Travelling Wavefronts in Nonlocal Diffusion Equations with Nonlocal Delay Effects

Abstract

This paper deals with the existence, monotonicity, uniqueness, and asymptotic behaviour of travelling wavefronts for a class of temporally delayed, spatially nonlocal diffusion equations.

1 Introduction

Travelling wavefront solutions play an important role in the description of the long-term behaviour of solutions to initial value problems in reaction–diffusion equations, both in the spatially continuous case and in spatially discrete situations. Such solutions are also of interest in their own right, for example to understand transitions between different states of a physical system, propagation of patterns, and domain invasion of species in population biology (see, e.g. [3, 4, 8, 13, 37]). In this paper, we study the existence, uniqueness, and asymptotic stability of travelling wavefronts of the equation:

$$\begin{aligned} u_t(x,t)=p u_{xx}(x,t)+{\hbox {d}}(J*u-u)(x,t)+f\left( u(x,t),(h**u)(x,t)\right) , \end{aligned}$$

(1)

where $x\in {\mathbb {R}}, d\ge 0, p\ge 0$, and

$$\begin{aligned} \begin{aligned} (J*u)(x,t):=&\int _{\mathbb {R}}J(x-y)u(y,t){\hbox {d}}y,\\ (h**u)(x,t):=&\int ^{t}_{-\infty }\int _{\mathbb {R}}h(x-y,t-s)u(y,s){\hbox {d}}y{\hbox {d}}s. \end{aligned} \end{aligned}$$

Equation (1) mixes a continuous Laplacian with a nonlocal diffusion $d(J*u-u)(x,t)$, which describes that the diffusion of density u at a point x and time t depends not only on u(x, t) but also on all the values of u in a neighbourhood of x through the convolution term $J*u$. In population dynamics, the reaction term $f(u, h**u)$ is usually used to describe the recruits of population, and $h**u$ represents a weighted average of the population density in both past time and space [6, 7]. The nonlinear functions f(u, v) and h(u) satisfy the following hypotheses:

• (F1) $f\in C([0,K]\times [0,K],{\mathbb {R}}), f(0,0)=f(K,K)=0$, $f(u,h**u)>0$ for all $u\in (0,K), \partial _2f(u,v)\ge 0$ for all $(u,v)\in [0,K]\times [0,K]$, where K is a positive constant.

• (F2) There exist some $M>0$ and $\sigma \in (0,1]$ such that $0\le \partial _1f(0,0)u+\partial _2f(0,0)v-f(u,v)\le M(u+v)^{1+\sigma }$ for all $(u,v)\in [0,K]\times [0,K]$ and $\partial _1f(K,K)+\partial _2f(K,K)<0$.

• (H1) Both $J(\cdot )$ and $h (\cdot ,t)$ are nonnegative, even, integrable, and satisfies $\int _{\mathbb {R}}J(x)dx=1$ and $\int _0^{\infty }\int _{\mathbb {R}}h(x,t)dxdt=1$.

• (H2) There exists some $\lambda _0>0$ (possibly equal to $\infty$) such that $\int ^{\infty }_0J(x)\exp \{\lambda x\}dx<\infty$ and $\int _0^{\infty }\int _{0}^{\infty }h(x,t)\exp \{\lambda (x-ct)\}dxdt<\infty$ for all $c\ge 0$ and $\lambda \in [0,\lambda _0)$.

Assumptions (F1) and (F2) are standard. From (F1), we can see that (1) has two equilibria 0 and K. Furthermore, condition (F2) together with (F1) and (H1) implies that $\partial _1f(0,0)+\partial _2f(0,0)\ge \frac{2}{K}f(\frac{K}{2},\frac{K}{2})=\frac{2}{K}f(\frac{K}{2},h**(\frac{K}{2}))>0$; hence, 0 is unstable and K is stable. In this article, we will not require that $\partial _2f(0,0)> 0$.

Since Eq. (1) involves a general diffusion kernel and delayed nonlinearity, it can be reduced to some well-known equations if J, h, and f are chosen to take some special form (see, e.g. [2, 16, 35, 39, 40, 43, 44]). In particular, special cases of (1) include a host–vector disease model, a nonlocal population model with age structure, and a nonlocal Nicholson’s blowflies model with delay; these cases are discussed in a second paper investigating the stability of the system [20]. For example, choosing $d=0, f(u,v)=-\tau u+\tau \beta ve^{-v}$, Eq. (1) can be reduced to the following Nicholson’s Blowflies equation with spatiotemporal delays

$$\begin{aligned} u_t(x,t)=p u_{xx}(x,t)-\tau u(x,t)+\tau \beta (h**u)(x,t)\exp \{-(h**u)(x,t)\}, \end{aligned}$$

which was studied by Li, Ruan, and Wang [26], and Lin [27]. If $p=0$ and $J(x)=\frac{1}{2}[\delta (-1)+\delta (1)]$ and $h(x,t)=k(x)\delta (t-\tau )$, where $\delta (\cdot )$ is the Dirac delta function, then (1) reduces to the discrete reaction–diffusion equation

$$\begin{aligned} u_t(x,t)= d\cdot \Delta _1u(x,t)+f \left( u(x,t),(k*u)(x,t-\tau )\right) , \end{aligned}$$

(2)

where $\Delta _1u(x,t)=\frac{1}{2}[u(x+1,t)-2u(x,t)+u(x-1,t)]$. If $f(u,v)=-au+b(v)$ and $k(x)=\delta (x)$, then (2) reduces to the local equation

$$\begin{aligned} u_t(x,t)=d\cdot \Delta _1u(x,t)-au(x,t)+b(u(x,t-\tau )),\quad x\in {\mathbb {R}},\,t\ge 0, \end{aligned}$$

(3)

where $b\in C^1([0,\infty ],{\mathbb {R}})$. If $f(u,v)=g(u)$ and g(u) denotes a Lipschitz continuous function satisfying $g(u)>0=g(0)=g(1)$ for all $u\in (0,1)$, Eq. (2) becomes

$$\begin{aligned} u_t(x,t)= d\cdot \Delta _1u(x,t)+g(u(x,t)). \end{aligned}$$

(4)

On the other hand, when $d=0$ and $h(x,t) =k(x)\delta (t-\tau )$, (1) reduces to the following reaction–diffusion equation with discrete time delay

$$\begin{aligned} u_t(x,t)= pu_{xx}(x,t)+f \left( u(x,t),(k*u)(x,t-\tau )\right) . \end{aligned}$$

(5)

Moreover, Eq. (2) is a spatially discrete version of (5) with p replaced by d. In recent years, spatially nonlocal differential equations such as (5) have attracted significant attention (see, e.g. [15, 19, 33, 37,38,39]). Under some monostable assumption, Wang et al. [39] investigated the existence, uniqueness, and global asymptotical stability of travelling wavefronts. We also refer to So et al. [37] for more details and some specific forms of f, obtained from integrating along characteristics of a structured population model, an idea from the work of Smith and Thieme [36]. See also [37] for a similar model and [18] for a survey on the history and the current status of the study of reaction–diffusion equations with nonlocal delayed interactions. In particular, when $f(u,v)= v(1-u)$ and $k(u)=\delta (u)$, Eq. (5) is delayed Fisher’s equation [17] or KPP equation [25], which arises in the study of gene development or population dynamics. When $f(u,v)=-au+b(v)$ and $k(u)=\delta (u)$, Eq. (5) is the local Nicholson’s blowflies equation and has been investigated in [19, 21, 28, 32]. When $f(u,v)=-au+b(1-u)v$, Eq. (5) is called the vector disease model as proposed by Ruan and Xiao [34]. When $f(u,v)=bv\exp \{-\gamma \tau \}-\delta u^2$ and $k(u)=\frac{1}{\sqrt{4\pi \alpha \tau }}\exp \{\frac{-y^2}{4\alpha \tau }\}$, Eq. (5) is the age-structured reaction–diffusion model of a single species proposed by Al-Omari and Gourley [1]. Existence and stability of travelling wavefronts for the reaction–diffusion equation (5) and its special forms have been extensively studied in the literature.

We are interested in wave propagation phenomena. In particular, we are interested in monotone travelling waves $u(x, t) = \phi (x + ct)$ for (1), with $\phi$ saturating at 0 and K. We call c the travelling wave speed and $\phi$ the profile of the wavefront. In order to address these questions, we need to find an increasing function $\phi (\xi )$, where $\xi = x + ct$, which is a solution of the associated travelling wave equation

$$\begin{aligned} \begin{aligned}&-c\phi '(\xi )+p\phi ''(\xi )+ d(J*\phi -\phi )(\xi )+f(\phi (\xi ), (h**\phi )(\xi ))=0,\quad \xi \in {\mathbb {R}},\\&\quad \lim \limits _{\xi \rightarrow -\infty }\phi (\xi )=0,\qquad \lim \limits _{\xi \rightarrow \infty }\phi (\xi )=K,\\&\qquad 0\le \phi (\xi )\le K,\quad \xi \in {\mathbb {R}}, \end{aligned} \end{aligned}$$

(6)

where

$$\begin{aligned} \begin{aligned} (J*\phi )(\xi )=&\int _{\mathbb {R}}J(y)\phi (\xi -y){\hbox {d}}y,\\ (h**\phi )(\xi )=&\int ^{\infty }_{0}\int _{\mathbb {R}} h(y,s)\phi (\xi -y-cs){\hbox {d}}y{\hbox {d}}s. \end{aligned} \end{aligned}$$

For convenience, we write $\phi (-\infty )$ and $\phi (\infty )$ as abbreviations for $\lim _{\xi \rightarrow -\infty }\phi (\xi )$ and $\lim _{\xi \rightarrow \infty }\phi (\xi )$, respectively. Travelling wavefronts of (3) have been intensively studied in recent years, see, e.g. [8,9,10,11,12,13, 22,23,24, 29,30,31, 42, 45,46,47]. Zinner et al. [47] addressed the existence and minimal speed of travelling wavefront for (4). Recently, based on [11, 12], Chen et al. [10] investigated the uniqueness and asymptotic behaviour of travelling waves for (2) with $d=2$. To the best of our knowledge, however, there are no results regarding the existence, uniqueness, monotonicity, asymptotic behaviour, and asymptotic stability of travelling waves for an equation as general as (1).

There is an enormous amount of work on related equations which is impossible even to sketch. We only mention the work of Coville and coworkers, where the nonlinearity is local, but general nonlocal expressions instead of the nonlocal diffusion equation are considered (e.g. [14]). Some methods are similar, such as the use of super- and subsolutions. For interesting work on a Fisher–KPP equation with a nonlocal saturation effect, where no maximum principle holds, we refer to [5].

We shall establish the existence, uniqueness, monotonicity, asymptotic behaviour of travelling waves for (1) under the assumptions (F1), (F2), (H1), and (H2).

Theorem 1.1

Under assumptions (F1), (F2), (H1), and (H2), there exists a minimal wave speed $c^*>0$ such that for each $c\ge c^*$, Eq. (1) has a travelling wavefront $\phi (x+ct)$ satisfying (6). Moreover,

• 1. the solution $\phi$ of (6) is unique up to a translation.

• 2. Every solution $\phi$ of (6) is strictly monotone, i.e. $\phi '(\xi )>0$ for all $\xi \in {\mathbb {R}}$.

• 3. Every solution $\phi$ of (6) satisfies $0<\phi (\cdot ) <K$ on ${\mathbb {R}}$.

• 4. Any solution of (6) satisfies $\lim _{\xi \rightarrow -\infty }\phi '(\xi )/\phi (\xi )=\lambda$, with $\lambda$ being the minimal positive root of

  $$\begin{aligned} c\lambda -p\lambda ^2-{\hbox {d}}\left[ H(\lambda )-1\right] -\partial _1f(0,0) -\partial _2f(0,0)G(c,\lambda )=0, \end{aligned}$$

  (7)

  where

  $$\begin{aligned} \begin{aligned} H(\lambda )=&\int _{\mathbb {R}}J(y)\exp \{-\lambda y\}{\hbox {d}}y,\\ G(c,\lambda )=&\int ^{\infty }_0\int _{\mathbb {R}}h(y,s)\exp \{-\lambda (y+cs)\}{\hbox {d}}y{\hbox {d}}s. \end{aligned} \end{aligned}$$

  for $\lambda \in {\mathbb {C}}$ with $\mathrm {Re}\lambda <\lambda _0$.

• 5. Any solution of (6) satisfies $\lim _{\xi \rightarrow \infty }\phi '(\xi )/[K-\phi (\xi )]=\gamma$, with $\gamma$ being the unique positive root of

  $$\begin{aligned} c\gamma +p\gamma ^2+{\hbox {d}}\left[ H(-\gamma )-1\right] +\partial _1f(K,K) +\partial _2f(K,K)G(c,-\gamma )=0. \end{aligned}$$

  (8)

We remark that if $c>c^*$, Eq. (7) has exactly two real roots, both positive.

This paper is organised as follows. In Sect. 2, we provide some preliminary results; in Sect. 3, we establish the existence of a travelling wavefront, using the monotone iteration method developed by Wu and Zou [42] with a pair of super- and subsolutions. In particular, Theorems 3.1 and 3.2 establish the existence part of Theorem 1.1. To derive the monotonicity and uniqueness of wave profiles (Sect. 5), we shall first apply Ikehara’s theorem in Sect. 4 to study the asymptotic behaviour of wave profiles. This idea originated in Carr and Chmaj’s paper [9], where the authors study the uniqueness of waves for a nonlocal monostable equation. Theorem 5.1 establishes the monotonicity part of Theorem 1.1, and uniqueness is discussed in Theorem 5.2, and nonexistence of travelling waves for $c < c^*$ is the content of Theorem 5.3.

2 Notation and Auxiliary Results

Throughout this paper, $C > 0$ denotes a generic constant, while $C_i (i = 1, 2, \ldots )$ represents a specific constant. Let I be an interval, typically $I = {\mathbb {R}}$. Let $T > 0$ be a real number and ${\mathcal {B}}$ be a Banach space. We denote by $C([0, T ], {\mathcal {B}})$ the space of the ${\mathcal {B}}$-valued continuous functions on [0, T], while $L^2([0, T ], {\mathcal {B}})$ is the space of the ${\mathcal {B}}$-valued $L^2$-functions on [0, T]. The corresponding spaces of the ${\mathcal {B}}$-valued functions on $[0,\infty )$ are defined similarly.

For a given travelling wave $\phi$ of (1) satisfying (6), define

$$\begin{aligned} \begin{aligned} G_j(\xi )&=\partial _jf\left( \phi (\xi ),\int _0^{\infty }\int _{\mathbb {R}}h(y,s) \phi (\xi -y-cs){\hbox {d}}y{\hbox {d}}s\right) ,\quad j=1,2,\\ B(\xi )&=\int _0^{\infty }\int _{\mathbb {R}}h(y,s)G_2(\xi +y+cs){\hbox {d}}y{\hbox {d}}s. \end{aligned} \end{aligned}$$

(9)

Obviously, $B(\xi )$ and $G_j(\xi ), j=1,2$ are nonincreasing and satisfy

$$\begin{aligned} G_1(\infty )=\partial _1f(K,K),\quad B(\infty )=G_2(\infty )=\partial _2f(K,K). \end{aligned}$$

(10)

Moreover, both $G(c,\lambda )$ and $H(\lambda )$ are twice differentiable in $\lambda \in [0, \lambda _0)$. Moreover, $G(c,0)=1, H(0)=1, H'(\lambda )>0, G_{\lambda \lambda }(c,\lambda )>0$, and $H''(\lambda )>0$. Set

$$\begin{aligned} \Delta (c,\lambda )=c\lambda -p\lambda ^2-d[H(\lambda )-1]-\partial _1f(0,0) -\partial _2f(0,0)G(c,\lambda ) \end{aligned}$$

(11)

and

$$\begin{aligned} {\widetilde{\Delta }}(c,\lambda )=c\lambda +p\lambda ^2+{\hbox {d}}[H(-\lambda )-1] +\partial _1f(K,K)+\partial _2f(K,K)G(c,-\lambda ) \end{aligned}$$

(12)

for all $c\in {\mathbb {R}}$ and $\lambda \in {\mathbb {C}}$ with $c\ge 0$ and $\mathrm {Re}\lambda <\lambda ^+$, where $\lambda ^+=\lambda _0$ if $\partial _2f(0,0)>0$ and $\lambda ^+=+\infty$ if $\partial _2f(0,0)=0$.

We require two simple technical statements.

Lemma 2.1

There exist $c^*>0$ and $\lambda ^*\in (0,\lambda ^+)$ such that $\Delta (c^*,\lambda ^*)=0$ and $\Delta _{\lambda }(c^*,\lambda ^*)=0$. Furthermore,

• (i) if $0<c<c^*$, then $\Delta (c,\lambda )<0$ for all $\lambda \ge 0$;

• (ii) if $c>c^*$, then the equation $\Delta (c,\cdot )=0$ has two positive real roots $\lambda _1(c)$ and $\lambda _2(c)$ with $0<\lambda _1(c)<\lambda ^*<\lambda _2(c)<\lambda ^+$ such that $\lambda '_1(c)<0, \lambda '_2(c)>0, \Delta (c,\lambda )>0$ for all $\lambda \in (\lambda _1(c),\lambda _2(c))$, and $\Delta (c,\lambda )<0$ for all $(-\infty ,\lambda ^+)\setminus [\lambda _1(c),\lambda _2(c)]$.

Proof

Note that for all $\lambda \in (0,\lambda ^+)$,

$$\begin{aligned} \Delta _{\lambda }(c,\lambda )&= c-2p\lambda -dH'(\lambda )-\partial _2f(0,0)G_{\lambda }(c,\lambda ), \\ \Delta _{\lambda \lambda }(c,\lambda )&=-2p -{\hbox {d}}H''(\lambda )-\partial _2f(0,0)G_{\lambda \lambda }(c,\lambda )<0, \\ \Delta _{c}(c,\lambda )&= \lambda -\partial _2f(0,0)G_c(c,\lambda )>0, \\ \Delta (c,0)&= -\partial _1f(0,0) -\partial _2f(0,0)<0, \\ \Delta (0,\lambda )&=-2p\lambda ^2 -{\hbox {d}}[H(\lambda )-1]-\partial _1f(0,0) -\partial _2f(0,0)G(0,\lambda )<0 \end{aligned}$$

and

$$\begin{aligned} \lim _{\lambda \rightarrow \lambda ^+-0}\Delta (c,\lambda )=-\infty . \end{aligned}$$

Then the conclusion of this lemma follows. $\square$

Lemma 2.2

Under assumptions (F1) and (F2), for each fixed $c\ge 0, {\widetilde{\Delta }}(c,\cdot )$ has exactly one positive zero $\upsilon (c)$.

Proof

In view of (F1) and (F2), we have

$$\begin{aligned} {\widetilde{\Delta }}(c,0)=\partial _1f(K,K)+\partial _2f(K,K)<0 \end{aligned}$$

and

$$\begin{aligned} \lim _{\lambda \rightarrow \lambda ^+-0}{\widetilde{\Delta }}(c,\lambda )=+\infty . \end{aligned}$$

Therefore, ${\widetilde{\Delta }}(c,\cdot )$ has at least one positive zero. Note that

$$\begin{aligned} {\widetilde{\Delta }}_{\lambda }(c,\lambda ) =c+2p\lambda -{\hbox {d}}H'(-\lambda )-\partial _2f(K,K)G_{\lambda }(c,-\lambda ) \end{aligned}$$

and for $\lambda >0$

$$\begin{aligned} G_{\lambda }(c,-\lambda )=\int ^{\infty }_0\int ^{\infty }_0h(y,s)\left[ (y-cs)e^{-\lambda (y+cs)}-(y+cs)e^{\lambda (y-cs)}\right] {\hbox {d}}y{\hbox {d}}s<0. \end{aligned}$$

Then we have ${\widetilde{\Delta }}_{\lambda }(c,\lambda )>0$ for all $\lambda \in (0,\lambda ^+)$. This implies that ${\widetilde{\Delta }}_{\lambda }(c,\lambda )$ is increasing in $\lambda$ and so it has exactly one positive zero. $\square$

We now define the notion of super- and subsolutions. For any absolutely continuous function $\varphi :{\mathbb {R}}\rightarrow {\mathbb {R}}$ satisfying that $\varphi '$ and $\varphi ''$ exist almost everywhere and are essentially bounded on ${\mathbb {R}}$, we set

$$\begin{aligned} N_c[\varphi ](\xi )\triangleq c\varphi '(\xi )-p\varphi ''(\xi ) -{\hbox {d}}(J*\phi -\phi )(\xi )-f(\varphi (\xi ), (h**\varphi )(\xi )). \end{aligned}$$

Given a positive constant c, a nondecreasing continuous function $\varphi ^+$ is called a super-solution of (6) if $\varphi ^+(-\infty )=0$ and $\varphi ^+$ is differentiable almost everywhere in ${\mathbb {R}}$ such that $N_c[\varphi ^+](\xi )\ge 0$ for almost every $\xi \in {\mathbb {R}}$. A continuous function $\varphi ^-$ is called a subsolution of (6) if $\varphi ^-(-\infty )=0, \varphi ^-(\xi )$ is not identically equal to 0 and $\varphi ^-$ is differentiable almost everywhere in ${\mathbb {R}}$ such that $N_c[\varphi ^-](\xi )\le 0$ for almost every $\xi \in {\mathbb {R}}$.

Next, we introduce the operator ${\mathcal {H}}_{\mu } :C({\mathbb {R}})\rightarrow C({\mathbb {R}})$ by

$$\begin{aligned} {\mathcal {H}}_{\mu }(\varphi )=d(J*\varphi -\varphi ) +f(\varphi , h**\varphi )+\mu \varphi . \end{aligned}$$

It is easy to see that $\varphi$ satisfies (6) if and only if $\varphi$ satisfies

$$\begin{aligned} \varphi (\xi )=T_{\mu }(\varphi )(\xi ), \end{aligned}$$

(13)

where

$$\begin{aligned} T_{\mu }(\varphi )(\xi )=\frac{1}{c}\int ^{\infty }_{0} \exp \left\{ -\frac{\mu x}{c}\right\} {\mathcal {H}}_{\mu }(\varphi )(\xi +x)dx \end{aligned}$$

if $p=0$, and

$$\begin{aligned} T_{\mu }(\varphi )(\xi )=\frac{1}{p(\varsigma _2-\varsigma _1)} \left[ \int _{-\infty }^{\xi } e^{\varsigma _1(\xi -x)}{\mathcal {H}}_{\mu }(\varphi )(x){\hbox {d}}x+ \int ^{\infty }_{\xi } e^{\varsigma _2(\xi -x)}{\mathcal {H}}_{\mu }(\varphi )(x){\hbox {d}}x\right] \end{aligned}$$

if $p>0$, and

$$\begin{aligned} \varsigma _1=\frac{c-\sqrt{c^2+4p\mu }}{2p}<0,\quad \varsigma _2=\frac{c+\sqrt{c^2+4p\mu }}{2p}>0. \end{aligned}$$

(14)

Choose $\mu >2d+\max \{|\partial _jf(u,v)|:\, (u,v)\in [0,K]\times [0,K],\, j=1,2\}$. Then the operator $T_{\mu }$ is well defined for functions $\phi$ of a growth rate less than $e^{\mu x}$. Furthermore, since f is monotone in the second argument by (F1), we have for $\varphi \le \psi$

$$\begin{aligned} {\mathcal {H}}_{\mu }(\varphi )- {\mathcal {H}}_{\mu }(\psi )= & {} \mu [\varphi -\psi ]+d[J*(\varphi -\psi )-(\varphi -\psi )]\\&+\,\, \partial _1f({\tilde{\varphi }}, h**\varphi ) [\varphi - \psi ]\le 0, \end{aligned}$$

where ${{\tilde{\varphi }}}(y)$ lies between $\varphi (y)$ and $\psi (y)$. Then the choice of $\mu$ shows that ${\mathcal {H}}_{\mu }(\varphi )$ is monotone in $\varphi$,

$$\begin{aligned} {\mathcal {H}}_{\mu }(\varphi )(\xi )\le {\mathcal {H}}_{\mu }(\psi )(\xi ) \quad \hbox {if } 0\le \varphi \le \psi \le K \hbox { in } {\mathbb {R}}. \end{aligned}$$

(15)

Thus, we have the following result on the monotonic travelling waves.

Lemma 2.3

Under assumptions (F1) and (H1), assume that there exists a super-solution $\varphi ^+$ and a subsolution $\varphi ^-$ of (6) such that $0\le \varphi ^-\le \varphi ^+\le K$ on ${\mathbb {R}}$. Then (6) has a solution $\varphi$ satisfying $\varphi '(\xi )\ge 0$ for all $\xi \in {\mathbb {R}}$.

Proof

Assume that there exist a super-solution $\varphi ^+$ and a subsolution $\varphi ^-$ of (6) such that $0\le \varphi ^-\le \varphi ^+\le K$ in ${\mathbb {R}}$. Define $\varphi _1=T_{\mu }(\varphi ^+)$. Then $\varphi _1$ is a well-defined $C^1$ function. From the definition of super-solution, we have

$$\begin{aligned} \varphi ^+\ge T_{\mu }(\varphi ^+)=\varphi _1. \end{aligned}$$

Also, by the definition of subsolution and the property (15) of ${\mathcal {H}}_{\mu }$, we get

$$\begin{aligned} \varphi ^-\le T_{\mu }(\varphi ^-)\le T_{\mu }(\varphi ^+)=\varphi _1. \end{aligned}$$

Hence $\varphi ^-(\xi )\le \varphi _1(\xi )\le \varphi ^+(\xi )$ for all $\xi \in {\mathbb {R}}$. Moreover, using the fact that $\varphi ^+$ is nondecreasing and $\mu >2d+\max \{|\partial _jf(u,v)|:\, u,v\in [0,K],\, j=1,2\}$, we have ${\mathcal {H}}_{\mu }(\varphi ^+)(s)\ge {\mathcal {H}}_{\mu }(\varphi ^+)(\xi )$ for all $s\ge \xi$ and hence $\varphi '_1(\xi )\ge 0$. Now define $\varphi _{n+1}=T_{\mu }(\varphi _n)$ for all $n\in {\mathbb {N}}$. By induction, it is easy to see that $0\le \varphi ^-\le \varphi _{n+1}\le \varphi _n\le \varphi ^+\le K$ and $\varphi '_{n+1}\ge 0$ on ${\mathbb {R}}$ for all $n\in {\mathbb {N}}$. Then the limit $\varphi (\xi )\triangleq \lim _{n\rightarrow \infty }\varphi _n(\xi )$ exists for all $\xi \in {\mathbb {R}}$ and $\varphi (\xi )$ is nondecreasing on ${\mathbb {R}}$. By Lebesgue’s dominated convergence theorem, $\varphi$ satisfies (13) and hence satisfies (6). $\square$

3 Existence

In this section, we shall establish the existence of travelling waves by constructing a suitable pair of super- and subsolutions. First, we derive two properties of possible solutions of (6).

Lemma 3.1

Under assumptions (F1) and (H1), every solution $(c, \varphi )$ of (6) satisfies $0<\varphi (\xi )< K$ for all $\xi \in {\mathbb {R}}$.

Proof

Let $(c,\varphi )$ be a solution of (6). Suppose that there exists $\xi _0\in {\mathbb {R}}$ such that $\varphi (\xi _0) = 0$. In view of $\varphi (\infty )=K$, without loss of generality, we may assume $\xi _0$ is the right-most point such that $\varphi (\xi _0) = 0$. Since $\varphi (\xi )\ge 0$ for all $\xi \ge \xi _0$, we have $\varphi '(\xi _0)=\varphi ''(\xi _0)=0$. It follows from (6) and (F1) that $\varphi (\xi )\equiv 0$ for all $\xi \in {\mathbb {R}}$, which contradicts the definition of $\xi _0$. Therefore, $\varphi >0$ on ${\mathbb {R}}$. Similarly, $\varphi <K$ on ${\mathbb {R}}$. This completes the proof. $\square$

Lemma 3.2

Under assumptions (F1) and (H1), every solution $(c, \varphi )$ of (6) satisfying $\varphi '\ge 0$ on ${\mathbb {R}}$ satisfies $\varphi '>0$ on ${\mathbb {R}}$.

Proof

Suppose on the contrary that there exists $\xi _0\in {\mathbb {R}}$ such that $\varphi '(\xi _0) = 0$. By differentiating (13) with respect to $\xi$, we obtain ${\mathcal {H}}_{\mu }(\varphi )(s)={\mathcal {H}}_{\mu }(\varphi )(\xi _0)$ for all $s\ge \xi _0$. Letting $s\rightarrow \infty$, we obtain ${\mathcal {H}}_{\mu }(\varphi )(\xi _0)=c\mu K$. This, together with $\varphi '(\xi _0)=0$ and (6), implies that $\varphi (\xi _0)=K$, which contradicts Lemma 3.1. Hence the lemma is proved. $\square$

Lemma 3.3

Assume that (F1), (F2), (H1), and (H2) hold. Let $c^*, \lambda _1(c)$ and $\lambda _2(c)$ be defined as in Lemma  2.1. Let $c > c^*$ be any number. Then for every $\gamma \in (0,\min \{\sigma \lambda _1(c),\lambda _2(c)-\lambda _1(c)\})$, there exists $Q(c,\gamma )>1$ such that for every $q>Q(c,\gamma )$, the functions $\phi ^{\pm }$ defined by

$$\begin{aligned} \phi ^{+}(\xi )=\min \left\{ K,\exp \{\lambda _1(c)\xi \} \right\} ,\quad \xi \in {\mathbb {R}} \end{aligned}$$

(16)

and

$$\begin{aligned} \phi ^{-}(\xi )= \exp \{\lambda _1(c)\xi \}\max \left\{ 0,1 -q\exp \{\gamma \xi \} \right\} , \quad \xi \in {\mathbb {R}} \end{aligned}$$

(17)

are a super-solution and a subsolution to (6), respectively.

Proof

We begin by proving that $\phi ^{\pm }$ are a pair of super- and subsolutions of (6). We only consider the case $\partial _2f(0,0)>0$ because the proof of the case $\partial _2f(0,0)=0$ is similar. It follows from (16) that $\phi ^{+}(\xi )\le \exp \{\lambda _1(c)\xi \}$ for all $\xi \in {\mathbb {R}}$ and hence

$$\begin{aligned} J*\phi ^{+}(\xi )\le \int _{{\mathbb {R}}}J(y)\exp \{\lambda _1(c)(\xi -y)\}{\hbox {d}}y= \exp \{\lambda _1(c)\xi \}H(\lambda _1(c)) \end{aligned}$$

and

$$\begin{aligned} h**\phi ^{+}(\xi )\le \int ^{\infty }_{0}\int _{{\mathbb {R}}}h(y,s)\exp \{\lambda _1(c)(\xi -y-cs)\}{\hbox {d}}y{\hbox {d}}s= \exp \{\lambda _1(c)\xi \}G(c,\lambda _1(c)). \end{aligned}$$

Moreover, there exists $\xi ^*>0$ satisfying $\exp \{\lambda _1(c)\xi ^*\}=K, \phi ^{+}(\xi )=K$ for $\xi >\xi ^*$ and $\phi ^+(\xi )=\exp \{\lambda _1(c)\xi \}$ for $\xi \le \xi ^*$. For $\xi >\xi ^*$, we have

$$\begin{aligned} N_c[\phi ^{+}](\xi )=-d[J*\phi ^{+}(\xi )-K]- f(K,(h**\phi ^{+})(\xi ) \ge -f(K,K)=0. \end{aligned}$$

For $\xi \le \xi ^*$, we have

$$\begin{aligned} N_c[\phi ^+](\xi )&\ge \phi ^{+}(\xi )\{c\lambda _1(c)-p\lambda ^2_1(c)- d[H(\lambda _1(c))-1]\}\\&\quad -f(\phi ^{+}(\xi ),(h**\phi ^{+})(\xi ))\\&= \phi ^{+}(\xi )\Delta (c,\lambda _1(c)) -f(\phi ^{+}(\xi ),(h**\phi ^{+})(\xi )) +\partial _1f(0,0)\phi ^{+}(\xi )\\&\quad {}+\phi ^{+}(\xi )\partial _2f(0,0) G(c,\lambda _1(c))\\&\ge -f(\phi ^{+}(\xi ),(h**\phi ^{+})(\xi )) +\partial _1f(0,0)\phi ^{+}(\xi )\\&\quad +\partial _2f(0,0) (h**\phi ^{+})(\xi )\ge 0, \end{aligned}$$

where we have used the condition (F2) in the last inequality. Therefore, $\phi ^{+}$ is a super-solution of (6).

It follows from (17) that $\exp \{\lambda _1(c)\xi \}\ge \phi ^{-}(\xi )\ge \exp \{\lambda _1(c)\xi \}(1 -q\exp \{\gamma \xi \})$ for all $\xi \in {\mathbb {R}}$ and hence

$$\begin{aligned} \begin{aligned} J*\phi ^{-}(\xi )&\ge \int _{{\mathbb {R}}}J(y)\exp \{\lambda _1(c)(\xi -y)\}(1 -q\exp \{\gamma (\xi -y)\}){\hbox {d}}y\\&=\exp \{\lambda _1(c)\xi \}H(\lambda _1(c))-q\exp \{[\gamma +\lambda _1(c)]\xi \}H(\gamma +\lambda _1(c)), \\ h**\phi ^{-}(\xi )&\ge \int ^{\infty }_{0}\int _{{\mathbb {R}}}h(y,s)\exp \{\lambda _1(c)(\xi -y-cs)\}(1 -q\exp \{\gamma (\xi -y-cs)\}){\hbox {d}}y{\hbox {d}}s\\&=\exp \{\lambda _1(c)\xi \}G(c,\lambda _1(c))-q\exp \{[\gamma +\lambda _1(c)]\xi \}G(c,\gamma +\lambda _1(c)). \end{aligned} \end{aligned}$$

Let $\xi _0=-\frac{1}{\gamma }\ln q$. Clearly, $\phi ^{-}(\xi )=0$ for $\xi >\xi _0$ and $\phi ^{-}(\xi )=\exp \{\lambda _1(c)\xi \}(1 -q\exp \{\gamma \xi \})$ for $\xi \le \xi _0$. For $\xi >\xi _0$, we have

$$\begin{aligned} N_c[\phi ^{-}](\xi )=-{\hbox {d}}J*\phi ^{-}(\xi ) -f(0,(h**\phi ^{-})(\xi ))\le -f(0,0)=0. \end{aligned}$$

For $\xi \le \xi _0$, we have

$$\begin{aligned} N_c[\phi ^{-}](\xi )&\le \exp \{\lambda _1(c)\xi \} \{c\lambda _1(c)-p\lambda ^2_1(c)-d[H(\lambda _1(c))-1]\}\\&\quad -q\exp \{[\gamma +\lambda _1(c)]\xi \} \{c[\gamma +\lambda _1(c)]-p[\gamma +\lambda _1(c)]^2\\&\quad -d[H(\gamma +\lambda _1(c))-1]\}\\&\quad -f(\phi ^{-}(\xi ),(h**\phi ^{-})(\xi ))\\&= \exp \{\lambda _1(c)\xi \}\Delta (c,\lambda _1(c)) -q\exp \{[\gamma +\lambda _1(c) ]\xi \}\Delta (c,\gamma +\lambda _1(c))\\&\quad -f(\phi ^{-}(\xi ),(h**\phi ^{-})(\xi )) +\partial _1f(0,0)\phi ^{-}(\xi )\\&\quad +\partial _2f(0,0)\exp \{\lambda _1(c)\xi \} G(c,\lambda _1(c))\\&\quad -q\partial _2(0,0)\exp \{[\gamma +\lambda _1(c) ]\xi \} G(c,\gamma +\lambda _1(c)) \\&\le -q\exp \{[\gamma +\lambda _1(c) ]\xi \} \Delta (c,\gamma +\lambda _1(c)) -f(\phi ^{-}(\xi ),(h**\phi ^{-})(\xi ))\\&\quad {}+\partial _1f(0,0) \phi ^{-}(\xi )+\partial _2f(0,0)(h**\phi ^{-})(\xi ). \end{aligned}$$

In view of (F2), we have

$$\begin{aligned} N_c[\phi ^{-}](\xi )&\le -q\exp \{[\gamma +\lambda _1(c) ]\xi \} \Delta (c,\gamma +\lambda _1(c))+M[\phi ^{-}(\xi ) +(h**\phi ^{-})(\xi )]^{1+\sigma }\\&\le -q\exp \{[\gamma +\lambda _1(c) ]\xi \} \Delta (c,\gamma +\lambda _1(c))\\&\quad +M[1 +G(c,\gamma +\lambda _1(c))]^{1+\sigma }\exp \{(1+\sigma )\lambda _1(c)\xi \}\\&\le \exp \{[\gamma +\lambda _1(c) ]\xi \} \Delta (c,\gamma +\lambda _1(c))\left\{ \frac{M[1 +G(c,\gamma +\lambda _1(c))]^{1+\sigma }}{ \Delta (c,\gamma +\lambda _1(c))} -q\right\} \\&\le 0, \end{aligned}$$

provided that

$$\begin{aligned} q\ge Q(c,\eta )\triangleq \max \left\{ 1,\frac{M[1 +G(c,\gamma +\lambda _1(c))]^{1+\sigma }}{ \Delta (c,\gamma +\lambda _1(c))}\right\} . \end{aligned}$$

Therefore, $\phi ^{-}$ is a subsolution of (6). The proof is complete. $\square$

As a consequence of Lemmas 3.1, 3.2, and 3.3, we have the following result on the existence of increasing travelling waves.

Theorem 3.1

Under the conditions (F1), (F2), (H1), and (H2), let $c^*, \lambda _1(c)$ and $\lambda _2(c)$ be defined as in Lemma 2.1. Then for each $c> c^*$, (6) admits a solution $(c, \phi )$ satisfying $0<\phi (\xi )<K, \phi '(\xi )>0$ for all $\xi \in {\mathbb {R}}$, and

$$\begin{aligned} \lim \limits _{\xi \rightarrow -\infty }\phi (\xi )\exp \{-\lambda \xi \}=1,\quad \lim \limits _{\xi \rightarrow -\infty }\phi '(\xi )\exp \{-\lambda \xi \}=\lambda , \end{aligned}$$

(18)

where $\lambda =\lambda _1(c)$ is the smallest positive zero of $\Delta (c,\cdot )$.

Proof

It follows from Lemmas 2.3, 3.1, 3.2, and 3.3 that there exists a strictly increasing solution $\phi (\xi )$ to (6), which will be denoted by $(c,\phi )$ and satisfies

$$\begin{aligned} \exp \{\lambda _1(c)\xi \}\left( 1-q\exp \{\gamma \xi \}\right) \le \phi (\xi )\le \exp \{\lambda _1(c)\xi \},\quad \xi \in {\mathbb {R}}. \end{aligned}$$

(19)

It then follows from (19) that

$$\begin{aligned} \lim \limits _{\xi \rightarrow -\infty }\left| \phi (\xi )\exp \{-\lambda _1(c)\xi \}-1\right| \le \lim \limits _{\xi \rightarrow -\infty }q\exp \{\gamma \xi \}=0. \end{aligned}$$

In view of condition (F2), we have

$$\begin{aligned}&\lim \limits _{\xi \rightarrow -\infty }\left| f(\phi (\xi ), (h**\phi )(\xi ))-\partial _1f(0,0)\phi (\xi ) -\partial _2f(0,0)(h**\phi ) (\xi )\right| \exp \{-\lambda _1(c)\xi \}\\&\quad \le M\lim \limits _{\xi \rightarrow -\infty }\left[ \phi (\xi )+(h**\phi )(\xi ) \right] ^{1+\sigma }\exp \{-\lambda _1(c)\xi \}\\&\quad = M\left[ \lim \limits _{\xi \rightarrow -\infty }\phi (\xi )e^{-\lambda _1(c)\xi /(1+\sigma )} +\lim \limits _{\xi \rightarrow -\infty }(h**\phi )(\xi )\exp \{-\lambda _1(c)\xi /(1+\sigma )\} \right] ^{1+\sigma }\\&\quad = M\left[ \lim \limits _{\xi \rightarrow -\infty }(h**\phi )(\xi )e^{-\lambda _1(c)\xi /(1+\sigma )} \right] ^{1+\sigma }\\&\quad \le M\left[ g'(0)\lim \limits _{\xi \rightarrow -\infty }(h**\phi )(\xi )e^{-\lambda _1(c)\xi /(1+\sigma )} \right] ^{1+\sigma } \end{aligned}$$

and

$$\begin{aligned}&\lim \limits _{\xi \rightarrow -\infty }(h**\phi )(\xi )\exp \{-\lambda _1(c)\xi /(1+\sigma )\}\\&\quad =\lim \limits _{\xi \rightarrow -\infty }\int ^{\infty }_{0}\int ^{\infty }_{-\infty } h(y,t)\phi (\xi -y-ct) \exp \{-\lambda _1(c)\xi /(1+\sigma )\}{\hbox {d}}y{\hbox {d}}t \\&\quad = \int ^{\infty }_{0}\int ^{\infty }_{-\infty }h(y,t)\left[ \lim \limits _{\xi \rightarrow -\infty }\phi (\xi -y-c t)\exp \{-\lambda _1(c)\xi /(1+\sigma )\}\right] {\hbox {d}}y{\hbox {d}}t= 0. \end{aligned}$$

Hence, if $p=0$ then for $c>c^*$, we have

$$\begin{aligned}&c\lim \limits _{\xi \rightarrow -\infty }\phi '(\xi )\exp \{-\lambda _1(c)\xi \}\\&\quad = \lim \limits _{\xi \rightarrow -\infty }\left[ d(J*\phi -\phi )(\xi )+f(\phi (\xi ), (h**\phi )(\xi ))\right] \exp \{-\lambda _1(c)\xi \}\\&\quad = d[H(\lambda _1(c))-1]+\lim \limits _{\xi \rightarrow -\infty }f(\phi (\xi ), (h**\phi )(\xi ))\exp \{-\lambda _1(c)\xi \}\\&\quad = d[H(\lambda _1(c))-1]+\lim \limits _{\xi \rightarrow -\infty }\left[ \partial _1f(0,0)\phi (\xi )+\partial _2f(0,0) (h**\phi )(\xi )\right] \exp \{-\lambda _1(c)\xi \}\\&\quad = d[H(\lambda _1(c))-1]+\partial _1f(0,0) +\partial _2f(0,0)G(c,\lambda _1(c)) = c\lambda _1(c). \end{aligned}$$

If $p\ne 0$, then for $c>c^*$, using $\phi '(-\infty )=0$ and integrating both sides of (6) from $-\infty$ to $\xi$, we have

$$\begin{aligned} \begin{aligned}&\lim _{\xi \rightarrow -\infty }\phi '(\xi )\exp \{-\lambda _1(c)\xi \}\\&\quad = \frac{c}{p}-\frac{1}{p}\lim _{\xi \rightarrow -\infty }\exp \{-\lambda _1(c)\xi \}\int ^{\xi }_{-\infty }\left[ d(J*\phi -\phi )(s)+f(\phi (s), (h**\phi )(s))\right] {\hbox {d}}s\\&\quad = \frac{c}{p}-\lim _{\xi \rightarrow -\infty }\frac{\exp \{-\lambda _1(c)\xi \} \left[ d(J*\phi -\phi )(\xi )+f(\phi (\xi ), (h**\phi )(\xi ))\right] }{p\lambda _1(c)}\\&\quad = \frac{c}{p}- \frac{d[H(\lambda _1(c))-1]+\partial _1f(0,0) +\partial _2f(0,0)G(c,\lambda _1(c))}{p\lambda _1(c)} = \lambda _1(c). \end{aligned} \end{aligned}$$

This completes the proof. $\square$

Remark 3.1

In Theorem 3.1, by Lebesgue’s dominated convergence theorem, we also have

$$\begin{aligned} \lim \limits _{\xi \rightarrow -\infty }(h**\phi )(\xi ) \exp \{-\lambda _1(c)\xi \}=G(c,\lambda _1(c)). \end{aligned}$$

Remark 3.2

Theorem 3.1 implies that the asymptotic behaviours of wave profiles of the travelling waves obtained by super- and subsolutions satisfy (18) for $c>c^*$. Furthermore, in the subsequent section, we shall show that the wave profile $\varphi$ of every travelling wave of (1) satisfying (6) has similar asymptotic behaviours.

Next, we prove that (6) has a solution $(c, \phi )$ with $0<\phi <K$ and $\phi '>0$ on ${\mathbb {R}}$ for $c = c^*$.

Theorem 3.2

Under the conditions (F1), (F2), (H1), and (H2), (6) has a solution $(c, \phi )$ with $0<\phi <K$ and $\phi '>0$ on ${\mathbb {R}}$ for $c = c^*$.

Proof

We choose a sequence $\{c_j\}\subseteq (c^*,\infty )$ such that $\lim _{j\rightarrow \infty }c_j=c^*$. Then for each j, there exists a strictly increasing travelling wave $(c_j,\phi _j)$ of (1) such that $\phi _j(-\infty )=0$ and $\phi _j(+\infty )=K$. Since $\phi _j(\cdot +\zeta ), \zeta \in {\mathbb {R}}$, is also a travelling wave, we can assume that $\phi _j(0)=\alpha$ and $\phi _j(x)\le K$ for a fixed $\alpha \in (0,K)$ and all $x\in {\mathbb {R}}$ and $j\ge 1$. Note that $\phi _j$ is a fixed point of operator $T_{\mu }$ in E with $c=c_j$ and $T_{\mu }(\phi _j)(\xi )$ can be differentiated with respect to $\xi$, where E is the Banach space of bounded and uniformly continuous functions on ${\mathbb {R}}$ equipped with the maximum norm. Moreover, we differentiate both sides of (6) with respect to $\xi$ to obtain

$$\begin{aligned} \begin{aligned} 0=&-c\phi ''_j(\xi )+p\phi '''_j(\xi )+d[J*\phi '_j-\phi '_j](\xi ) +\partial _1f(\phi _j(\xi ), (h**\phi _j)(\xi ))\phi '_j(\xi )\\&+\partial _2f(\phi _j(\xi ), (h**\phi _j)(\xi ))h**\phi '_j(\xi ). \end{aligned} \end{aligned}$$

By the definition of ${\mathcal {H}}_{\mu }$, it follows that there exist three positive numbers $N_1, N_2$, and $N_3$ (if $p\ne 0$) such that

$$\begin{aligned} |\phi '_j(\xi )|\le N_1,\quad |\phi ''_j(\xi )|\le N_2,\quad |\phi '''_j(\xi )|\le N_3 \end{aligned}$$

for all n and $\xi$. Therefore, $\phi '_j, \phi ''_j$ and $\phi '''_j$ (if $p\ne 0$) are uniformly bounded and equi-continuous sequences of functions on ${\mathbb {R}}$. Then the Arzelà-Ascoli theorem implies that there exists a subsequence of $\{c_j\}$ (for simplicity, denoted again by $\{c_j\}$), such that $\lim _{j\rightarrow \infty }c_j=c^*$, and $\phi '_j, \phi ''_j$ and $\phi '''_j$ (if $p\ne 0$) converge uniformly on every bounded and closed subset of ${\mathbb {R}}$. Thus, $\phi '_j, \phi ''_j$ and $\phi '''_j$ (if $p\ne 0$) converge pointwise on ${\mathbb {R}}$ to $\phi '_*, \phi ''_*$ and $\phi '''_*$ (if $p\ne 0$), respectively. By Lebesgue’s dominated convergence theorem, letting $j\rightarrow \infty$ in the equation $\phi _j=T_{\mu }(\phi _j)$, we then get $\phi _{*}=T_{\mu }(\phi _{*})$. Thus, $\phi _*$ is a solution of (6) in the case where $c=c_*$. Clearly, $\phi _*$ is monotonically increasing on ${\mathbb {R}}, \phi _{*}(0)=\alpha$ and $\phi _{*}(x)\le K$ for all $x\in {\mathbb {R}}$. One can easily verify that $\phi _{*}(-\infty )=0$ and $\phi _{*}(+\infty )=K$. Thus, (1) has a monotone travelling wave solution connecting 0 and K with the wave speed $c=c_*$. This completes the proof. $\square$

4 Asymptotic Behaviour

In this section, we always assume that (F1), (F2), (H1), and (H2) hold, and that $c^*, \lambda ^*$, and $\lambda _1(c)$ are defined as in Lemma 2.1. We shall follow a method of Carr and Chmaj [9] and Wang, Li, and Ruan [39] to establish the exact asymptotic behaviour of the profile $\phi (\xi )$ as $\xi \rightarrow -\infty$. For this purpose, we need Ikehara’s theorem on the asymptotic behaviour of a positive decreasing function whose Laplace is of a certain given shape. The proof of Ikehara’s theorem can be found for example in [9, 41].

Theorem 4.1

(Ikehara’s theorem) Let ${\mathcal {L}}[u](\mu )=\int ^{\infty }_{0}\exp \{-\mu \xi \}u(\xi ){\hbox {d}}\xi$ be the Laplace transform of u, with u being a positive nondecreasing function. Assume that ${\mathcal {L}}[u]$ has the representation

$$\begin{aligned} {\mathcal {L}}[u](\mu )=\frac{E(\mu )}{(\mu +\alpha )^{k+1}}, \end{aligned}$$

where $k>-1$ and E is analytic in the strip $-\alpha \le \mathrm {Re}\mu <0$. Then

$$\begin{aligned} \lim \limits _{\xi \rightarrow \infty }\frac{u(\xi )}{\xi ^ke^{-\alpha \xi }} =\frac{E(-\alpha )}{\Gamma (\alpha +1)}, \end{aligned}$$

where $\Gamma$ is the Gamma function.

Lemma 4.1

Assume that (F1), (F2), (H1), and (H2) hold. Let $(c, \varphi )$ be a solution of (6). Then there exists $\gamma >0$ such that

$$\begin{aligned} \sup \limits _{\xi \in {\mathbb {R}}}\varphi (\xi )\exp \{-\gamma \xi \}<\infty \quad \text{ and }\quad \sup \limits _{\xi \in {\mathbb {R}}}\Phi (\xi )\exp \{-\gamma \xi \}<\infty , \end{aligned}$$

(20)

where $\Phi (\xi )\triangleq \int ^{\xi }_{-\infty }\varphi (s){\hbox {d}}s$.

Proof

For each $\xi \in {\mathbb {R}}$, define

$$\begin{aligned} \begin{aligned} \psi (\xi )\triangleq&(h**\varphi )(\xi )= \int _0^{\infty }\int _{{\mathbb {R}}}h(y,\tau ) \varphi (\xi -y-c\tau ){\hbox {d}}y{\hbox {d}}\tau . \end{aligned} \end{aligned}$$

In view of $\partial _2f(0,0)+\partial _1f(0,0)>0$, there exist $\delta _0\in (0,K)$ such that

$$\begin{aligned} \varepsilon _1\triangleq [\partial _2f(0,0)+\partial _1f(0,0)]/4\ge M(u+v)^{\sigma } \end{aligned}$$

(21)

for all $u,v\in [0,\delta _0]$. In view of (22) and $\varphi (-\infty )=0$, there exists $\xi _0<0$ such that for all $\xi <\xi _0$, both $\varphi (\xi )$ and $\psi (\xi )$ lie in the interval $(0,\delta _0)$, where $\delta _0$ is defined as (21). Thus, for every $\xi <\xi _0$,

$$\begin{aligned} \begin{aligned} f(\varphi (\xi ),\psi (x))&\ge \partial _1f(0,0)\varphi (\xi )+\partial _2f(0,0)\psi (\xi )-M[\varphi (\xi )+\psi (\xi )]^{1+\sigma }\\&\ge \partial _1f(0,0)\varphi (\xi )+\partial _2f(0,0)\psi (\xi )-\varepsilon _1[\varphi (\xi )+\psi (\xi )]\\&= (\varepsilon _1-2\varepsilon _2)\varphi (\xi )+(\varepsilon _1+2\varepsilon _2)\psi (\xi ) \end{aligned} \end{aligned}$$

(22)

for all $u,v\in [0,\delta _0]$, where $\varepsilon _2 = [\partial _2f(0,0)-\partial _1f(0,0)]/4$. Therefore,

$$\begin{aligned} c\varphi '(\xi )-p\varphi ''(\xi )-d(J*\varphi -\varphi )(\xi )\ge (\varepsilon _1-2\varepsilon _2)\varphi (\xi ) +(\varepsilon _1+2\varepsilon _2)\psi (\xi ) \end{aligned}$$

(23)

for all $\xi <\xi _0$. By using a similar argument as in the proof of Theorem 3.2, we can prove that $\varphi (\xi )$ and $\psi (\xi )$ are both integrable on $(-\infty ,0]$.

By Fubini’s theorem and Lebesgue’s dominated convergence theorem

$$\begin{aligned} \int ^{\xi }_{-\infty }\psi (s){\hbox {d}}s&= \int ^{\xi }_{-\infty }\left[ \int _0^{\infty }\int _{{\mathbb {R}}}h(y,\tau ) \varphi (s-y-c\tau ){\hbox {d}}y{\hbox {d}}\tau \right] ds\\&= \lim \limits _{z\rightarrow -\infty }\int ^{\xi }_{z}\left[ \int _0^{\infty } \int _{{\mathbb {R}}}h(y,\tau ) \varphi (s-y-c\tau ){\hbox {d}}y{\hbox {d}}\tau \right] ds\\&= \lim \limits _{z\rightarrow -\infty }\int _0^{\infty }\int _{{\mathbb {R}}}h(y,\tau )\left[ \int ^{\xi }_{z} \varphi (s-y-c\tau ){\hbox {d}}s\right] {\hbox {d}}y{\hbox {d}}\tau \\&= \int _0^{\infty }\int _{{\mathbb {R}}}h(y,\tau )\left[ \int ^{\xi }_{-\infty }\varphi (s-y-c\tau ){\hbox {d}}s \right] dyd\tau \\&= \int _0^{\infty }\int _{{\mathbb {R}}}h(y)\Phi (\xi -y-c\tau ){\hbox {d}}y{\hbox {d}}\tau . \end{aligned}$$

Integrating (23) from $-\infty$ to $\xi$ with $\xi <\xi _0$, we have (using again Fubini’s theorem)

$$\begin{aligned} \begin{aligned}&c\varphi (\xi )-p\varphi '(\xi )-d(J*\Phi -\Phi )(\xi )\\&\quad \ge (\varepsilon _1-2\varepsilon _2)\Phi (\xi )+(\varepsilon _1+2\varepsilon _2) \int _0^{\infty }\int _{{\mathbb {R}}}h(y,\tau )\Phi (\xi -y-c\tau ){\hbox {d}}y{\hbox {d}}\tau \\&\quad = \varepsilon _1\Phi (\xi )+\varepsilon _1\int _0^{\infty }\int _{{\mathbb {R}}}h(y,\tau ) \Phi (\xi -y-c\tau ){\hbox {d}}y{\hbox {d}}\tau \\&\qquad +2\varepsilon _2\int _0^{\infty } \int _{{\mathbb {R}}}h(y,\tau )\left[ \Phi (\xi -y-c\tau ) -\Phi (\xi )\right] {\hbox {d}}y{\hbox {d}}\tau . \end{aligned} \end{aligned}$$

(24)

Note that

$$\begin{aligned}&\int ^{\xi }_{z}\int _0^{\infty }\int _{{\mathbb {R}}}h(y,\tau ) \left[ \Phi (s-y-c\tau )-\Phi (s) \right] {\hbox {d}}y{\hbox {d}}\tau {\hbox {d}}s \\&\quad = -\int _0^{\infty }\int _{{\mathbb {R}}}(y+c\tau )h(y,\tau )\int ^1_0 \left[ \Phi (\xi -t(y+c\tau ))-\Phi (z-t(y+c\tau ))\right] dtdyd\tau \\&\qquad \rightarrow -\int _0^{\infty }\int _{{\mathbb {R}}}(y+c\tau )h(y,\tau ) \int ^1_0\Phi (\xi -t(y+c\tau )){\hbox {d}}t{\hbox {d}}y{\hbox {d}}\tau \end{aligned}$$

as $z\rightarrow -\infty$. Thus, (24) means that $\Phi (\xi )$ and $\int _0^{\infty }\int _{{\mathbb {R}}}h(y,\tau )\Phi (s-y-c\tau ){\hbox {d}}y{\hbox {d}}\tau$ are integrable on $(-\infty ,\xi ]$. Moreover,

$$\begin{aligned}&c\Phi (\xi )-p\varphi (\xi )-d\int ^{\xi }_{-\infty }(J*\Phi -\Phi )(s){\hbox {d}}s\nonumber \\&\qquad +2\varepsilon _2\int _0^{\infty }\int _{{\mathbb {R}}}(y+c\tau )h(y,\tau ) \int ^1_0\Phi (\xi -t(y+c\tau )){\hbox {d}}t{\hbox {d}}y{\hbox {d}}\tau \nonumber \\&\quad \ge \varepsilon _1\int ^{\xi }_{-\infty }\Phi (s){\hbox {d}}s +\varepsilon _1\int ^{\xi }_{-\infty }\int _0^{\infty } \int _{{\mathbb {R}}}h(y,\tau )\Phi (s-y-c\tau ){\hbox {d}}y{\hbox {d}}\tau {\hbox {d}}s. \end{aligned}$$

(25)

Since $\Phi (\xi )$ is increasing, for any $y\in {\mathbb {R}}$ we have

$$\begin{aligned} (y+c\tau )h(y,\tau )\Phi (\xi )\ge (y+c\tau )h(y,\tau )\int ^1_0\Phi (\xi -t(y+c\tau )){\hbox {d}}t. \end{aligned}$$

If $\varepsilon _2\ge 0$, then it follows from (25) that

$$\begin{aligned} \begin{aligned}&c\Phi (\xi )-p\varphi (\xi )-d\int ^{\xi }_{-\infty }(J*\Phi -\Phi )(s){\hbox {d}}s +2\varepsilon _2\Phi (\xi )\int _0^{\infty } \int _{{\mathbb {R}}}(y+c\tau )h(y,\tau ){\hbox {d}}y{\hbox {d}}\tau \\&\quad \ge \varepsilon _1\int ^{\xi }_{-\infty }\Phi (s){\hbox {d}}s. \end{aligned} \end{aligned}$$

(26)

The mean value theorem for integrals implies that for each $y>0$, there exist $\xi _1(y)\in (\xi ,\xi +y)$ and $\xi _2(y)\in (\xi -y,\xi )$ such that $\int ^{\xi +y}_{\xi }\Phi (s){\hbox {d}}s=y\Phi (\xi _1(y))$ and $\int ^{\xi }_{\xi -y}\Phi (s){\hbox {d}}s=y\Phi (\xi _2(y))$. It follows from the monotonicity of $\Phi$ that

$$\begin{aligned} \int ^{\xi }_{-\infty }(J*\Phi -\Phi )(s){\hbox {d}}s&=\int ^{\xi }_{-\infty }\int _{{\mathbb {R}}}J(y)[\Phi (x-y)-\Phi (x)]{\hbox {d}}y{\hbox {d}}x\nonumber \\&=\int _{{\mathbb {R}}}J(y)\int ^{\xi -y}_{\xi }\Phi (x){\hbox {d}}x{\hbox {d}}y\nonumber \\&= \int _0^{\infty }J(y)\int ^{\xi -y}_{\xi }\Phi (x){\hbox {d}}x{\hbox {d}}y+\int ^0_{-\infty }J(y)\int ^{\xi -y}_{\xi }\Phi (x){\hbox {d}}x{\hbox {d}}y\nonumber \\&= \int _0^{\infty }J(y)\int ^{\xi -y}_{\xi }\Phi (x){\hbox {d}}x{\hbox {d}}y+\int _0^{\infty }J(-y)\int ^{\xi +y}_{\xi }\Phi (x)dxdy\nonumber \\&= \int _0^{\infty }J(y)\left[ \int ^{\xi +y}_{\xi }\Phi (x){\hbox {d}}x-\int _{\xi -y}^{\xi }\Phi (x){\hbox {d}}x\right] {\hbox {d}}y\nonumber \\&= \int _0^{\infty }yJ(y)[\Phi (\xi _1(y))-\Phi (\xi _2(y))]{\hbox {d}}y>0. \end{aligned}$$

(27)

This, together with (26), implies that

$$\begin{aligned} \left[ c+2\varepsilon _2\int _0^{\infty } \int _{{\mathbb {R}}}(y+c\tau )h(y,\tau ){\hbox {d}}y{\hbox {d}}\tau \right] \Phi (\xi ) \ge \varepsilon _1\int ^{\xi }_{-\infty }\Phi (s){\hbox {d}}s+p\varphi (\xi ). \end{aligned}$$

(28)

If $\varepsilon _2<0$, that is, $\partial _1f(0,0)>\partial _2f(0,0)\ge 0$, then there exists $\xi _0'<\xi _0$ such that

$$\begin{aligned} \begin{aligned} c\varphi '(\xi )-p\varphi ''(\xi )-d(J*\varphi -\varphi )(\xi )&= f(\varphi (\xi ),\psi (\xi )) \ge f(\varphi (\xi ),0)\\&\ge \frac{1}{2}\partial _1f(0,0)\varphi (\xi )>\varepsilon _1\varphi (\xi ) \end{aligned} \end{aligned}$$

for all $\xi <\xi '_0$. Thus

$$\begin{aligned} c\Phi (\xi ) \ge \varepsilon _1\int ^{\xi }_{-\infty }\Phi (s){\hbox {d}}s+p\varphi (\xi ). \end{aligned}$$

(29)

Combing (28) and (29), we have

$$\begin{aligned}&\left[ c+2|\varepsilon _2|\int _0^{\infty } \int _{{\mathbb {R}}}|y+c\tau |h(y,\tau ){\hbox {d}}y{\hbox {d}}\tau \right] \Phi (\xi ) \ge \varepsilon _1\int ^{0}_{-\infty }\Phi (s+\xi ){\hbox {d}}s \\&\quad \ge \varepsilon _1\int ^0_{-r}\Phi (s+\xi ){\hbox {d}}s\ge \varepsilon _1r\Phi (\xi -r) \end{aligned}$$

for all $r>0$ and $\xi <\xi '_0$. Thus there exists $r_0>0$ and some $\theta \in (0,1)$ such that

$$\begin{aligned} \gamma \triangleq \frac{1}{r_0}\ln \frac{1}{\theta }\in (\lambda _1(c),\lambda _0) \end{aligned}$$

(30)

and

$$\begin{aligned} \Phi (\xi -r_0)\le \theta \Phi (\xi ) \end{aligned}$$

uniformly in $\xi$. Thus

$$\begin{aligned} \Phi (\xi -r_0)\exp \{-\gamma (\xi -r_0)\}\le \Phi (\xi )\exp \{-\gamma \xi \}. \end{aligned}$$

This, together with $\lim _{\xi \rightarrow \infty }\Phi (\xi )e^{-\gamma \xi }=0$, implies that

$$\begin{aligned} \sup \limits _{\xi \in {\mathbb {R}}}\{\Phi (\xi )\exp \{-\gamma \xi \}\}<\infty . \end{aligned}$$

(31)

Moreover,

$$\begin{aligned} \begin{aligned} \exp \{-\gamma \xi \}\int ^{\xi }_{-\infty }\Phi (s){\hbox {d}}s=&\exp \{-\gamma \xi \}\int ^{0}_{-\infty } \Phi (\xi +s){\hbox {d}}s\\ =&\int ^{0}_{-\infty }\Phi (\xi +s)\exp \{-\gamma (\xi +s)\}\exp \{\gamma s\}{\hbox {d}}s\\ \le&\frac{1}{\gamma } \sup \limits _{\xi \in {\mathbb {R}}}\{\Phi (\xi )\exp \{-\gamma \xi \}\}. \end{aligned} \end{aligned}$$

Thus, it follows from (28), (29), and (31) that

$$\begin{aligned} \sup \limits _{\xi \in {\mathbb {R}}}\{\varphi (\xi )\exp \{-\gamma \xi \}\}<\infty . \end{aligned}$$

(32)

This completes the proof. $\square$

Lemma 4.2

Assume that (F1), (F2), (H1), and (H2) hold. Let $(c, \varphi )$ be a solution of (6). Then $\lim _{\xi \rightarrow -\infty }\varphi (\xi )\exp \{-\lambda _1(c)\xi \}$ exists for each $c>c^*$.

Proof

Define a bilateral Laplace transform of $\varphi (\xi )$ by

$$\begin{aligned} L[\varphi ](\lambda )=\int _{{\mathbb {R}}}\exp \{-\lambda \xi \}\varphi (\xi ){\hbox {d}}\xi . \end{aligned}$$

By Lemma 4.1 and Fubini’s theorem, we have

$$\begin{aligned} \int _{{\mathbb {R}}}e^{-\lambda \xi }\psi (\xi ){\hbox {d}}\xi&= \int _{{\mathbb {R}}}\left[ e^{-\lambda \xi }\int _0^{\infty } \int _{{\mathbb {R}}} h(y,\tau )\varphi (\xi -y-c\tau ){\hbox {d}}y\right] {\hbox {d}}\xi {\hbox {d}}\tau \\&= \int _0^{\infty }\int _{{\mathbb {R}}}h(y,\tau ) \left[ \int _{{\mathbb {R}}}e^{-\lambda \xi } \varphi (\xi -y-c\tau ){\hbox {d}}\xi \right] {\hbox {d}}y {\hbox {d}}\tau \\&= L(\lambda )\int _0^{\infty }\int _{{\mathbb {R}}}h(y,\tau ) e^{-\lambda (y+c\tau )}{\hbox {d}}y{\hbox {d}}\tau = L(\lambda )G(c,\lambda ). \end{aligned}$$

Take the bilateral Laplace transform of (6) with respect to $\xi$, we have (with $\Delta (c,\lambda )$ defined in (11))

$$\begin{aligned} \Delta (c,\lambda )L[\varphi ](\lambda )={\mathcal {R}}(\lambda ), \end{aligned}$$

(33)

where ${\mathcal {R}}(\lambda )$ is the Laplace transform of the function $f(\varphi (\xi ),\psi (\xi ))-\partial _1f(0,0)\varphi (\xi ) -\partial _2f(0,0)\psi (\xi )$. It is not difficult to see that ${\mathcal {R}}(\lambda )$ is defined for $\lambda$ with $0< \mathrm {Re}\lambda < \gamma$. In addition, it is easy to see that $\Delta (c,\cdot )$ has no zero $\lambda$ with $\mathrm {Re}\lambda =\lambda _1(c)$ other than $\lambda =\lambda _1(c)$. This implies that $(\lambda -\lambda _1(c))/\Delta (c,\lambda )$ is analytic in the strip $0<\mathrm {Re}\lambda \le \lambda _1(c)$. If there exists some $\xi _0>0$ such that $\varphi (\xi )$ is increasing for all $\xi \in (-\infty ,-\xi _0)$, then $u(\xi )=\varphi (-\xi )$ is a positive decreasing function on $(\xi _0,\infty )$. Moreover, it follows from (33) that

$$\begin{aligned} \int ^{\infty }_{\xi _0}e^{\lambda \xi }u(\xi ){\hbox {d}}\xi =\int _{-\infty }^{-\xi _0}e^{-\lambda \xi }\varphi (\xi ){\hbox {d}}\xi =\frac{{\mathcal {E}}(\lambda )}{\lambda -\lambda _1(c)} \end{aligned}$$

(34)

with

$$\begin{aligned} {\mathcal {E}}(\lambda ) =\frac{(\lambda -\lambda _1(c)){\mathcal {R}}(\lambda )}{\Delta (c,\lambda )} -(\lambda -\lambda _1(c))\int ^{\infty }_{-\xi _0}e^{-\lambda \xi }\varphi (\xi ){\hbox {d}}\xi , \end{aligned}$$

which is analytic in the strip $0<\mathrm {Re}\lambda \le \lambda _1(c)$ because $\int ^{\infty }_{-\xi _0}\exp \{-\lambda \xi \}\varphi (\xi ){\hbox {d}}\xi$ is analytic for all $\mathrm {Re}\lambda >0$. By means of Theorem 4.1, $\lim _{\xi \rightarrow \infty }u(\xi )\exp \{\lambda _1(c)\xi \}$, which is equal to $\lim _{\xi \rightarrow -\infty }\varphi (\xi )\exp \{-\lambda _1(c)\xi \}$, exists.

If $\varphi (\xi )$ is not monotone on any interval $(-\infty , \xi _0)$ with $|\xi _0|$ sufficiently large, let $\chi (\xi )=\exp \{q\xi \}\varphi (\xi )$, where $q=d/c$ if $p=0$ and

$$\begin{aligned} q=\frac{-c+\sqrt{c^2+4pd}}{2p} \end{aligned}$$

if $p> 0$. Then

$$\begin{aligned} (c+2pq)\chi '(\xi )-p\chi ''(\xi )=\exp \{q\xi \}\left[ d J*\varphi (\xi ) +f(\varphi (\xi ),\psi (\xi ))\right] \ge 0. \end{aligned}$$

Suppose that there exists $\xi _1<\xi _2$ such that $\chi (\xi _1)>\chi (\xi _2)$. Note that $\lim _{\xi \rightarrow \infty }\chi (\xi )=+\infty$; thus, there exists $\xi _3>\xi _1$ such that $\chi '(\xi _3)=0$ and $\chi ''(\xi _2)\ge 0$, which contradicts the equation above. Thus, we have $\chi '(\xi )\ge 0$. Then for the bilateral Laplace transform of $\chi (\xi ), L[\chi ](\lambda )=L(\lambda -q)$. It follows from (33) that

$$\begin{aligned} \Delta (c,\lambda -q)L_1(\lambda )={\mathcal {R}}(\lambda -q). \end{aligned}$$

Using a similar argument as above, we see that

$$\begin{aligned} \lim \limits _{\xi \rightarrow -\infty }\varphi (\xi )\exp \{-\lambda _1(c)\xi \}=\lim \limits _{\xi \rightarrow -\infty }\chi (\xi )\exp \{-[q+\lambda _1(c)]\xi \} \end{aligned}$$

exists. This completes the proof. $\square$

Using a similar argument as in the proof of the previous lemma, we can verify the following result.

Lemma 4.3

Assume that (F1), (F2), (H1), and (H2) hold. Let $(c^*, \varphi )$ be a solution of (6). Then $\lim _{\xi \rightarrow -\infty }\varphi (\xi )\xi ^{-1}\exp \{-\lambda ^*\xi \}$ exists.

Now we are ready to summarise the asymptotic behaviour of wave profile $\varphi$ as follows.

Theorem 4.2

Under assumptions (F1), (F2), (H1) and (H2), for each solution of $(c, \varphi )$ of (6), there exists $\eta =\eta (\varphi )$ such that

$$\begin{aligned} \lim \limits _{\xi \rightarrow -\infty }\frac{\varphi (\xi +\eta )}{\exp \{\lambda _1(c)\xi \}}=1 \quad \text{ for } \quad c>c^* \end{aligned}$$

(35)

and

$$\begin{aligned} \lim \limits _{\xi \rightarrow -\infty } \frac{\varphi (\xi +\eta )}{\xi \exp \{\lambda _1(c)\xi \}} =1\quad \text{ for } \quad c=c^*. \end{aligned}$$

(36)

Moreover,

$$\begin{aligned} \lim \limits _{\xi \rightarrow -\infty }\frac{\varphi '(\xi )}{\varphi (\xi )} =\lambda _1(c)\quad \text{ for } \quad c\ge c^*. \end{aligned}$$

(37)

Proof

Both (35) and (36) follow easily from Lemmas 4.2 and 4.3. It follows from (6) that

$$\begin{aligned} \lim \limits _{\xi \rightarrow -\infty }\frac{c \varphi '(\xi )}{\varphi (\xi )}&=p\lim \limits _{\xi \rightarrow -\infty } \frac{\varphi ''(\xi )}{\varphi (\xi )}+ d \lim \limits _{\xi \rightarrow -\infty }\left[ \frac{J*\varphi (\xi )}{\varphi (\xi )} -1\right] +\lim \limits _{\xi \rightarrow -\infty } \frac{f(\varphi (\xi ),\psi (\xi ))}{\varphi (\xi )}\\&=p\lambda ^2_1(c)+ d\left[ H(\lambda _1(c))-1\right] +\partial _1f(0,0) +\partial _2f(0,0)\lim \limits _{\xi \rightarrow -\infty }\frac{\psi (\xi )}{\varphi (\xi )}\\&=p\lambda ^2_1(c)+ d\left[ H(\lambda _1(c))-1\right] +\partial _1f(0,0) \\&\qquad {} +\partial _2f(0,0)\lim \limits _{\xi \rightarrow -\infty } \int _0^{\infty }\int _{{\mathbb {R}}}h(y,\tau ) \frac{\varphi (\xi -y-c\tau )}{\varphi (\xi )}{\hbox {d}}y{\hbox {d}}\tau \\&=p\lambda ^2_1(c)+ d\left[ H(\lambda _1(c))-1\right] +\partial _1f(0,0) +\partial _2f(0,0)G(c,\lambda _1(c))\\&= c\lambda _1(c). \end{aligned}$$

This completes the proof. $\square$

Theorem 4.3

Under assumptions (F1), (F2), (H1), and (H2), for each solution of $(c, \varphi )$ of (6), there exists $\eta =\eta (\varphi )$ such that

$$\begin{aligned} \lim \limits _{\xi \rightarrow \infty }\frac{K-\varphi (\xi +\eta )}{\exp \{-\upsilon (c)\xi \}}=1, \end{aligned}$$

(38)

where $\upsilon (c)$ is the unique positive zero of ${\widetilde{\Delta }}(c,\cdot )$, according to Lemma 2.2. Moreover,

$$\begin{aligned} \lim \limits _{\xi \rightarrow \infty }\frac{\varphi '(\xi )}{K-\varphi (\xi )}=\upsilon (c). \end{aligned}$$

(39)

Proof

Define $\Phi (\xi )\triangleq K-\varphi (-\xi )$ and $\Psi (\xi )\triangleq K-(h**\varphi )(-\xi )$. Obviously, $\Phi (\xi )$ satisfies that $\Phi (-\infty )=0, \Phi (\infty )=K, 0<\Phi (\xi )<K$, and

$$\begin{aligned} c\Phi '(\xi ) = -p\Phi ''(\xi )+ f(K-\Phi (\xi ),K-\Psi (\xi ))-d(J*\Phi -\Phi )(\xi ). \end{aligned}$$

(40)

Then for any $\mu >2d+\max \{|\partial _1f(u,v)|:\,u,v\in [0,K]\}$, we have $\left[ \Phi (\xi )e^{-\mu \xi }\right] '<0$ for all $\xi$. Then, using the bilateral Laplace transform $L[\Phi ]$ of $\Phi (\xi )$, we have, using Fubini’s theorem again,

$$\begin{aligned} \int _{{\mathbb {R}}}e^{-\lambda \xi }\Psi (\xi ){\hbox {d}}\xi&= \int _{{\mathbb {R}}}e^{-\lambda \xi }\left[ K-\int _0^{\infty } \int _{{\mathbb {R}}}h(y,\tau ) \varphi (-\xi -y-c\tau ){\hbox {d}}y {\hbox {d}}\tau \right] {\hbox {d}}\xi \\&= \int _{{\mathbb {R}}}\left[ e^{-\lambda \xi }\int _0^{\infty } \int _{{\mathbb {R}}}h(y,\tau ) \Phi (\xi +y+c\tau ){\hbox {d}}y{\hbox {d}}\tau \right] {\hbox {d}}\xi \\&= \int _0^{\infty }\int _{{\mathbb {R}}}h(y,\tau )\left[ \int _{{\mathbb {R}}}e^{-\lambda \xi } \Phi (\xi +y+c\tau ){\hbox {d}}\xi \right] {\hbox {d}}y{\hbox {d}}\tau \\&= L[\Phi ](\lambda )\int _0^{\infty }\int _{{\mathbb {R}}}h(y,\tau ) e^{\lambda (y+c\tau )}{\hbox {d}}y{\hbox {d}}\tau = L[\Phi ](\lambda )G(c,-\lambda ). \end{aligned}$$

Take Laplace transform of (40) with respect to $\xi$, we have

$$\begin{aligned} {\widetilde{\Delta }}(c,\lambda ){\widetilde{L}}(\lambda ) =\widetilde{{\mathcal {R}}}(\lambda ), \end{aligned}$$

(41)

where $\widetilde{{\mathcal {R}}}(\lambda )$ is the Laplace transform of the function $f(K-\Phi (\xi ),K-\Psi (\xi ))-\partial _1f(K,K)\Phi (\xi )-\partial _2f(K,K)\Psi (\xi )$. Using a similar arguments as that in the proof of Lemma 4.2, we see that there exists $\eta =\eta (\varphi )$ such that

$$\begin{aligned} \lim \limits _{\xi \rightarrow -\infty }\frac{\Phi (\xi +\eta )}{\exp \{\upsilon (c)\xi \}}=1 \quad \text{ and } \quad \lim \limits _{\xi \rightarrow -\infty }\frac{\Phi '(\xi )}{\Phi (\xi )}=\lambda _1(c), \end{aligned}$$

(42)

from which (38) and (39) follow. The proof is completed. $\square$

5 Monotonicity and Uniqueness

In this section, we investigate the monotonicity and uniqueness (up to a translation) of the travelling wavefront of (6) by using the sliding method developed in Chen and Guo [12].

Theorem 5.1

Under assumptions (F1), (F2), (H1), and (H2), every solution $(c,\varphi )$ of (6) satisfies $\varphi '(\xi )>0$ for all $\xi \in {\mathbb {R}}$.

Proof

It follows from Lemma 3.1 and Theorems 4.2 and 4.3 that there exists $M>0$ such that $\varphi '(\xi )>0$ for all $|\xi |\ge M$. It thus suffices to show that $\varphi '(\xi )>0$ for all $\xi \in [-M,M]$. Suppose on the contrary that $\varphi '(\xi )\le 0$ for some $\xi _0\in [-M,M]$. By continuity of $\varphi '(\xi )$, there exists $\xi _1\in [-M,M]$ such that $\varphi '(\xi _1)=0$. Then, using the similar arguments as in the proof of Lemma 3.2, we obtain a contradiction. This completes the proof. $\square$

In order to prove the uniqueness up to a translation, we shall need the following strong comparison principle.

Lemma 5.1

Let $(c,\phi _1)$ and $(c,\phi _2)$ be solutions of (6) with $\phi _1\ge \phi _2$ on ${\mathbb {R}}$. Then either $\phi _1\equiv \phi _2$ or $\phi _1>\phi _2$ on ${\mathbb {R}}$.

Proof

Suppose that there exists some $\xi _0\in {\mathbb {R}}$ such that $\phi _1(\xi _0)=\phi _2(\xi _0)$. In view of $\phi _1\ge \phi _2$ on ${\mathbb {R}}$, it follows that ${\mathcal {H}}_{\mu }(\phi _1)(x)={\mathcal {H}}_{\mu }(\phi _2)(x)$ for all $x\ge \xi _0$. It follows from the monotonicity of ${\mathcal {H}}_{\mu }$ that $\phi _1\equiv \phi _2$ on ${\mathbb {R}}$. $\square$

Lemma 5.2

Under assumptions (F1) and (F2), there exists $\varepsilon _0\in (0,K)$ such that

$$\begin{aligned} f((1+s)u,(1+s)v)-(1+s)f(u,v)<0 \end{aligned}$$

(43)

for all $s\in (0,\varepsilon _0)$ and $(u,v)\in {\mathbb {R}}^2$ satisfying $|K-u|<\varepsilon _0$ and $|K-v|<\varepsilon _0$.

Proof

For each $s\ge 0$, define

$$\begin{aligned} F(s,u,v)=f((1+s)u,(1+s)v)-(1+s)f(u,v). \end{aligned}$$

Then $F(0,u,v)=0$ and $F_s(0,u,v)=u\partial _1f(u,v)+v\partial _2f(u,v)-f(u,v)$ for all $(u,v)\in {\mathbb {R}}^2$. In view of assumption (F2), we have $F_s(0,K,K)=K\partial _1f(K,K)+K\partial _2f(K,K)<0$. Therefore, there exists $\varepsilon _0>0$ such that $F(s,u,v)<0$ for all $s\in (0,\varepsilon _0)$ and $(u,v)\in {\mathbb {R}}^2$ satisfying $|K-u|<\varepsilon _0$ and $|K-v|<\varepsilon _0$. $\square$

Lemma 5.3

Assume that (F1), (F2), (H1), and (H2) hold. Let $(c,\phi _1)$ and $(c,\phi _2)$ be solutions of (6). Suppose there exists a constant $\varepsilon \in (0,\varepsilon _0]$ such that $(1+\varepsilon )\phi _1(x-\kappa \varepsilon )\ge \phi _2(x)$ on ${\mathbb {R}}$, where

$$\begin{aligned} \kappa =\sup \left\{ \frac{\phi _1(x)}{\phi '_1(x)}:\, \phi _1(x)\le K-\varepsilon _0\right\} . \end{aligned}$$

Then $\phi _1 \ge \phi _2$ on ${\mathbb {R}}$.

Proof

Define $W(\varepsilon ,x)=(1+\varepsilon )\phi _1(x-\kappa \varepsilon )-\phi _2(x)$ and $\varepsilon ^*=\inf \{\varepsilon \ge 0: W(\varepsilon ,x)\ge 0$ for all $x\in {\mathbb {R}}\}$. By continuity of $W, W(\varepsilon ^*,x)\ge 0$ for all $x\in {\mathbb {R}}$. We claim $\varepsilon ^*=0$. Suppose on the contrary that $\varepsilon \in (0,\varepsilon _0]$. Then, by the definition of $\kappa$,

$$\begin{aligned} W_{\varepsilon }(\varepsilon ,x)=\phi _1(x-\kappa \varepsilon ) -\kappa (1+\varepsilon )\phi '_1(x-\kappa \varepsilon )<0 \end{aligned}$$

on $\{x\in {\mathbb {R}}:\, \phi _1(x-\kappa \varepsilon )\le K-\varepsilon _0\}$. Noting that $W(\varepsilon ^*,\infty )=\varepsilon ^*K>0$, we can find $x_0$ with $\phi _1(x_0-\kappa \varepsilon ^*)>K-\varepsilon _0$ such that

$$\begin{aligned} 0=W(\varepsilon ^*,x_0)=W_{x}(\varepsilon ^*,x_0)=W_{xx}(\varepsilon ^*,x_0). \end{aligned}$$

Thus, $(1+\varepsilon ^*)\phi _1(\xi _0)=\phi _2(x_0), (1+\varepsilon ^*)\phi '_1(\xi _0)=\phi '_2(x_0)$, and $(1+\varepsilon ^*)\phi ''_1(\xi _0)= \phi ''_2(x_0)$, where $\xi _0=x_0-\kappa \varepsilon ^*$. This, together with (43), implies

$$\begin{aligned} 0&= -c\phi '_2(x_0)+p\phi ''_2(x_0)+d(J*\phi _2-\phi _2)(x_0)+f(\phi _2(x_0), (h**\phi _2)(x_0))\\&\le -c(1+\varepsilon ^*)\phi '_1(\xi _0)+p(1+\varepsilon ^*)\phi ''_1(\xi _0)+d(1+\varepsilon ^*)(J*\phi _1- \phi _1)(\xi _0)\\&\quad +f((1+\varepsilon ^*)\phi _1(\xi _0),\{h**[(1+\varepsilon ^*)\phi _1]\} (\xi _0))\\&=f((1+\varepsilon ^*)\phi _1(\xi _0),(1+\varepsilon ^*)(h**\phi _1) (\xi _0)) -(1+\varepsilon ^*)f(\phi _1(\xi _0), (h**\phi _1)(\xi _0))\\&< 0, \end{aligned}$$

a contradiction. Hence $\varepsilon ^*=0$ and so $\phi _1\ge \phi _2$ on ${\mathbb {R}}$.

Theorem 5.2

Assume that (F1), (F2), (H1), and (H2) hold. For each $c \ge c^*$, let $(c,\phi _1)$ and $(c,\phi _2)$ be two solutions to (6). Then there exists $\gamma \in {\mathbb {R}}$ such that $\phi _1(\cdot )=\phi _2(\cdot +\gamma )$, i.e. travelling waves are unique up to a translation.

Proof

By translating $\phi _2$ if necessary, we can assume that $0<\phi _1(0)=\phi _2(0)<K$. By Theorem 4.2, we have

$$\begin{aligned} \lim \limits _{x\rightarrow -\infty }\frac{\phi _2(x)}{\phi _1(x)}=e^{\lambda _1(c)\theta } \end{aligned}$$

for some $\theta \in {\mathbb {R}}$. Without loss of generality, we assume that $e^{\lambda _1(c)\theta }\le 1$, for otherwise we can exchange $\phi _1$ and $\phi _2$. Then

$$\begin{aligned} W(\xi )=\lim \limits _{x\rightarrow -\infty }\frac{\phi _2(x)}{\phi _1(x+\xi )}<1 \end{aligned}$$

for all $\xi >0$. Fix $\xi =1$; then there exists $x_1>0$ such that

$$\begin{aligned} \phi _1(x+1)>\phi _2(x) \text{ for } \text{ all } x\in (-\infty ,-x_1). \end{aligned}$$

(44)

Since $\phi _1(\infty )=K$, there exists $x_2\gg 1$ such that $\phi _1(x)\ge K/(1+\varepsilon _0)$ for all $x>x_2$. It follows that

$$\begin{aligned} (1+\varepsilon _0)\phi _1(x)\ge K\ge \phi _2(x) \text{ for } \text{ all } x>x_2. \end{aligned}$$

(45)

Let $\eta =\max \{\phi _2(x):\, x\in [-x_1,x_2]\}\in (0,K)$. In view of $\phi _1(\infty )=K$, there exists $x_3\gg 1$ such that $\phi _1(x)\ge \eta$ for all $x>x_3$. Thus, for $x\in [-x_1,x_2]$, we have $x+x_1+x_3\in [x_3,x_1+x_2+x_3]$ and hence

$$\begin{aligned} \phi _1(x+x_1+x_3)\ge \eta \ge \phi _2(x). \end{aligned}$$

(46)

Set $z=1+x_1+x_3+\kappa \varepsilon _0$. It follows from (44)–(46) that

$$\begin{aligned} (1+\varepsilon _0)\phi _1(x+z-\kappa \varepsilon _0)\ge \phi _2(x) \text{ for } \text{ all } x\in {\mathbb {R}}. \end{aligned}$$

(47)

By monotonicity of $\varphi _1$ and Lemma 5.3, $\phi _1(x+z)\ge \phi _2(x)$ for all $x\in {\mathbb {R}}$. Set

$$\begin{aligned} \xi ^*=\inf \{z>0:\, \phi _1(x+z)\ge \phi _2(x) \text{ for } \text{ all } x\in {\mathbb {R}}\}. \end{aligned}$$

We claim that $\xi ^*=0$. If not, then $\xi ^*>0$ and so we have $\phi _1(x+\xi ^*)\ge \phi _2(x)$ for all $x\in {\mathbb {R}}$. It follows from $W(\xi ^*/2)<1$ that there exists $x_4>0$ such that

$$\begin{aligned} \phi _1(x+\xi ^*/2)\ge \phi _2(x) \hbox {for } x\le -x_4. \end{aligned}$$

Consider the function $(1+\varepsilon )\phi _1(x+\xi ^*-2\kappa \varepsilon )$. Since $\phi _1(\infty )=K$ and $\phi '_1(\infty )=0$, there exists $x_5\gg 1$ such that

$$\begin{aligned} \frac{d}{{\hbox {d}}\varepsilon }\left\{ (1+\varepsilon ) \phi _1(x+\xi ^*-2\kappa \varepsilon )\right\} =\phi _1(x+\xi ^*-2\kappa \varepsilon )- 2\kappa (1+\varepsilon )\phi '_1(x+\xi ^*-2\kappa \varepsilon )>0 \end{aligned}$$

for all $x\ge x_5$ and $\varepsilon \in [0,1]$. That is, for all $x\ge x_5$ and $\varepsilon \in [0,1]$,

$$\begin{aligned} (1+\varepsilon )\phi _1(x+\xi ^*-2\kappa \varepsilon )\ge \phi _1(x+\xi ^*)\ge \phi _2(x). \end{aligned}$$

Now we consider the interval $[-x_4,x_5]$, since $\phi _1(\cdot +\xi ^*)\ge \phi _2(\cdot )$, by Lemma 5.1, $\phi _1(\cdot +\xi ^*)>\phi _2(\cdot )$ in $[-x_4,x_5]$. Thus, there exists $\varepsilon \in (0,\min \{\varepsilon _0,\xi ^*/(4\kappa )\})$ such that $\phi _1(\cdot +\xi ^*-2\kappa \varepsilon )\ge \phi _2(\cdot )$ on $[-x_4,x_5]$. Therefore, combining the estimates on $(-\infty ,-x_4], [-x_4,x_5]$, and $[x_5,\infty )$, we conclude that $(1+\varepsilon )\phi _1(\cdot +\xi ^*-2\kappa \varepsilon )\ge \phi _2(\cdot )$ on ${\mathbb {R}}$. It follows from Lemma 5.3 that $\phi _1(\cdot +\xi ^*-\kappa \varepsilon )\ge \phi _2(\cdot )$ on ${\mathbb {R}}$. This contradicts the definition of $\xi ^*$. Hence, $\xi ^*=0$, i.e. $\phi _1(\cdot )\ge \phi _2(\cdot )$. Since $\phi _1(0)=\phi _2(0)$, we have $\phi _1\equiv \phi _2$ on ${\mathbb {R}}$. $\square$

Finally, we show that no travelling wave solution of speed $c<c^*$ exists. The usual approach is to combine the comparison method and the finite time-delay approximation to establish the existence of the spreading speed $c^*$ for the solutions with initial functions having compact supports. In fact, $c^*$ coincides with the minimal wave speed for monotone travelling waves of (1). Thus, the nonexistence of travelling waves with the wave speed $c<c^*$ is a straightforward consequence of the spreading speed. In what follows, however, we shall employ a different method to investigate the nonexistence of travelling waves with the wave speed $c<c^*$.

Theorem 5.3

Assume that (F1), (F2), (H1), and (H2) hold. Let $c^*$ be defined as in Lemma 2.1. Then for every $c\in (0,c^*)$, (1) has no travelling wavefront with $(c,\varphi )$ satisfying (6).

Proof

In view of Theorem 5.1, every solution $(c,\varphi )$ of (6) satisfies $\varphi '(\xi )>0$ for all $\xi \in {\mathbb {R}}$. Take a sequence $\xi _n\rightarrow -\infty$ such that $\varphi (\xi _n)\rightarrow 0$ and set $v_n(\xi ) = \varphi (\xi +\xi _n)/\varphi (\xi _n)$. As $\varphi$ is bounded and satisfies (6), the Harnack’s inequality implies that the sequence $v_n$ is locally uniformly bounded. This function $v_n$ satisfies

$$\begin{aligned}&-cv'_n(\xi )+pv''_n(\xi )+d(J*v_n-v_n)(\xi )+\partial _1f(0,0)v_n(\xi )\nonumber \\&\quad +\partial _2f(0,0)0 (h**v_n)(\xi )+R_n(\xi )=0 \end{aligned}$$

(48)

for $\xi \in {\mathbb {R}}$, where

$$\begin{aligned} R_n(\xi )=\frac{f(\varphi (\xi +\xi _n),(h**\varphi )(\xi +\xi _n)) -\partial _1f(0,0)\varphi (\xi +\xi _n)- \partial _2f(0,0)(h**\varphi )(\xi +\xi _n)}{\varphi (\xi _n)}. \end{aligned}$$

The Harnack’s inequality implies that the shifted functions $R_n(\xi )$ converge to zero locally uniformly in $\xi$. Thus one may assume, up to extraction of a subsequence, that the sequence $v_n$ converges to a function v that satisfies:

$$\begin{aligned} -cv'(\xi )+pv''(\xi )+d(J*v-v)(\xi )+\partial _1f(0,0)v(\xi ) +\partial _2f(0,0) (h**v)(\xi )=0,\quad \xi \in {\mathbb {R}},\nonumber \\ \end{aligned}$$

(49)

Moreover, v is positive since it is nonnegative and $v(0) = 1$. Equation (49) admits such a solution if and only if $c\ge c^*$. Therefore, for every $c\in (0,c^*)$, (1) has no travelling wavefront with $(c,\varphi )$ satisfying (6). This completes the proof. $\square$