phi ^n(x,s)\! = \!\underline{u}(x,s), &{}{}\quad \text { for } x\in \mathbb {R}, \ s\in [\!-\!n\!-\!\tau ,-\!n]. \end{array} \right. \end{aligned} \end{aligned}$$

(4.1)

By Lemmas 2.2, 3.2 and 3.3, we have

$$\begin{aligned} \text{ and }\quad \begin{array}{ll} \underline{u} (x,t) \le \Phi ^n(x,t) \le \Phi ^{n+1}(x,t)\le \overline{u}(x,t), &{} \quad \text { for }x\in \mathbb {R},\ -n\le t<T, \\ \underline{u} (x,t) \le \Phi ^n(x,t) \le K, &{} \quad \text { for }x\in \mathbb {R},\ t>-n. \end{array} \end{aligned}$$

(4.2)

Then, there exists a function $\Phi (x,t)$ such that $\lim \limits _{n\rightarrow \infty }\Phi ^n(x,t)=\Phi (x,t)$ for any $(x,t)\in \mathbb {R}^2$. Moreover, for any given $t_{0}\in \mathbb {R}$, there exists some $n\in \mathbb {N}$ such that $t_{0}>-n$ and $\Phi ^n(x,t)$ satisfies

$$\begin{aligned} \Phi ^n(t)(x)=T(t-t_{0})\Phi ^n(t_{0})(x)+\int _{t_0}^tT(t-r)F\big [(\Phi ^n)^r\big ](x)dr, \end{aligned}$$

where $T(t)$ and $F[\cdot ]$ are defined as in Sect. 2.1. By Lebesgue’s dominated convergence theorem, we obtain

$$\begin{aligned} \Phi (t)(x)=T(t-t_{0})\Phi (t_{0})(x)+\int _{t_0}^tT(t-r)F\big [(\Phi )^r\big ](x)dr. \end{aligned}$$

This implies that $\Phi (x,t)$ is continuous and differentiable with respect to $t$. In addition, one can show that

$$\begin{aligned} \Phi _t = D \big [ J*\Phi -\Phi \big ] -d \Phi (x,t) + \big (G\star b( \Phi ) \big )(x,t). \end{aligned}$$

Therefore, $\Phi _{\omega _1,\omega _2}(x,t):=\Phi (x,t)$ is an entire solution of (1.2). Now we prove the assertions of (1)–(4) in the sequel.

(1) By (4.2), it’s obvious that

$$\begin{aligned} \text{ and }\quad \begin{array}{ll} \underline{u} (x,t) \le \Phi _{\omega _1,\omega _2}(x,t) \le \overline{u} (x,t), &{} \quad \text { for all }x\in \mathbb {R},\ t<T,\\ \underline{u} (x,t) \le \Phi _{\omega _1,\omega _2} (x,t) \le K, &{} \quad \text { for all }(x,t)\in \mathbb {R}^2. \end{array} \end{aligned}$$

(4.3)

Clearly, $\Phi _{\omega _1,\omega _2}(x,t)>0$ for all $(x,t)\in \mathbb {R}^2$. Since

$$\begin{aligned} \Phi ^n(x,t) \ge \underline{u} (x,t)\ge \underline{u} (x,s)= \Phi ^n(x,s) \end{aligned}$$

for $(x,t)\in \mathbb {R} \times [-n,+\infty )$ and $s\in [-n-\tau ,-n]$, by Lemma 2.2, we have $\partial _t\Phi ^n(x,t)\ge 0$ for $(x,t)\in \mathbb {R} \times (-n,+\infty )$. This yields to $\partial _t\Phi _{\omega _1,\omega _2}(x,t)\ge 0$ for all $(x,t)\in \mathbb {R}^2.$

Moreover, from (4.1), we have

$$\begin{aligned} \partial _{tt}\Phi _{\omega _1,\omega _2} =&D \big [J* \partial _{t}\Phi _{\omega _1,\omega _2} - \partial _{t}\Phi _{\omega _1,\omega _2} \big ] -d \partial _{t}\Phi _{\omega _1,\omega _2} \\&+ \int _0^\tau \int _{-\infty }^{+ \infty }G(x-y,s)b'\big ( \Phi _{\omega _1,\omega _2}(y,t-s)\big ) \partial _{t}\Phi _{\omega _1,\omega _2}(y,t-s)dyds\\ \ge&-(D+d) \partial _{t}\Phi _{\omega _1,\omega _2},\ \ \text{ for } (x,t)\in \mathbb {R}^2, \end{aligned}$$

which implies

$$\begin{aligned} \partial _t\Phi _{\omega _1,\omega _2}(x,t) \ge \partial _t\Phi _{\omega _1,\omega _2}(x,s) e^{-(D+d) (t-s)} , {\ } \forall s<t . \end{aligned}$$

(4.4)

Suppose that the first part of (1) is false, then there exists a $(x_0,t_0)\in \mathbb {R}^2$ such that $\partial _t\Phi _{\omega _1,\omega _2}(x_0,t_0)=0$ and it follows from (4.4) that $\partial _t\Phi _{\omega _1,\omega _2}(x_0,t)=0$ for all $t\le t_0$. Hence $\Phi _{\omega _1,\omega _2}(x_0,t)=\Phi _{\omega _1,\omega _2}(x_0,t_0)$ for all $t\le t_0$, which implies that

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }\Phi _{\omega _1,\omega _2}(x_0,t)=\Phi _{\omega _1,\omega _2}(x_0,t_0). \end{aligned}$$

On the other hand, from (4.3), we have

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }\Phi _{\omega _1,\omega _2}(x_0,t)=0\ \hbox { and } \Phi _{\omega _1,\omega _2}(x_0,t_0)>0. \end{aligned}$$

This contradiction yields that $\partial _t\Phi _{\omega _1,\omega _2}(x,t)>0$ for all $(x,t)\in \mathbb {R}^2$. Moreover, we can show that $\Phi _{\omega _1,\omega _2}(x,t)<K$ for all $(x,t)\in \mathbb {R}^2$.

(2) Noting that $U'(z)>0$ and $0<U(z)<1$ for $z\in \mathbb {R}$, then it follows that $\Phi _{\omega _1,\omega _2}(x,t)\ \text{ is } \text{ increasing } \text{ with } \text{ respect } \text{ to } (\omega _1,\omega _2).$

(3) & (4) Using (4.3), the proofs of these parts are straightforward and thus omitted. The proof is complete. $\square$

4.1.2 Uniqueness and stability of entire solutions

In order to prove the uniqueness, stability and continuous dependence on the shift parameters $\omega _1,\omega _2$ of the entire solution $\Phi _{\omega _1,\omega _2}(x,t)$, we construct different sub-supersolution pairs of (1.2) to trap the entire solution.

Lemma 4.2

There exist $\delta _0\in (0,K)$, $\rho _0>0$ and $\sigma _0>0$ such that for any $\gamma \in \mathbb {R}$, $\delta \in (0,\delta _0]$ and $\sigma \ge \sigma _0$, the functions $u^\pm (x,t)$ defined by

$$\begin{aligned} u^\pm (x,t)= \Phi _{\omega _1,\omega _2}\left( x,t+\gamma \pm \sigma \delta \big (1-e^{- \rho _0t}\big )\right) \pm \delta e^{- \rho _0t} \end{aligned}$$

constitute a pair of super- and subsolution of (1.2) on $[0,+\infty )$.

Proof

We only prove that $u^+(x,t)$ is a supersolution of (1.2) on $[0,+\infty )$. Following the same arguments, we can also show that $u^-(x,t)$ is a subsolution. Since

$$\begin{aligned} \text{ and }\ \ \begin{array}{ll} &{}\displaystyle \lim _{(\rho ,\bar{\omega })\rightarrow (0,b'(0))}\big [-\!\rho +d-e^{\rho \tau }\bar{\omega }\big ]=d-b'(0)>0,\\ &{}\displaystyle \lim _{(\rho ,\bar{\omega })\rightarrow (0,b'(K))}\big [-\!\rho +d-e^{\rho \tau }\bar{\omega } \big ]=d-b'(K)>0,\qquad \end{array} \end{aligned}$$

we can fix $\rho _0>0$ and $0<\delta _1\ll K$ such that

$$\begin{aligned} \text{ and }\quad \begin{array}{ll} &{}-\rho _0+d-e^{\rho _0 \tau }\bar{\omega }>0,\quad \text { for }\bar{\omega }\in \big [b'(0)-\delta _1,b'(0)+\delta _1\big ],\\ &{}-\rho _0+d-e^{\rho _0 \tau }\bar{\omega }>0,\quad \text { for }\bar{\omega }\in \big [b'(K)-\delta _1,b'(K)+\delta _1\big ]. \end{array} \end{aligned}$$

(4.5)

Now we choose $\delta _0\in (0,\delta _1)$ and $\nu \in (0,K)$ such that $\delta _0e^{\rho _0 \tau }L_2\le {\delta _1}/{4},$

$$\begin{aligned} \text{ and }\quad \begin{array}{ll} &{}b'(u)\in \big [b'(0)-{\delta _1}/{2}, b'(0)+{\delta _1}/{2}\big ],\quad \text { for }u\in [0,\nu ], \\ &{}b'(u)\in \big [b'(K)-{\delta _1}/{2}, b'(K)+{\delta _1}/{2}\big ],\quad \text { for }u\in [K-\nu ,K+\nu ]. \end{array} \end{aligned}$$

(4.6)

By assumption $\mathrm (G)$, there exists an $M>0$ such that

$$\begin{aligned}&L_1\left\{ \int _0^\tau \int _{-\infty }^{-M}+\int _0^\tau \int _{M}^{+\infty } \right\} G(y,s) dyds\in (0,{\delta _1}/{4}),\end{aligned}$$

(4.7)

$$\begin{aligned}&\quad \int _0^\tau \int _{-M}^{M} G(y,s) dyds\ge {\left( b'(K)-\frac{3}{4}\delta _1\right) }/{\left( b'(K)-\frac{1}{2}\delta _1\right) }, \qquad \text { if } b'(K)>0, \end{aligned}$$

(4.8)

$$\begin{aligned}&\quad \int _0^\tau \int _{-M}^{M} G(y,s) dyds\ge {\left( b'(0)-\frac{3}{4}\delta _1\right) }/{\left( b'(0)-\frac{1}{2}\delta _1\right) }, \quad \text { if } b'(0)>0. \end{aligned}$$

(4.9)

Take $X>0$ such that

$$\begin{aligned} \text{ and }\quad \begin{array}{ll} U(x)\in (0,{\nu }/{4}), &{}\quad \text { for }x\le -X+M \\ U(x)\in \left( K-{\nu }/{4},K+{\nu }/{4}\right) ,&{}\quad \text { for }x\ge X-M-c\tau . \end{array} \end{aligned}$$

(4.10)

Since

$$\begin{aligned} \lim _{t\rightarrow +\infty } \sup _{x\in \mathbb {R}} \left| \int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)b'\big (\Phi _{\omega _1,\omega _2}(x-y,t-s) \big )dyds -b'(K)\right| =0, \end{aligned}$$

there exists a $T_1>\tau$ such that

$$\begin{aligned} \int _0^\tau \int _{-\infty }^{+\infty }G(y,s)b'\big (\Phi _{\omega _1,\omega _2}(x-y,t-s) \big )dyds \in \big [b'(K)-{\delta _1}/{2}, b'(K)+{\delta _1}/{2}\big ], \end{aligned}$$

(4.11)

for any $t>T_1$ and $x\in \mathbb {R}$. In view of $\lim \limits _{t\rightarrow -\infty }[p_i(t)-ct-\omega _i]=0$, $i=1,2,$ we can take $T_2\le T$, where $T<0$ is defined in Lemma 3.2, such that

$$\begin{aligned} 2\max _{i=1,2}|p_i(t)-ct-\omega _i|\max _{x\in \mathbb {R}} U'(x)\in (0, {\nu }/{4}),\quad \text{ for } t\le T_2-\tau . \end{aligned}$$

(4.12)

Letting $\kappa _1:=\min _{|x|\le X}U'(x)>0$, then there exists a $\sigma _1>0$ such that

$$\begin{aligned} \frac{1}{2}\kappa _1\sigma _1\rho _0-\rho _0+d- e^{\rho _0\tau }L_1>0. \end{aligned}$$

(4.13)

Set $\Psi (x,t):=\Phi _{\omega _1}^+(x,t)+\Phi _{\omega _2}^-(x,t)$. One can easily show that

$$\begin{aligned} \lim _{t\rightarrow -\infty }\sup _{x\in \mathbb {R}}\big \Vert \Phi _{\omega _1,\omega _2}(x,\cdot )-\Psi (x,\cdot ) \big \Vert _{C^0((-\infty ,t])}=0. \end{aligned}$$

Since $0< \Phi _{\omega _1,\omega _2}(x,t)<K$, $\forall (x,t)\in \mathbb {R}^2$, one can verify that

$$\begin{aligned} 0<\partial _{t}\Phi _{\omega _1,\omega _2}(x,t)\le L_3:=(D+d)K\text { and }| \partial _{tt}\Phi _{\omega _1,\omega _2} (x,t)|\le L_4:= [2D+d+L_1]L_3, \end{aligned}$$

for $(x,t)\in \mathbb {R}^2$. Similarly, we have $|\Psi _{tt}(x,t)|\le 2L_4/c^2$ for $(x,t)\in \mathbb {R}^2$. Then, by the interpolation $\Vert \cdot \Vert _{C^1}\le 2\sqrt{\Vert \cdot \Vert _{C^0}\Vert \cdot \Vert _{C^2}}$, we obtain

$$\begin{aligned} \lim _{t\rightarrow -\infty }\sup _{x\in \mathbb {R}}\big \Vert \Phi _{\omega _1,\omega _2}(x,\cdot )-\Psi (x,\cdot ) \big \Vert _{C^1((-\infty ,t])}=0. \end{aligned}$$

Thus, there exists a $T_3\le T_2$ such that

$$\begin{aligned} \sup _{x\in \mathbb {R}}\big \Vert \Phi _{\omega _1,\omega _2}(x,\cdot )-\Psi (x,\cdot ) \big \Vert _{C^1((-\infty ,t])}<\kappa _1/2,\quad \text{ for } \text{ any } t\le T_3. \end{aligned}$$

(4.14)

Since

$$\begin{aligned} \lim _{|x|\rightarrow +\infty } \max _{t\in [T_3,T_1]} \left| \int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)b'\big (\Phi _{\omega _1,\omega _2}(x-y,t-s) \big )dyds -b'(K)\right| =0, \end{aligned}$$

we can take $X_1>0$ such that (4.11) holds for any $|x|>X_1$ and $t\in [T_3,T_1]$. In addition, let $\kappa _2:=\min \limits _{|x|\le X_1,t\in [T_3,T_1] }\partial _t\Phi _{\omega _1,\omega _2}(x,t)>0$ and take $\sigma _0>\sigma _1$ such that

$$\begin{aligned} \kappa _2\sigma _0\rho _0-\rho _0+d- e^{\rho _0\tau }L_1>0. \end{aligned}$$

(4.15)

Then, for $\gamma \in \mathbb {R}$, $\delta \in (0,\delta _0]$ and $\sigma \ge \sigma _0$, we denote $\xi (t):=t+\gamma + \sigma \delta (1-e^{- \rho _0t})$. Clearly, $\xi (t-s)\le \xi (t)-s$ for $t\ge 0$ and $s\in [0,\tau ]$. Since ${\partial _t}\Phi _{\omega _1,\omega _2}(x,t)>0$ for $(x,t)\in \mathbb {R}^2$ and $|b'(u)-b'(v)|\le L_2|u-v|$ for $u,v\in [0,2K],$ direct computations show that

$$\begin{aligned}&\mathcal {F}(u^+)(x,t) \nonumber \\&\quad := u^+_t-D \big [(J*u^+)(x,t)-u^+(x,t)\big ]+d u^+- \big (G\star b(u^+)\big )(x,t) \nonumber \\&\quad = \partial _t \Phi _{\omega _1,\omega _2}(x,\xi (t)) \big (1+\sigma \delta \rho _0 e^{- \rho _0t}\big ) -\rho _0 \delta e^{- \rho _0t}\nonumber \\&\qquad -D\Big [\big (J*\Phi _{\omega _1,\omega _2}\big )(x,\xi (t))-\Phi _{\omega _1,\omega _2}(x,\xi (t))\Big ]\nonumber \\&\qquad + d\Phi _{\omega _1,\omega _2}(x,\xi (t))+d\delta e^{- \rho _0t}-G\star b\big (u^+\big )(x,t)\nonumber \\&\quad = \delta e^{- \rho _0t} \big [\sigma \rho _0 \partial _t\Phi _{\omega _1,\omega _2}(x,\xi (t))-\rho _0+d \big ]+ \big (G\star b(\Phi _{\omega _1,\omega _2})\big )(x,\xi (t)) \nonumber \\&\qquad - \int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)b\big (\Phi _{\omega _1,\omega _2}(x-y,\xi (t-s)) +\delta e^{- \rho _0(t-s)} \big )dyds \nonumber \\&\quad \ge \delta e^{- \rho _0t} \Big [\sigma \rho _0 \partial _t\Phi _{\omega _1,\omega _2}(x,\xi (t))-\rho _0+d \Big ]\nonumber \\&\qquad + \int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)b\big (\Phi _{\omega _1,\omega _2}(x-y,\xi (t)-s) \big )dyds\nonumber \\&\qquad - \int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)b\big (\Phi _{\omega _1,\omega _2}(x-y,\xi (t)-s) +\delta e^{- \rho _0(t-s)} \big )dyds \nonumber \\&\quad \ge \delta e^{- \rho _0t} \Big [\sigma \rho _0 \partial _t \Phi _{\omega _1,\omega _2}(x,\xi (t))-\rho _0+d \nonumber \\&\qquad \quad - e^{\rho _0 \tau }\int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)b'\big (\Phi _{\omega _1,\omega _2}(x-y,\xi (t)-s) +\theta _1\delta e^{- \rho _0(t-s)} \big )dyds \Big ] \end{aligned}$$

(4.16)

$$\begin{aligned}&\quad \ge \delta e^{- \rho _0t} \left\{ \sigma \rho _0 \partial _t\Phi _{\omega _1,\omega _2}(x,\xi (t))-\rho _0+d -e^{\rho _0 \tau } \Big [ \delta e^{\rho _0 \tau }L_2 \nonumber \right. \\&\left. \qquad \quad +\int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)b'\big (\Phi _{\omega _1,\omega _2}(x-y,\xi (t)-s) \big )dyds \Big ]\right\} \nonumber \\&\quad \ge \delta e^{- \rho _0t} \left\{ -\rho _0+d -e^{\rho _0 \tau } \Big [ \frac{\delta _1}{4}+\int _0^\tau \int _{-\infty }^{+ \infty }G(y,s)b'\big (\Phi _{\omega _1,\omega _2}(x-y,\xi (t)-s) \big )dyds \Big ]\right\} , \end{aligned}$$

(4.17)

where $\theta _1\in (0,1)$. Moreover, for $\xi (t)\le T_3$, Lemmas 3.2 and 3.3 imply that

$$\begin{aligned}&\ \max \big \{ U(x-y+c\xi (t)-cs+\omega _1),U(-x+y+c\xi (t)-cs+\omega _2)\big \} \nonumber \\&\quad \le \Phi _{\omega _1,\omega _2}(x-y,\xi (t)-s) \le U(x-y+p_1(\xi (t)-s))+U(-x+y+p_2(\xi (t)-s)) \nonumber \\&\quad \le 2\max _{i=1,2}\big |p_i(\xi (t)-s)-c(\xi (t)-s) -\omega _i\big |\max _{x\in \mathbb {R}}U'(x) \nonumber \\&\qquad + U(x-y+c\xi (t)-cs+\omega _1)+U(-x+y+c\xi (t)-cs+\omega _2) . \end{aligned}$$

(4.18)

Now, we consider the following seven cases.

• (1) $\xi (t)>T_1$. Following (4.5), (4.11) and (4.17), we have $\mathcal {F}(u^+)(x,t) \ge 0$.

• (2) $\xi (t)\le T_3$ and $x+c\xi (t)+\omega _1 \ge X$. Then $-x<-X$. It follows from (4.7) and (4.17) that

  $$\begin{aligned} \mathcal {F}(u^+)(x,t)\ge&\delta e^{- \rho _0t} \Big \{-\rho _0+d-e^{\rho _0 \tau } \big [ {\delta _1}/{2}+ \nonumber \\&\quad \qquad \int _0^\tau \int _{-M}^{ M}G(y,s)b'\big (\Phi _{\omega _1,\omega _2}(x-y,\xi (t)-s)\big )dyds \big ]\Big \}. \end{aligned}$$

  (4.19)

  Moreover, from (4.10), (4.12) and (4.18), we have

  $$\begin{aligned} \Phi _{\omega _1,\omega _2}(x-y,\xi (t)-s)\in (K-\nu ,K+\nu ), \hbox { for } y\in [-M,M] \hbox { and } s\in [0,\tau ]. \end{aligned}$$

  If $b'(K)>0$, it then follows from (4.6) and (4.8) that

  $$\begin{aligned} \int _0^\tau \int _{-M}^{ M}G(y,s)b'\big (\Phi _{\omega _1,\omega _2}(x-y,\xi (t)-s) \big )dyds \in \big [b'(K)-{3\delta _1}/{4}, b'(K)+{\delta _1}/{2}\big ] . \end{aligned}$$

  Moreover, if $b'(K)=0$, then

  $$\begin{aligned} \int _0^\tau \int _{-M}^{ M}G(y,s)b'\big (\Phi _{\omega _1,\omega _2}(x-y,\xi (t)-s) \big )dyds \in \big [b'(K), b'(K)+{\delta _1}/{2}\big ] . \end{aligned}$$

  By (4.5) and (4.19), we conclude that $\mathcal {F}(u^+)(x,t) \ge 0$.

• (3) $\xi (t)\le T_3$ and $-x+c\xi (t)+\omega _2 \ge X$. Similar to case (2), we can prove that $\mathcal {F}(u^+)(x,t) \ge 0$.

• (4) $\xi (t)\le T_3$, $x+c\xi (t)+\omega _1 \le -X$ and $-x+c\xi (t)+\omega _2 \le -X$. Using (4.10), (4.12) and (4.18), we have

  $$\begin{aligned} \Phi _{\omega _1,\omega _2}(x-y,\xi (t)-s)\in (0,\nu ), \quad \hbox { for } y\in [-M,M] \hbox { and } s\in [0,\tau ]. \end{aligned}$$

  Similar to case (2), we can show that $\mathcal {F}(u^+)(x,t) \ge 0$ by using (4.5), (4.6), (4.9) and (4.19).

• (5) $\xi (t)\le T_3$ and $x+c\xi (t)+\omega _1 \in [-X,X]$ or $-x+c\xi (t)+\omega _2 \in [-X,X]$. According to (4.14), we have

  $$\begin{aligned} \partial _t\Phi _{\omega _1,\omega _2}(x,\xi (t))&\ge \partial _t\Psi (x,\xi (t))-\frac{\kappa _1}{2}\\&= U'(x+c\xi (t)+\omega _1)+U'(-x+c\xi (t)+\omega _2 )-\frac{\kappa _1}{2}\ge \frac{\kappa _1}{2}. \end{aligned}$$

  Then, by (4.13) and (4.16), we obtain $\mathcal {F}(u^+)(x,t) \ge 0$.

• (6) $T_3\le \xi (t)\le T_1$ and $|x|>X_1$. Noting that (4.11) holds for any $|x|>X_1$ and $t\in [T_3,T_1]$, it follows from (4.5), (4.11) and (4.17) that $\mathcal {F}(u^+)(x,t) \ge 0$.

• (7) $T_3\le \xi (t)\le T_1$ and $|x|\le X_1$. Following (4.15) and (4.16), it must be $\mathcal {F}(u^+)(x,t) \ge 0$.

Summing up the above seven cases, we see that $\mathcal {F}(u^+)(x,t)\ge 0$ for $(x,t)\in \mathbb {R}\times [0,+\infty )$, i.e. $u^+(x,t)$ is a supersolution of (1.2) on $[0,+\infty )$. The proof is complete. $\square$

Lemma 4.3

There exist $\delta _*>0$, $\rho _*>0$ and $\sigma _*>0$ such that for any $\gamma \in \mathbb {R}$, $\delta \in (0,\delta _*]$ and $\sigma \ge \sigma _*$, the functions $V^\pm (x,t)$ defined by

$$\begin{aligned} V^\pm (x,t):= U\Big (-x+ct+\gamma \pm \sigma \delta \big (1-e^{- \rho _*t}\big )\Big ) \pm \delta e^{- \rho _*t} \end{aligned}$$

constitute a pair of super- and subsolutions of (1.2) on $[0,+\infty )$.

Proof

The proof is similar to that of Lemma 4.2, we omit it. $\square$

Let $\sigma _0$, $\rho _0$, $\delta _0$ and $\sigma _*$, $\rho _*$, $\delta _*$ be the positive constants given in Lemmas 4.2 and 4.3, respectively. We have the following results.

Theorem 4.4

Let $\Phi _{\omega _1,\omega _2}(x,t)$ be the entire solution of (1.2) decided in Theorem 4.1, then the following statements hold.

$\mathrm (1)$ :

If $\widetilde{\Phi }(x,t)$ is an entire solution of (1.2) satisfying the first property of (3) of Theorem 4.1, then $\widetilde{\Phi }(x,t)= \Phi _{\omega _1,\omega _2}(x,t)$.

$\mathrm (2)$ :

For any $(x,t)\in \mathbb {R}^2$, $\Phi _{\omega _1,\omega _2}(x,t) \text { converges to }\Big \{ \begin{array}{ll} \Phi _{\omega _2}^+(x,t) \ as\ \omega _1\rightarrow -\infty ;\\ \Phi _{\omega _1}^-(x,t) \ as\ \omega _2\rightarrow -\infty . \end{array}$

$\mathrm (3)$ :

$\Phi _{\omega _1,\omega _2}(x,t)$ depends continuously on $(\omega _1,\omega _2)\in (-\infty ,\bar{\omega }]^2$.

$\mathrm (4)$ :

$\Phi _{\omega _1,\omega _2}(x,t)$ is Liapunov stable in the sense of part $\mathrm (7)$ of Theorem 1.1.

Proof

(1) Suppose that $\widetilde{\Phi }(x,t)$ is an entire solution of (1.2) satisfying the first property of (3) of Theorem 4.1. Given any $t_1<0$, we define

$$\begin{aligned} \eta := \sup _{x \in \mathbb {R}}\big \Vert \widetilde{\Phi }(x,\cdot +t_1) -\Phi _{\omega _1,\omega _2}(x,\cdot +t_1)\big \Vert _{L^\infty [-\tau ,0]}. \end{aligned}$$

It suffices to show that $\eta =0.$ By our assumptions, for any $\delta \in (0,\delta _0]$, there exist a $t_2<t_1-\tau$ such that $\sup _{x \in \mathbb {R} }\Vert \widetilde{\Phi }(x,\cdot +t_2)-\Phi _{\omega _1,\omega _2}(x,\cdot +t_2)\Vert _{L^\infty [-\tau ,0]}<\delta .$ Hence,

$$\begin{aligned}&\Phi _{\omega _1,\omega _2}\big (x,s+t_2- \sigma _0\delta \left( e^{ \rho _0\tau }-e^{- \rho _0s}\right) \big ) -\delta e^{- \rho _0s}\\&\quad \le \widetilde{\Phi }(x,s+t_2)\le \Phi _{\omega _1,\omega _2}\big (x,s+t_2+ \sigma _0\delta \left( e^{ \rho _0\tau }-e^{- \rho _0s}\right) \big ) +\delta e^{- \rho _0s}, \end{aligned}$$

for $x\in \mathbb {R}, \ s\in [-\tau ,0].$ In addition, by Lemmas 2.2 and 4.2, we have

$$\begin{aligned}&\Phi _{\omega _1,\omega _2}\big (x,t+t_2- \sigma _0\delta \left( e^{ \rho _0\tau }-e^{- \rho _0t}\right) \big )- \delta e^{- \rho _0t}\\&\quad \le \widetilde{\Phi }(x,t+t_2) \le \Phi _{\omega _1,\omega _2}\big (x,t+t_2+ \sigma _0\delta \left( e^{ \rho _0\tau }-e^{- \rho _0t}\right) \big )+ \delta e^{- \rho _0t} \end{aligned}$$

for $x\in \mathbb {R}$ and $t\ge 0$. Noting that $s+t_1-t_2>0$, then we obtain

$$\begin{aligned}&\Phi _{\omega _1,\omega _2}\big (x,s+t_1- \sigma _0\delta (e^{ \rho _0\tau }-e^{ \rho _0(s+t_1-t_2)})\big )- \delta \\&\quad \le \widetilde{\Phi }(x,s+t_1)\le \Phi _{\omega _1,\omega _2}\big (x,s+t_1+ \sigma _0\delta (e^{ \rho _0\tau }-e^{ \rho _0(s+t_1-t_2)})\big )+ \delta . \end{aligned}$$

Therefore, for all $x\in \mathbb {R}$, it follows that

$$\begin{aligned} \Phi _{\omega _1,\omega _2}\big (x,s+t_1- \sigma _0\delta e^{ \rho _0\tau }\big )- \delta \le \widetilde{\Phi }(x,s+t_1) \le \Phi _{\omega _1,\omega _2}\big (x,s+t_1+ \sigma _0\delta e^{ \rho _0\tau }\big )+ \delta . \nonumber \\ \end{aligned}$$

(4.20)

On the other hand, since $0< \partial _t\Phi _{\omega _1,\omega _2}(x,t)\le M_3:=(D+d)K$ for any $(x,t)\in \mathbb {R}^2$, then (4.20) implies that

$$\begin{aligned} \sup _{x\in \mathbb {R}}\left\| \widetilde{\Phi }(x,s+t_1)- \Phi _{\omega _1,\omega _2}(x,s+t_1) \right\| _{L^\infty [-\tau ,0]}\le \left( 1+M_3\sigma _0e^{ \rho _0\tau }\right) \delta . \end{aligned}$$

Thus we have $\eta \le (1+M_3\sigma _0e^{ \rho _0\tau })\delta$. By the arbitrariness of $\delta$, we see that $\eta =0$.

(2) Let $\{(\omega _1,\omega _{2}^k)\}_{k\in \mathbb {N}}$ be a sequence satisfying $(\omega _1,\omega _{2}^k)\in (-\infty ,\bar{\omega }]^2$, $\omega _{2}^{k+1}<\omega _{2}^k<\omega _1$ and $\omega _{2}^k\rightarrow -\infty$ as $k\rightarrow -\infty$. According to Theorem 4.1, for each $k\in \mathbb {N}$, there exist an entire solution $\Phi _{\omega _1,\omega _{2}^k}(x,t)$ of (1.2) such that for any $x\in \mathbb {R}$ and $t<T$, there holds

$$\begin{aligned} \Phi _{\omega _1}^-(x,t)&\le \max \left\{ \Phi _{\omega _1}^-(x,t),\Phi _{\omega _{2}^k}^+(x,t) \right\} \le \Phi _{\omega _1,\omega _{2}^{k+1}}(x,t)\le \Phi _{\omega _1,\omega _{2}^k}(x,t) \nonumber \\&\le U \left( -x+p_1\big (t;\omega _1,\omega _{2}^k\big )\right) + U\left( x+p_2\big (t;\omega _1,\omega _{2}^k\big )\right) \nonumber \\&= U \left( -x+\widetilde{p}_1(t;\omega _1)\right) + U\left( x+\widetilde{p}_2\big (t;\omega _1,\omega _{2}^k\big )\right) . \end{aligned}$$

(4.21)

By the monotonicity of $\Phi _{\omega _1,\omega _{2}^k}(x,t)$ on $k$, there exists a function $\Psi (x,t)$ such that $\lim \limits _{k\rightarrow +\infty }\Phi _{\omega _1,\omega _{2}^k}(x,t)=\Psi (x,t)$. It then follows from (4.21) that

$$\begin{aligned} \Phi _{\omega _1}^-(x,t)\le \Psi (x,t)\le U\big (-x+\widetilde{p}_1(t;\omega _1)\big ),\quad \text{ for } \text{ any } x\in \mathbb {R} \text{ and } t<T. \end{aligned}$$

(4.22)

Moreover, given any $t_3<T$, we define

$$\begin{aligned} \bar{\eta }:= \sup _{x\in \mathbb {R}}\big \Vert \Psi (x,t_3+\cdot )-U(-x+c(t_3+\cdot )+\omega _1)\big \Vert _{L^\infty [-\tau ,0]}. \end{aligned}$$

For any $\delta \in (0,\delta _*]$, since $\widetilde{p}_1(t;\omega _1)-ct- \omega _1\rightarrow 0$ as $t\rightarrow -\infty$, it follows from (4.22) that there exists $t_4<t_3-\tau$ such that for any $x\in \mathbb {R}$ and $s\in [-\tau ,0]$, there holds

$$\begin{aligned} \Phi _{\omega _1}^-(x,s+t_4)\le \Psi (x,s+t_4)\le U\left( -x+c(s+t_4)+\omega _1+\sigma \delta \big (e^{\rho _*\tau }-e^{- \rho _*s}\big )\right) + \delta e^{- \rho _*s}. \end{aligned}$$

By comparison principle and Lemma 4.3, we have

$$\begin{aligned} \Phi _{\omega _1}^-(x,t)\le \Psi (x,t ) \le U\left( -x+ct+\omega _1+ \sigma \delta \left( e^{\rho _*\tau }-e^{- \rho _*(t-t_4)}\right) \right) + \delta e^{- \rho _*(t-t_4)}, \end{aligned}$$

for any $x\in \mathbb {R}$ and $t>t_4$. Then it follows that

$$\begin{aligned} \Phi _{\omega _1}^-(x,t_3+s)\le&\Psi (x, t_3+s) \\ \le&U\left( -x+c(t_3+s)+\omega _1+ \sigma \delta \left( e^{\rho _*\tau }-e^{- \rho _*(t_3+s-t_4)}\right) \right) + \delta e^{- \rho _*(t_3+s-t_4)}\\ \le&U\left( -x+c(t_3+s)+\omega _1+ \sigma \delta e^{\rho _*\tau }\right) + \delta ,\quad \text{ for } x\in \mathbb {R}, \end{aligned}$$

which implies that

$$\begin{aligned} \sup _{x\in \mathbb {R}}\big \Vert \Psi (x,(t_3+\cdot ))-\Phi _{\omega _1}^-(x,t_3+\cdot )\big \Vert _{L^\infty [-\tau ,0]}\le \delta +\sigma \delta e^{\rho _*\tau }\max _{ z\in \mathbb {R}}U'(z) . \end{aligned}$$

According to the arbitrariness of $\delta$, we obtain $\bar{\eta }=0$. Thus, $\Psi (x,t)=\Phi _{\omega _1}^-(x,t)$ for any $(x,t)\in \mathbb {R}^2$. Since $\Phi _{\omega _1,\omega _2}(x,t)$ is increasing with respect to $\omega _2$, we obtain

$$\begin{aligned} \lim \limits _{\omega _2\rightarrow -\infty }\Phi _{\omega _1,\omega _2}(x,t)=\Phi _{\omega _1}^-(x,t), for\,any\,(x,t)\in \mathbb {R}^2. \end{aligned}$$

Similarly, we can show the other assertion of this part.

(3) Given any $(\omega _1^0,\omega _2^0)\in (-\infty ,\bar{\omega }]^2$, we choose two sequences $\{(\omega _{\pm ,1}^k,\omega _{\pm ,2}^k)\}$ with $(\omega _{\pm ,1}^k,\omega _{\pm ,2}^k)\in \mathbb {R}^2$ such that $\lim \limits _{k\rightarrow +\infty }(\omega _{\pm ,1}^k,\omega _{\pm ,2}^k)\rightarrow (\omega _1^0,\omega _2^0)$ and

$$\begin{aligned} \big (\omega _{-,1}^k,\omega _{-,2}^k\big )\le \big (\omega _{-,1}^{k+1},\omega _{-,2}^{k+1}\big )< \big (\omega _1^0,\omega _2^0\big ) < \big (\omega _{+,1}^{k+1},\omega _{+,2}^{k+1}\big ) \le \big (\omega _{+,1}^k,\omega _{+,2}^k\big ), \quad \forall k\in \mathbb {N}. \end{aligned}$$

From Theorem 4.1, there exist entire solutions $\Phi _{\omega _1^0,\omega _2^0}(x,t)$ and $\Phi _{\omega _{\pm ,1}^k,\omega _{\pm ,2}^k}(x,t)$ of (1.2) for $k\in \mathbb {N}$ which satisfy

$$\begin{aligned} 0\le \Phi _{\omega _{-,1}^{k+1},\omega _{-,2}^{k+1}}\le \Phi _{\omega _{-,1}^k,\omega _{-,2}^k}\le \Phi _{\omega _1^0,\omega _2^0}\le \Phi _{\omega _{+,1}^k,\omega _{+,2}^k}\le \Phi _{\omega _{+,1}^{k+1},\omega _{+,2}^{k+1}}\le K \end{aligned}$$

for $(x,t)\in \mathbb {R}^2$. Then, there exist $\Phi _\pm (x,t)$ such that $\lim _{k\rightarrow \infty }\Phi _{\omega _{\pm ,1}^k,\omega _{\pm ,2}^k}(x,t)= \Phi _\pm (x,t)$ and $\Phi _\pm (x,t)$ are entire solutions of (1.2). Since $0<U'(z)\le M_3/c$ for $z\in \mathbb {R}$, $0<p_i(t;\omega _{+,1}^k,\omega _{+,2}^k)-c_it-\omega _{+,i}^k \le R_0e^{c\lambda _{12} t}$ for $t\le 0$ and

$$\begin{aligned} \max \big \{\Phi _{\omega _1^0}^-(x,t),\Phi _{\omega _2^0}^+(x,t) \big \} \le&\max \big \{\Phi _{\omega _{+,1}^k}^-(x,t),\Phi _{\omega _{+,2}^k}^+(x,t)\big \} \le \Phi _{\omega _{+,1}^k,\omega _{+,2}^k}(x,t)\\ \le&U\big (-x+p_1\big (t;\omega _{+,1}^k,\omega _{+,2}^k\big )\big )+U\big (x+p_2\big (t;\omega _{+,1}^k,\omega _{+,2}^k\big )\big ) \end{aligned}$$

for any $t<T$, $x\in \mathbb {R}$ and $k\in \mathbb {N}$, we can easily show that

$$\begin{aligned} \lim _{t\rightarrow -\infty }\big \{\sup _{x\le 0}|\Phi _+(x,t)-\Phi _{\omega _1^0}^-(x,t)| +\sup _{x\ge 0}|\Phi _+(x,t)-\Phi _{\omega _2^0}^+(x,t)|\big \}=0. \end{aligned}$$

By the uniqueness of entire solutions, we have $\Phi _+(x,t)=\Phi _{\omega _1^0,\omega _2^0}(x,t)$. Similarly, one can prove that $\Phi _-(x,t)=\Phi _{\omega _1^0,\omega _2^0}(x,t)$. Hence, we can easily show that $\Phi _{\omega _1,\omega _2}(x,t)$ depends continuously on $(\omega _1,\omega _2)$.

(4) Given any $\epsilon >0$, let us define $\tilde{\delta } :=\tilde{\delta } (\epsilon )=\epsilon /(2M_3)>0$. Then, for all $|z|\le \tilde{\delta }$, it follows that

$$\begin{aligned} \sup _{x ,t\in \mathbb {R}}\big |\Phi _{\omega _1,\omega _2}(x,t)-\Phi _{\omega _1,\omega _2}(x,t+z)\big |\le \sup _{x ,t\in \mathbb {R}}\big |\partial _t\Phi _{\omega _1,\omega _2}(x,t)\big ||z|\le M_3\tilde{\delta } \le {\epsilon }/{2}. \nonumber \\ \end{aligned}$$

(4.23)

Let $\bar{\delta }:=\min \{\epsilon /2,\tilde{\delta } /(\sigma _0e^{ \rho _0\tau }),\delta _0\}$. For any $\varphi \in \mathcal {C}_{[{ 0},{ {K}}]}$ satisfying

$$\begin{aligned} \sup _{x\in \mathbb {R}}\left\| \varphi (x,\cdot )-\Phi _{\omega _1,\omega _2}(x+x_0,\cdot +t_0)\right\| _{L^\infty [-\tau ,0]}<\bar{\delta }, \end{aligned}$$

we have

$$\begin{aligned}&\ \Phi _{\omega _1,\omega _2}\big (x+x_0,s+t_0-\sigma _0\bar{\delta }\left( e^{ \rho _0\tau }-e^{- \rho _0s}\right) \big )-\bar{\delta }e^{- \rho _0s}\\&\quad \le \varphi (x,s)\le \Phi _{\omega _1,\omega _2}\big (x+x_0,s+t_0+\sigma _0\bar{\delta }\left( e^{ \rho _0\tau }-e^{- \rho _0s}\right) \big )+\bar{\delta }e^{- \rho _0s} \end{aligned}$$

for $x\in \mathbb {R}$ and $s\in [-\tau ,0]$. By comparison principle and Lemma 4.2, we obtain

$$\begin{aligned}&\Phi _{\omega _1,\omega _2}\big (x+x_0,t+t_0- \sigma _0\bar{\delta }\left( e^{ \rho _0\tau }-e^{- \rho _0t}\right) \big )- \bar{\delta }e^{- \rho _0t} \nonumber \\&\quad \le u(x,t;\varphi ) \le \Phi _{\omega _1,\omega _2}\big (x+x_0,t+t_0+ \sigma _0\bar{\delta }\left( e^{ \rho _0\tau }-e^{- \rho _0t}\right) \big )+ \bar{\delta }e^{- \rho _0t} \end{aligned}$$

(4.24)

for $x\in \mathbb {R}$ and $t\ge 0$. It then follows form (4.23) and (4.24) that

$$\begin{aligned} \big |u(x,t;\varphi ) - \Phi _{\omega _1,\omega _2}(x+x_0,t+t_0)\big |\le M_3\sigma _0\bar{\delta }e^{ \rho _0\tau }+\bar{\delta }\le \epsilon , \end{aligned}$$

(4.25)

for all $x\in \mathbb {R}$ and $t\ge 0.$ The proof is complete. $\square$

Based on the results of the previous subsection, we are ready to prove Theorem 1.1.

4.2 Proof of Theorem 1.1

For any $\theta _1,\theta _2\in \mathbb {R}$, there exists a $T_0<0$ such that $c T_0+\theta _1< \bar{\omega }$ and $c T_0+\theta _2< \bar{\omega }$. Take

$$\begin{aligned} \omega _1:=c T_0+\theta _1 \text { and } \omega _2:=c T_0+\theta _2. \end{aligned}$$

Clearly, $(\omega _1,\omega _2)\in (-\infty , \bar{\omega }]^2$. Therefore, there exists an entire solution $\Phi _{\omega _1,\omega _2}(x,t)$ satisfying the assertions of Theorems 4.1 and 4.4. Let

$$\begin{aligned} \Phi _{\theta _1,\theta _2}(x,t):=\Phi _{\omega _1,\omega _2}(x,t-T_0), \end{aligned}$$

then $\Phi _{\theta _1,\theta _2}(x,t)$ is also an entire solution of (1.2) which satisfies the assertions of Theorem 1.1.