Adoption of the combined assurance model by South African state-owned enterprises (SOEs)

1.   Introduction

In the present paper, we focus on the nonlocal dispersal equation with spatially dependent bistable nonlinearity

where the nonlinearity $f(x,u)$ and kernel function $J(x)$ satisfy the following assumptions.

(F) The nonlinearity $f$ satisfies that

where $L>0$, $f_1$ and $f_2$ are two given bistable nonlinearities satisfying that

(J) The kernel function $J\in C^1(\mathbb{R})$ satisfies that

A typical example is $f(x,u) = u(1-u)(a(x)-u)$.

The nonlocal dispersal equation has attracted so much attention because of its extensive applications to account for diffusion phenomena involving jumps in biology, physics and chemistry [1,12]. Traveling waves as one kind of special solutions with invariant profile and fixed speed as well as entire solutions have been adequately investigated. It is well-known that the importance of the study of entire solutions of reaction-diffusion equations (nonlocal dispersal equations) is frequently recalled in the literature. Since the pioneering works of Hamel and Nadirashvili [13,14], there have been tremendous advances in studying the existence of entire solutions for various models. In particular, when the nonlinearity is homogeneous (i.e. $L = 0$ and $f_1 = f_2$), the theories of traveling waves and entire solutions for nonlocal dispersal equation 1 with various types of nonlinearities have been well established, the related results refer to [2,6,7,8,9,15,18,23,24,26,27] and references therein. Specifically, when the kernel $J(x)$ is compactly supported, Sun et al. [23] constructed a two-dimensional manifold of entire solutions which behave as two traveling wave solutions coming from both directions for bistable nonlocal dispersal equation.

However, for the inhomogeneous nonlinearity, several works are devoted to transition fronts (see [3,19]) of nonlocal dispersal equations [17,20] and forced waves in shifting habitats [25]. Traveling waves and spreading speed of monostable nonlocal dispersal equations with space periodic nonlinearity were studied in [21]. Li et al. [16] further obtained the existence of entire solutions for space periodic nonlinearity. Particularly, Eberle [10,11] constructed a heteroclinic orbit connecting two traveling waves for bistable local dispersal equation 1 in cylinders. Meanwhile, Berestycki and Rodríguez [5] considered a bistable nonlocal dispersal equation with a gap in one dimension. As far as we know, there is no result on entire solutions for the nonlocal dispersal equation 1 with inhomogeneous nonlinearity $f(x,u)$ satisfying $(F)$.

In this paper, we aim to construct an entire solution connecting traveling waves with the two different nonlinearities, motivated by [10,11]. By constructing suitable sub- and super-solutions, we establish the existence and uniqueness of the entire solution behaving as the traveling wave coming from one side and eventually going to the other side, which is different with the one constructed in [23] for the case $L = 0$ and $f_2 = f_1$. It should be pointed out that there are many differences to [4], though the similar method is applied. More precisely, since the nonlocal operator is not a compact operator as the Laplacian operator in [4], we have to establish the Lipschitz continuity of entire solutions in order to prove its uniqueness as in [23]. Compared to [23], we drop the assumption the kernel $J(x)$ is compactly supported.

The crucial part of this paper is to figure out the long time asymptotic behavior of the entire solution. Since the lack of compactness of the nonlocal operator, we can not use the Lyapunov function argument as [10,11] to show that the entire solution converges to a translation of the other traveling wave as time goes to positive infinity. However, inspired by the idea in [7], we use a "squeezing" technique to address this issue. Furthermore, we also apply sub- and super-solutions method with comparison principle to establish Lyapunov stability of the entire solution. This can be done because that we can obtain a positive estimate on the derivative of the entire solution with respect to $t$ when it is not so close to $0$ as well as $1$ and $t$ is large enough.

Now we state the main results of this paper as follows.

Theorem 1.1. Let assumptions $(F)$ and $(J)$ hold. Then there exists a unique entire solution $u(x,t)$ of 1 with $0<u(x,t)<1$, $u_t(x,t)>0$ for all $t\in\mathbb{R}$ and $x\in\mathbb{R}$ such that

and

for some $\beta\in\mathbb{R}$, where $(\phi_i,c_i)(i = 1,2)$ are the traveling wave solutions solving

Theorem 1.2. The entire solution $u(x,t)$ constructed in Theorem 1.1 is Lyapunov stable in the following sense: For any given $\epsilon>0$, there exists $\delta>0$ such that for any uniformly continuous $v_0(x)\in[0,1]$ with $\sup_{x\in\mathbb{R}}|v_0(x)-u(x+a,t_0)|<\delta$, the solution $v(x,t,v_0(x))$ of 1 with initial value $v_0(x)$ satisfies

for $x\in\mathbb{R}$ and $t\geq0$, where $a,\; t_0\in\mathbb{R}$ are two constants.

The rest of this paper is organized as follows. In Section 2, we recall some results of the bistable traveling waves for homogeneous nonlinearities and prove the comparison principle for 1. Section 3 is devoted to constructing the unique entire solution. In Section 4, we study the asymptotic profile as time goes to positive infinity. Finally, we establish Lyapunov stability of the entire solution in Section 5.

2.   Preliminaries

In this section, some known results on the traveling waves of 4 are outlined and the comparison principle is established.

2.1.   Traveling waves

It follows from Theorem 2.7 in [2] and Theorem 2.7 in [23] that 4 admits a solution $\phi_1(z)$ satisfying

where $\alpha_0,\; \alpha_1,\; \beta_0$ and $\beta_1$ are positive constants, $\lambda_0$ and $\lambda_1$ are the positive roots of

respectively. Moreover, we have

for some constants $\tilde{\alpha}_0,\; \tilde{\alpha}_1,\; \tilde{\beta}_0$ and $\tilde{\beta}_1>0$. At last, note that $f_1\in C^{1,1}([0,1])$, there exists some $L_f>0$ such that

2.2.   Comparison principle

We show that the following comparison principle holds by a contradiction argument.

Proposition 1. Suppose that assumptions $(F)$ and $(J)$ hold. Furthermore, let $u(x,t)$, $v(x,t)$ be continuous and bounded functions on $\mathbb{R}\times[0,T]$ for some $T>0$ satisfy

respectively. Then,

Furthermore, if $u(x,0)\not\equiv v(x,0)$ for $x\in\mathbb{R}$, then $u(x,t)>v(x,t)$ for $x\in\mathbb{R},\; t\in[0,T].$

Proof of Proposition 1. Let $\bar{w}(x,t) = u(x,t)-v(x,t)$. In fact, it is sufficient to show $\bar{w}(x,t)\geq0$ for $(x,t)\in\mathbb{R}\times[0,\epsilon_0T]$ with $\epsilon_0\in(0,1)$. Otherwise, suppose that $\inf_{\mathbb{R}\times [0,\epsilon_0T]}\bar{w}(x,t)<0$. Denote $\check{w}(x,t) = e^{Zt}\bar{w}(x,t)$, where $Z = \|f_u(x,u)\|_{\infty}+1$. It follows that $\inf_{\mathbb{R}\times[0,\epsilon_0T]}\check{w}(x,t)<0.$ Then there exists a sequence $(x_n,t_n)\in\mathbb{R}\times(0,\epsilon_0T]$ such that

Observe that

where $u_\theta(x,t)$ is between $v(x,t)$ and $u(x,t)$. This implies that

Letting $n$ converge to infinity, we have

Since $\inf_{\mathbb{R}\times[0,\epsilon_0T]}\check{w}(x,t)<0$, we can choose $\epsilon_0$ sufficiently small such that

Thus, we get a contradiction. Therefore, we obtain $u(x,t)\geq v(x,t)$ for $x\in\mathbb{R}$ and $t\in[0,T]$. The remaining part of this proposition can be proved similarly by replacing the auxiliary function $\check{w}(x,t) = e^{Zt}\bar{w}(x,t)$ with $\check{w}(x,t) = e^{Zt}\bar{w}(x,t)-\epsilon t$ for small $\epsilon>0$. In fact, we can similarly obtain

Here, we can also choose $\epsilon_0$ such that

This finishes the proof.

3.   Construction of the entire solution

In this section, we focus on the construction of the entire solution which behaves like a traveling wave approaching from infinity. The main idea is to establish suitable sub- and super-solutions, which are defined as follows.

and

here $\xi(t)$ is the solution of the following equation

where $M,\; \lambda$ and $T$ are positive constants to be specified later. A direct calculation yields that

For the function $\xi(t)$ to be defined, one must have $1-c_1^{-1}Me^{\lambda c_1t}>0$. Besides, there is

where $T: = \frac{1}{\lambda c_1}\ln\frac{c_1+M}{c_1}$.

Now we verify that $W^{\mp}$ are sub- and super-solutions of 1 for $t\leq-T$. Define

3.1.   Sub-solution

We first deal with the sub-solution $W^{-}(x,t)$. Since it is obvious for $x<0$, we only consider the case $x\geq0$. A straightforward calculation yields that

Recall that $f(x,u) = f_1(u)$ for $x\geq0$ and $\phi'_1>0$, it follows that

Then we continue to show $\mathcal{L}W^{-}(x,t)\leq0$ in two cases $0<x<-c_1t+\xi(t)$ and $x\geq-c_1t+\xi(t)$.

Case 1. For $0<x<-c_1t+\xi(t)$, similar to [4], we have

for some $m>0$. Consequently,

Therefore, if we choose $M\geq\frac{L_f\alpha_0}{m}$ then $\mathcal{L}W^{-}(x,t)\leq0$.

Case 2. For $x\geq-c_1t+\xi(t)$, if $\lambda_0\geq\lambda_1$, then we obtain

Since $c_1t+\xi(t)<0$ for $t<-T$ and $\lambda<\min\{\lambda_0,\lambda_1\}$, $\mathcal{L}W^{-}(x,t)\leq0$ provided $M\geq\frac{2L_f\alpha_0}{\tilde{\beta_1}}$.

For $\lambda_0<\lambda_1$, which means $f'_1(0)>f'_1(1)$, we have

Moreover, if $x+c_1t-\xi(t)>L$ for some $L\gg1$ such that $\phi_1(x+c_1t-\xi(t))\in[1-\sigma,1]$ and $\phi_1(-x+c_1t-\xi(t))\in[0,\sigma]$. It follows that

Since $t<-T$ such that $c_1t+\xi(t)\ll-1$, we have $\mathcal{L}W^{-}(x,t)\leq0$. Finally, if $0<x+c_1t-\xi(t)\leq L$, then, from the above case $\lambda_0\geq\lambda_1$, there holds

Thanks to $t<-T$, $\lambda<\min\{\lambda_0,\lambda_1\}$ and $M>\frac{2L_f\alpha_0}{\tilde{\beta_1}}e^{(\lambda_1-\lambda_0)L}$, we obtain

3.2.   Super-solution

We intend to testify the super-solution $W^+(x,t)$ in two steps.

Step 1. $W^+(x,t) = 2\phi_1(c_1t+\xi(t))$ for $x<0$. Following a direct calculation, we have

Denote

Then, it follows that

Let

It follows from assumption $(J)$ that there exists $K_J>0$ such that

Furthermore, there hold

Therefore, according to $t<-T$ such that $\phi_1(c_1t+\xi(t))\ll1$, we get

Since

consequently, we have $\mathcal{L}W^+(x,t)\geq0$ for $t<-T$.

Step 2. For $x\geq0$, $W^+(x,t) = \phi_1(x+c_1t+\xi(t))+\phi_1(-x+c_1t+\xi(t))$. Then we can obtain

Denote

We consider two cases.

Case 1. $0\leq x\leq-c_1t-\xi(t)$. Without loss of generality, let $\phi_1(0) = \theta_1$. Since $\phi'_1>0$ and $t<-T$ such that $\phi_1(c_1t+\xi(t))<\frac{1}{2}\theta_1$, we have

The second inequality follows from

which can be easily obtained by 5 and 6. As a consequence,

The last inequality holds since $\lambda<\min\{\lambda_0,\lambda_1,\eta\}$. Thus $\mathcal{L}W^+(x,t)\geq0$, provided that

Case 2. Here $x>-c_1t-\xi(t)$. From Case 1, we know

Moreover, if $\lambda_1\leq\lambda_0$, then

Remember that $\lambda<\min\{\lambda_0,\lambda_1,\eta\}$ and $t<-T$. As a matter of fact, take $M$ sufficiently large such that

This yields that $\mathcal{L}W^+(x,t)\geq0.$

If $\lambda_0<\lambda_1$, we have $f'_1(0)>f'_1(1)$. In addition, for $x>-c_1t-\xi(t)+\mathcal{M}$ with $\mathcal{M}\gg1$ such that $\phi_1(x+c_1t+\xi(t))\in[1-\sigma,1]$, we have

Meanwhile,

Then $\mathcal{L}W^+(x,t)\geq0$ by 7 and $-x<c_1t+\xi(t)-\mathcal{M}\ll-1$ such that

When $-c_1t-\xi(t)<x\leq-c_1t-\xi(t)+\mathcal{M}$, it follows from the case $\lambda_0\geq\lambda_1$ that

Similar to the case $\lambda_1\leq\lambda_0$, we can obtain $\mathcal{L}W^+(x,t)\geq0$ for $t<-T$ by letting $M$ be sufficiently large.

Proof of Theorem 1.1. Let $u_n(x,t)$ be the unique solution of the Cauchy problem

Since $W^-(x,-n) = u_n(x,-n)\leq W^+(x,-n)$, the comparison principle yields for $n>T$

Moreover, since $m: = \min_{\mathbb{R}\times[0,1]}(1-f_u(x,u))>0$ by the assumption $f_u(x,u)<1$, it is not difficult to see that $\delta u_n(x,t) = u_n(x+\eta,t)-u_n(x,t)$ with $\eta\in\mathbb{R}$ is a sub-solution of

where $L\geq\int_{\mathbb{R}}|J'(y)|dy$ and $M$ is some positive constant. It is then similar to the proof of Proposition 2.4 in [23] that $u_n(x,t)$ and $(u_n)_t(x,t)$ are Lipschiz continuous with respect to $x$. Besides, a direct calculation gives that $|u_n(x,t)|<C$ and $|(u_n)_t(x,t)|<C$ for some positive constant $C$. Then applying Arzela-Ascoli Theorem, there exists a subsequence of $\{u_n\}_{n = 1}^{+\infty}$ and $\{(u_n)_t\}_{n = 1}^{+\infty}$, still denoted by $\{u_n\}_{n = 1}^{+\infty}$ and $\{(u_n)_t\}_{n = 1}^{+\infty}$, such that $\{u_n\}_{n = 1}^{+\infty}$ and $\{(u_n)_t\}_{n = 1}^{+\infty}$ converge to a function $u(x,t)$ and $u_t(x,t)$. Now we have obtained the entire solution. In fact, the regularity of this entire solution is not so well. But we can show the entire solution is Lipschitz continuous in $x\in\mathbb{R}$, which is important to show the uniqueness of the entire solution.

Proposition 2. Let $u(x,t)$ be the entire solution in Theorem 1.1. Then $u(x,t)$ satisfies

and

for $M',\; M">0$.

The proof is similar to that of Proposition 3.1 in [23] by virtue of the fact $f_u(x,u)<1$ for $x\in\mathbb{R}$ and $u\in[0,1]$, so we omit it here.

As for the uniqueness, since it is easy to see $u(x,t)\rightarrow\phi_1(x+c_1t)$ as $t\rightarrow-\infty$ by the sub- and super-solutions established above, we can similarly show that the conclusion of Lemma 3.1 in [4] also holds true here. Then referring to the process of Section 3 in [4], we can obtain the uniqueness of the entire solution by constructing similar sub- and super-solutions. Thus we have shown the first part of Theorem 1.1.

4.   Long time asymptotic profile

In this section, we are going to show the entire solution established in previous section converges to a shift of $\phi_2(x+c_2t)$ as $t\rightarrow+\infty$. The main idea here is first to construct suitable sub-and super-solutions taking advantage of the traveling wave $\phi_2(x+c_2t)$ to get proper lower and upper bounds, which is a classical and regular method in study of traveling waves. Then we apply a "squeezing" technique which is used to prove the asymptotic stability of the traveling waves for nonlocal dispersal equation in [7] to obtain the convergence of the entire solution. Here, we assume that for some $X>0,\; \omega,\; \sigma>0$ and

there hold

besides,

Theorem 4.1. Let assumptions $(F)$ and $(J)$ hold, and $u(x,t)$ be the unique entire solution of 1 with $0<u(x,t)<1$, $u_t(x,t)>0$ for all $t\in\mathbb{R}$ and $x\in\mathbb{R}$ such that

Then

for some $\beta\in\mathbb{R}$.

Lemma 4.2. Suppose assumptions $(F)$ and $(J)$ hold. Then there exist $T_-$, $\beta_-$, $\; \delta_-,\; \omega>0$ such that

for $x\in\mathbb{R}$ and $t\geq T_-$.

Proof of Lemma 4.2. In fact, it follows from 8 that there exists $t_-\in\mathbb{R}$ such that

for any $\epsilon_->0$ and $x\in\mathbb{R}$. Denote

where

We first show $\tilde{u}(x,0)\geq\underline{u}(x,0)$ for $x\in\mathbb{R}$. Then there exists $x_1>0$ such that

As $u_t>0$ and $\phi'_i>0$ $(i = 1,2)$, we have

for $t\geq t_-,\; x\geq x_1$. Then,

Meanwhile, there exists $x_2\in\mathbb{R}$ such that

Particularly, there holds

In addition, $\min\limits_{x\in[x_1,x_2]}u(x,t)>0$ since $0<u(x,t)<1$. Then we can choose $\beta^0>0$ such that

From all the discussion above we obtain $\tilde{u}(x,0)\geq\underline{u}(x,0)$. Next, we prove

In fact, we only need to consider $\underline{u}(x,t) = \phi_2(\xi_-(x,t))-C_ve^{-\omega t}$ because $0$ is obviously a sub-solution of 1. Hence,

In the following, we continue the proof in three cases.

Case $\xi_-(x,t)\leq-X$. In this case,

Then, by $\omega<\frac{|f'_2(0)|}{4}$,

Case $\xi_-(x,t)\geq X$. For this case, by $2C_v<\sigma$,

Then, since $\omega<\frac{|f'_2(1)|}{4}$,

Case $\xi_-\in[-X,X]$. Let $\tau_0 = \min\limits_{\xi\in[-X,X]}\phi'_2(\xi)$. Then,

Choose $C^v = \tau^{-1}_0\omega_0^{-1}(\|f'_2\|_{\infty}C_v++C_v\omega)$. This gives $\mathcal{L}\underline{u}(x,t)\leq0$. We finish the proof by letting $\beta_- = \beta^0+C^v$ and $\delta_- = C_v$.

Similarly, we can prove the following lemma.

Lemma 4.3. Under the assumptions of Lemma 4.2, there exist $T_+>T_-$, $\; \beta_+$, $\; C_+,\; \omega>0$ such that

for $x\in\mathbb{R},\; t\geq T_+$, where $\omega,\; T_-$ are defined in Lemma 4.2.

Now we are in a position to establish an important lemma.

Lemma 4.4. Under the assumptions of Lemma 4.2, there exist $T>\max\{T_+,T_-\}$, $\; \beta,\; C,\; \omega>0$ such that

for $x\in\mathbb{R},\; t\geq T$, where $\omega,\; T_-,\; T_+$ are defined in Lemmas 4.2 and 4.3.

Proof of Lemma 4.4. Define $\hat{u}(x,t) = u(x,t+T)$ and $\overline{u}(x,t) = \min\{\phi_2(\xi_+(x,t))+v(t),1\}$, where

and

for $0<\epsilon_1,\; C_3<\frac{\sigma}{4}$ and $t\geq0$. Here, $\lambda^1$ is the positive root of

Then

and

Now we intend to show $\hat{u}(x,0)\leq\overline{u}(x,0)$. Recall the fact that $|1-\phi_2(x)|\leq C^2e^{-\lambda^1 x}$ for $x\in\mathbb{R},\; C^2>0$. Meanwhile, we assume that $0<\omega<\lambda^1c_2$. Additionally, by the assumptions of $f(x,u)$, there exists $C_0>0$ such that

Since $\phi_i(-\infty) = 0$ and $\phi_i(+\infty) = 1$, there exists $x_3<0$ such that

as well as $x_4>0$ such that

for $x>x_4$. Note that we can take $T>0$ sufficiently large such that

Therefore, it is obvious that

Furthermore, since $\max\limits_{x\in[x_3,x_4]}u(x,T)<1$, one can choose $\beta^+>0$ such that

This yields that $\hat{u}(x,0)\leq\overline{u}(x,0)$.

In the following, we are going to prove

Here we only need to consider $\overline{u}(x,t) = \phi_2(\xi_+(x,t))+v(t)$ since $1$ is obviously a super-solution of 1. A direct calculation gives that

We continue to prove $\mathcal{L}\overline{u}(x,t)\geq0$ in three cases.

Case 1. If $\xi_+>X$ and $x\geq-L$, then

Particularly, choose $C_3$ such that

which means

As a consequence,

In addition, if $\xi_+>X$ and $x<-L$, then

Case 2. If $\xi_+<-X$, then $x<-L$. Similar to Case 1 with $x<-L$, we can obtain $\mathcal{L}\overline{u}(x,t)\geq0.$

Case 3. For $\xi_+\in[-X,X]$, we know $x<-L$. Therefore,

where $\tau_0$ is defined as in Lemma 4.2. It follows from

that $\mathcal{L}\overline{u}(x,t)\geq0.$ This finishes the proof by letting $\beta = \beta^++\epsilon_1+C_3$ and $C = C_3+\epsilon_1$.

Now we are ready to show that the entire solution converges to a shift of $\phi_2(x+c_2t)$ using a "squeezing" technique [7]. We should declare that assumption (A) (see (B2) in [7]) holds.

$(\textbf{A})$ There exists a positive nonincreasing function $\zeta(n)$ defined on $[1,+\infty)$ such that for any $u_1(x,t),\; u_2(x,t)$ satisfying $-1\leq u_1(x,t),\; u_2(x,t)\leq2$, $\mathcal{L}u_1(x,t)\geq0$, $\mathcal{L}u_2(x,t)\leq0$ and $u_1(x,0)\geq u_2(x,0)$, there holds

In fact, it follows from the comparison principle that $u(x,t)\geq v(x,t)$ for $x\in\mathbb{R}$ and $t\geq0.$ Furthermore, denote $\breve{w}(x,t) = e^{Kt}[u(x,t)-v(x,t)]\geq0$ with $K\geq1+\max_{s\in[-1,2]}|f_s(x,s)|$. Then

which implies that $\breve{w}(x,t)\geq\breve{w}(x,0)$ for $t\geq0$. Therefore, $\breve{w}_t(x,t)\geq J*\breve{w}(x,0)$ and $\breve{w}(x,t_*)\geq t_*J*\breve{w}(x,0)$. Repeating the same progress on $[t_*,2t_*],\; ...,\; [(N-1),Nt_*]$, we have $\breve{w}(x,Nt_*)\geq t^N_*J*\breve{w}(x,0)$ for $N\geq1$. In addition, since $J(x)\geq0$ and $\int_{\mathbb{R}}J(y)dy = 1$, there exists $N_0 = N_0(M)\geq1$ such that by letting $t_* = \frac{T}{N_0}$ we have

where

Thus, there exists a positive function $\eta(x,t)\in C([0,\infty),[0,\infty))$ such that

which implies the assumption (A) holds.

Now, we start to prove the following lemma, which plays an important role in the proof of Theorem 4.1.

Lemma 4.5. Suppose that assumptions $(F)$ and $(J)$ hold. Then there exists a small $\epsilon_0$ such that if for some $\tau\geq0,\; \xi\in\mathbb{R},\; \delta\in(0,\frac{\sigma}{2}]$, and $h>0$, there holds

then for every $t>\tau+1$, there exist $\tilde{\xi}(t),\; \tilde{\delta}(t)$ and $\tilde{h}(t)$ satisfying

such that 9 holds with $(\tau,\xi,\delta,h)$ replaced by $(t,\tilde{\xi}(t),\tilde{\delta}(t),\tilde{h}(t))$. Here, $C_m: = \max\{\delta_-,C\}$ and $\gamma: = \max\left\{C^v,\frac{K_V}{\omega}\right\}$, where the parameters are defined in Lemmas 4.2 and 4.4.

Proof of Lemma 4.5. In view of Lemmas 4.2 and 4.4, it is easy to see that

Furthermore, as in the proofs of Lemmas 4.2 and 4.4,

and

Denote

In addition, by the definition of $C_m$, $\gamma$ and letting $\check{u}(x,t) = u(x-\kappa,t)$, we have

Let $\overline{h} = \min\{h,1\}$ and $\vartheta_0 = \frac{1}{2}\min\limits_{[0,2]}\phi'_2(x)$. Then

Therefore, at least one of the following two inequalities is true

Next, we consider the case (i) since the case (ii) is similar. According to the assumption (A), for $\zeta = \zeta(M+c_2+2)$ with $M\gg1$ such that $\frac{\gamma}{C_m}\phi'_2(x)<1$ for $|x|\geq M$, and for every $x\in[-M-c_2-\gamma,M+c_2+\gamma]$, there holds

Now define

Accordingly, there exists $\tilde{\theta}\in(0,1)$ such that

for all $x\in[-M-c_2-(\gamma-1),M+c_2+(\gamma-1)].$ Hence,

for $x\in[-M-c_2-(\gamma-1),M+c_2+(\gamma-1)]$.

For $|x|\geq M+c_2+(\gamma-1)$, by the choice of $M$, we know that

Then, it follows that

Note that $p: = \delta e^{-\omega}+\epsilon_0\overline{h}\leq\sigma$, repeat the operation above with $1$ replaced by $1+t'$, which implies that

Thus we finish the proof by setting

Now we shall prove Theorem 4.1.

Proof of Theorem 4.1. We shall divide the proof in three steps.

Step 1. Following from Lemmas 4.2 and 4.4, there exist $T^*,\; M^*>0$ such that

Here, $C_m$ is defined as in the proof of Lemma 4.4. Define

Meanwhile, fix $t^*\geq2$ such that

Then, replace $C_m$ with $\epsilon^*$ and denote $M^*,\; T^*$ by $\frac{h_0}{2},\; T_0$. Assume that $h_0\geq1$, otherwise, we directly go to Step 2.

By 10, applying Lemma 4.5 with $\tau = T_0$, $\xi = -\frac{h_0}{2}$, $h = h_0$, $\delta = \epsilon^*$, 9 holds with $\tau = T_0+t^*$, some $\xi\in\left[-\frac{h_0}{2}-\frac{\gamma\epsilon^*}{C_m},\frac{h_0}{2}+\frac{\gamma\epsilon^*}{C_m}\right]$, $\delta = \epsilon^*$ and $h = h_0-k^*$, by the definition of $k^*,\; t^*$, it follows that

Repeat the same process, it yields that 9 holds for $\tau = T_0+Nt^*$, $\delta = \epsilon^*$ and $h = h_0-Nk^*$ with $N$ such that $h_0-(N-1)t^*\geq1$. Thus, there exists $T_1>T_0$ such that 9 holds for $\tau = T_1$, $\delta = \epsilon^*$ and $h = 1$ and some $\xi\in\mathbb{R}$.

Step 2. In this step, we use a mathematical induction to show that for every nonnegative integer $k$, 9 holds for $\xi = \xi^k\in\mathbb{R}$ and

It is obvious that the assertion holds for $k = 0$ by Step 2. Then suppose that the assertion is true for $k = l\geq0$. We show that it is true for $k = l+1$. In fact, as can be seen in Lemma 4.5 with $\tau = T^l$ and $t = T^{l+1}$, one can obtain that 9 holds with $(\tau,\xi,\delta,h)$ replaced by $(\hat{\tau},\hat{\xi},\hat{\delta},\hat{h})$ satisfying

by the definition of $\epsilon^*,\; k^*$ and $t^*$. This means that 9 holds for $\tau = T^{l+1}$, some $\xi = \xi^{l+1}\in\left[\xi^l-\delta^l\frac{\gamma}{C_m},\xi^l+\delta^l\frac{\gamma}{C_m}\right]$, $\delta = [1-k^*]^{l+1}\epsilon^*$ and $h = [1-k^*]^{l+1}$. Now we finish the mathematical induction.

Step 3. So far, we have known that 9 holds for $(\tau,\xi,\delta,h) = (T^k,\xi^k,\delta^k,h^k)$ for all $k = 0,1...$. Furthermore, 9 holds with $\tau\in[T^k,\infty)$, $\delta = \delta^k$, $h = h^k+2\delta^k\frac{\gamma}{C_m}$ and $\xi = \xi^k-\delta^k\frac{\gamma}{C_m},\; k = 0,1...$.

Now we define

for $t\in\left[T^k,T^{k+1}\right],\; k = 0,1...$. Then,

It follows from the definition of $\delta(t)$ and $h(t)$ that

Moreover, since for any $t\geq\tau\geq T_1$,

there holds

which implies that $\xi(\infty): = \lim\limits_{t\rightarrow+\infty}\xi(t)$ exists and

Therefore, we have that

Furthermore, the convergence is exponential. Then we complete the proof.

5.   Lyapunov stability of the entire solution

We investigate the Lyapunov stability of the entire solution obtained in Theorem 1.1 in this section. That is, the aim here is to prove Theorem 1.2. The following lemma plays an important role in proving Theorem 1.2.

Lemma 5.1. Let $u(x,t)$ be the unique entire solution in Theorem 1.1. Then for any $\varphi\in(0,\frac{1}{2}]$, there exist constants $T_\varphi = T_\varphi(\varphi)>1$ and $K_\varphi = K_\varphi(\varphi)>0$ such that

where

Proof of Lemma 5.1. It is easy to choose $T_\varphi$ and $M_\varphi$ such that

Now suppose there exist sequences $t_k\in[T_\varphi,+\infty)$ and $x^k\in\Omega_{\varphi}(t)$ such that

Here only two cases happen, $t_k\rightarrow+\infty$ or $t_k\rightarrow t_*$ for some $t_*\in[T_\varphi,+\infty)$ as $k\rightarrow+\infty$.

For the former case, denote

By Proposition 2, $\{u_k(x,t)\}_{k = 1}^\infty$ is equicontinuous in $x\in\mathbb{R}$ and $t\in\mathbb{R}$. Furthermore, there exists a subsequence still denoted by $\{u_k(x,t)\}_{k = 1}^\infty$ such that

for some function $u_*$ satisfying 1 with $\frac{\partial u_*(0,0)}{\partial t} = 0$ and $(u_*(x,t))_t\geq0$. By the comparison principle, we further have

However, this is impossible because by Theorem 4.1

For the second case, $x^k$ remains bounded by the definition of $\Omega_{\varphi}(t)$. Therefore, we assume that $x^k\rightarrow x^*$ as $k\rightarrow+\infty$ and let

Then, each $u_k(x,t)$ is defined for all $(x,t)\in\{x\in\mathbb{R}\mid x\leq-1\}\times[T_\varphi,+\infty)$ by the definition of $\Omega_\varphi(t)$. Similarly, there exists a subsequence, again denoted by $\{u_k\}^{\infty}_{k = 1}$, such that

for some function $u^*$ satisfying 1 with $\frac{\partial u^*}{\partial t}(0,t_*) = 0,\; (u^*(x,t))_t\geq0.$ Then by the comparison principle, we obtain $\frac{\partial u^*}{\partial t}(x,t)\equiv0$ for $t\geq t_*$, but this is impossible since by Theorem 4.1,

This ends the proof.

Proof of Theorem 1.2. We first define a pair of sub- and super-solutions as follows.

where $\omega$ is defined as in Section 4 and $t_0\geq T_\varphi$ and $\tilde{\delta},\; \varpi>0$ are constants. Besides, we claim that $U^{\pm}$ are super- and sub-solutions of 1. Then, by the comparison principle, we have

In view of that for all $|\tau|\leq\varpi$ with $\varpi = :\varpi(\epsilon) = \frac{\epsilon}{2\tilde{M}}$ and $\tilde{M} = 2+\max\limits_{u\in[0,1]}(f_L(x,u))_u$, we have

It then follows from 11 and 12 that

for $x\in\mathbb{R}$ and $t\geq0$.

Now we prove the claim. We show $U^+(x,t)$ is a super-solution of 1 for $t\geq0$. Then it can be similarly shown that $U^-(x,t)$ is a sub-solution of 1 for $t\geq0$. Since it is easy to see that $U^+(x,0)\geq v_0(x)\; \text{for}\; x\in\mathbb{R}\; \text{and}\; \delta<\frac{\varpi}{2},$ we only need to show that

We go further to show $\mathcal{L}U^+(x,t)\geq0$ in two cases.

Case 1. For any $u = u(x+a,t+\gamma+\tilde{\delta}\varpi(1-e^{-\omega t}))\in[0,\sigma]\cup[1-\sigma,1]$ with $\sigma$ defined as in Section 4,

Case 2. For $u\in[\sigma,1-\sigma]$, it follows from Lemma 5.1 that there exists $K_\sigma$ such that $u_t\geq K_\sigma$. Therefore,

Let $\tilde{\delta}$ be sufficiently large such that

Thus, we have $\mathcal{L}U^+(x,t)\geq0$. Now, we obtain the claim. The proof of Theorem 1.2 is completed.

Remark 1. In this paper, we have considered the existence, uniqueness, asymptotic behavior and Lyapunov stability of entire solutions of the nonlocal dispersal equation 1 under the assumption $J(x) = J(-x)$, $x\in\mathbb{R}$. It is well-known from [8,22,28,29] that the asymmetry of $J$ has a great influence on the profile of the traveling waves and the sign of the wave speeds, which further makes the properties of the entire solution more diverse. Naturally, an interesting problem is to consider entire solutions of 1 under asymmetric conditions.

Remark 2. The method used here can be also applied to consider the bistable lattice differential equations with $f_i$ satisfying $(F)$

The existence, uniqueness, asymptotic behavior and the Lyapunov stability of entire solutions to 13 can be similarly obtained.

While for bistable random diffusion equations with $f(x,u)$ satisfying $(F)$

the construction of the entire solution behaving as the traveling wave pertaining to $f_1$ coming from infinity is similar. In fact, since regularity of the entire solution is known, the Lyapunov function method can be used to show the approach of the entire solution to the traveling wave pertaining to $f_2$, one can refer to [10,11].