Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect

1.   Introduction

In this paper, we will consider the following Cauchy problem

where

and $\phi(x)$ satisfies the following assumptions

The model (1) is described as the population dynamics with nonlocal consumption of resources. Where $u(x,t)$ represents the density of population at the position $x$ and time $t$. The coefficients $k$, $b$ are positive. The term $ku^{2}(1-\phi*u)$ is interpreted as the reproduction of the population, which is proportional to the square of density $u$, and to available resources $1-\phi*u$. The convolution term $\phi*u$ represents the nonlocal consumption of resources, specific is that the consumption of resources at the space point $x$ is determined by the individuals located at $y\in(-\infty,\infty)$. The last term $-bu$ represents mortality of the population. Moreover, the nonlocal model (1) can be used to describe the emergence and evolution of biological species and the process of speciation [7, 12].

When $\phi(x)$ tends to be a $\delta$-function, the model (1) can be changed to the classical reaction-diffusion equation

if $k>4b$, the equation (2) has three zeros

there are two stable steady-states $u_{-}$ and $u_{+}$, $u_{0}$ is unstable point, which implies that the model (2) is bistable. The traveling wave solution, asymptotic speed of propagation and Cauchy problem of the equation (2) have been studied extensively, see [22]. Since the consumption of resources with nonlocal makes more reasonable for describing the behavior of various biological phenomena. Apreutesei et al. [5] introduced the nonlocal consumption of resources into equation (2), that is the equation (1), they explored the Fredholm operator and used it to prove the existence of travelling waves in case that the integral is sufficiently small. For more results of (1) can refer to [2, 7, 22]. When the support of the function $\phi$ is not small, Demin and Volpert [9] and Alfaro et al. [4] proved the existence of traveling wave solution when the nonlinearities with the form of $u(\phi*u)(1-u)-\alpha u$ and $u(u-\theta)(1-\phi*u)$, respectively. Moreover, Li et al. [19] and Han et al. [14] obtained the general results for the problem (1) when the nonlocal term without limit by using monotone iteration method and Leray-Schauder degree theory, respectively. But it is only partially solved and much work in this area remains open questions. It is worth noting that two points should be pointed out through the above researches, one is that the difficulties caused by different nonlocal locations are different, and the other is that the difficulties caused by different nonlocal strength are also different. Therefore, the methods of studying the problem are quite different.

Actually, most of the research on the nonlocal reaction-diffusion equation is focused on monostable case (the dynamic behavior of the solution is relatively simple), such as Hamel and Ryzhik [13], Ai [1], Faye and Holzer [11] and others have done a vest of researches, and have obtained plenty of meaningful results in the traveling wave solution [3, 20, 21], the asymptotic propagation speed of the solution [1, 6, 13], the spatial dynamic behavior of the solution corresponding to the Cauchy problem [8], the bifurcation [11].

Especially in recent decades, there are many results about the spatial dynamics behavior of the Cauchy problem [10, 15, 16, 17, 18, 19, 23]. Deng and Wu [10] analyzed the global stability for Cauchy problem

and shown the existence and uniqueness of the solution by establishing comparison principle and constructing monotone sequences. Han and Yang [17] further considered the nonlocal reaction-diffusion-mutation model

and obtained the well-posedness of solutions, including the existence, uniqueness and global stability. From the above researches, it can be seen that the study of the spatial dynamics of the solution has a great enlightenment and guidance on understanding the nonlocal effect. To date, however, the research on the Cauchy problem of the nonlocal bistable reaction-diffusion equation is still blank. So a very natural question, what is the solution of the nonlocal bistable reaction-diffusion equation corresponding to the Cauchy problem?

Inspired by [4, 5, 10, 17], we will try to solve (or partially solve) the global dynamics of the solution of problem (1) in this paper. The main difficulty is that due to the introduction of nonlocal term, the maximum principle of problem (1) is not valid and the maximum modulus estimate of solution cannot be obtained. For such a difficulty, we will define suitable super- and sub-solutions and construct monotonic iterative sequences to obtain the existence of the solution. Furthermore, the uniqueness is given by using fundamental solution and Gronwall's inequality. Finally, we obtain the uniform boundedness of the solution by means of auxiliary function.

This paper is organized as follows. In section 2, some preparations including the notion of super- and sub-solutions, as well as the comparison principle will be given. Then we consider the existence and uniqueness of the global solution of the Cauchy problem (1) in Section 3. The results of the asymptotic behavior of solution will be obtained in Section 4.

2.   Preliminaries

In this section, we will do the preparation works. First, we will review the notion of the super- and sub-solutions of the problem (1), and then give the order of the super- and sub-solutions. For convenience, we define $I_T = \mathbb{R}\times(0,T)$ and $B_T = \mathbb{R}\times[0,T)$. We now make the following definition, which will be used frequently in our paper.

Definition 2.1. Assume $\overline{u}(x,t)$, $\underline{u}(x,t)\in C^{2,1}(I_T)\cap C_{B}(B_T)$, where $C_B$ is a bounded continuous space. They are called super- and sub-solutions of Cauchy problem (1), respectively, if $\overline{u}(x,t)$ and $\underline{u}(x,t)$ satisfy

and

Under the definition of super- and sub-solutions, the following result is obtained.

Lemma 2.2. Suppose that $\overline{u}(x,t)$ and $\underline{u}(x,t)$ are super- and sub-solutions of problem (1), respectively, where $\overline{u}(x,t)$ and $\underline{u}(x,t)$ are nonnegative bounded functions. Then,

Proof. Let $u(x,t) = \overline{u}(x,t)-\underline{u}(x,t)$. Then for any $(x,t)\in I_T$, we have

This yields that

where

Take $\sigma>0$ large enough such that

Denote $\widetilde{u} = e^{-\sigma t}u$, from (5) we can see that

Since the functions $\overline{u},\; \underline{u}$ are nonnegative bounded in $B_T$, then, there exists $M>0$ such that

Further, we know that $d_2(x,t)$ is nonnegative bounded in $B_T$. To complete our proof, it suffices to show that

where $T_{0} = \min \left\{T,\frac{s}{kM^{2}}\right\}$ and $s$ is positive which will be determined by (6). For contradiction we assume that $\widetilde{u}<0$ at some points in $I_{T_0}$, then, we get

since $\widetilde{u}$ is bounded, then there exists a positive constant $s$ and a point $\left(x^*,t^*\right)\in I_{T_0}$, such that

and

We now define

where $\zeta$ is a positive constant. Then, we obtain

thus,

By the definition of $w(x,t)$ that $\lim_{|x|\rightarrow+\infty}w(x,t) = 0$ and together with (7), hence there exists a point $\left(\widetilde{x},\widetilde{t}\; \right)\in I_{T_0}$, such that $w$ reaches the negative minimum $w_{min}$, one has,

Combining with (6), we have

equivalent to

Due to $w_t\leq0, w_x = 0$ and $w_{xx}\geq0$ at $\left(\widetilde{x},\widetilde{t}\; \right)$, together with (7), we get

Furthermore, we deduce that

together with (8), one has

then,

further,

clearly

since $x^*$ is independent of $\zeta$, if choosing $\zeta$ large enough that the inequality (9) doesn't hold, which is a contradiction, this yields

If $T>T_0$, we repeat the above process with the initial time $t = T_0$, thus

This implies that $u\geq0$, that is, $\overline{u}(x,t)\geq\underline{u}(x,t)$ in $B_T$. This completes the proof.

3.   Well-posedness of the solution about (1.1)

In this section, we study the well-posedness of problem (1). The existence and uniqueness of global solution is based on the comparison principle. Firstly, we investigate the existence of solution for the problem (1) by constructing two monotone sequences. Later, we prove the uniqueness of the solution for the problem (1) by using Gronwall's inequality and fundamental-solution. Finally, we will study the uniform boundedness of the solution for the problem (1).

Theorem 3.1. Suppose that $\overline{u},\; \underline{u}$ are nonnegative functions and $\overline{u},\; \underline{u}$ are super- and sub-solutions of problem (1) in $I_T$, respectively. Then the problem (1) has a solution $u(x,t)$ in $I_T$ and satisfies

Proof. Since $\overline{u}, \; \underline{u}$ are nonnegative and bounded functions, then there exists $N>0$ such that

Moreover, we choose $L>0$ large enough, such that

where $\theta,\; \tilde{\theta},\; \hat{\theta}$ will be determined later.

Denote $\overline{u}^{(0)} = \overline{u}\; \mbox{and}\; \underline{u}^{(0)} = \underline{u}$, there are two sequences $\left\{\overline{u}^{(m)}\right\}_{m = 0}^{\infty}\; \mbox{and} \left\{\underline{u}^{(m)}\right\}_{m = 0}^{\infty}$ through the following iteration format for $m = 1,2,\cdots,$

and

First of all, we claim that

Let $\widehat{u} = \underline{u}^{(1)}-\underline{u}$, then from (4) and (11), we have

by applying the comparison principle, we know that $\widehat{u}\geq0$, namely, $\underline{u}^{(1)}\geq\underline{u}$ in $B_T$. Similarly, let $\widetilde{u} = \overline{u}-\overline{u}^{(1)}$, from (3) and (10) we obtain

Using the comparison principle, one has $\widetilde{u}\geq0$, which means $\overline{u}\geq\overline{u}^{(1)}$ in $B_T$. Finally, we prove $\overline{u}^{(1)}\geq\underline{u}^{(1)}$, let $\overline{\widetilde{u}} = \overline{u}^{(1)}-\underline{u}^{(1)}$, then from (10) and (11), $\overline{\widetilde{u}}$ satisfies

where $\theta\in(\underline{u}, \overline{u})$, then we have

We obtain $\overline{\widetilde{u}}\geq0$ by means of the comparison principle, that is, $\overline{u}^{(1)}\geq\underline{u}^{(1)}$ in $B_T$. Thus, $\underline{u}\leq\underline{u}^{(1)}\leq\overline{u}^{(1)}\leq\overline{u}$ in $B_T$.

Next, we prove that $\overline{u}^{(1)}$ and $\underline{u}^{(1)}$ is a pair of super- and sub-solutions for problem (1). From (10) $\overline{u}^{(1)}$ satisfies

where $\tilde{\theta}\in\left(\overline{u}^{(1)},\overline{u}\right)$. On the other hand, from (11) $\underline{u}^{(1)}$ satisfies

where $\hat{\theta}\in\left(\underline{u},\underline{u}^{(1)}\right)$. Thanks to the Definition 2.1, together with (13) and (14), we obtain that $\overline{u}^{(1)}$ and $\underline{u}^{(1)}$ are super- and sub-solutions of (1) respectively.

We assume $\overline{u}^{(m)}$ and $\underline{u}^{(m)}$ are super- and sub-solutions of the problem (1) respectively. By repeating above processes, one has

and

Likewise, $\overline{u}^{(m+1)}$ and $\underline{u}^{(m+1)}$ are super- and sub-solutions of (1) respectively. Thus, combine with (12), we have that, for $m = 0,1,2,\cdots,$

Therefore, there exist $u$ and $\mathfrak{u}$ such that

Additionally, It is clearly that $\mathfrak{u}$ and $u$ are super- and sub-solutions of (1), respectively, and there is $u = \mathfrak{u}$, thus $u$ is the bounded solution of (1). This completes the proof.

Thanks to Theorem 3.1, we further have the following result immediately.

Lemma 3.2. For any nonnegative bounded initial $u_0(x)$, the solution of the problem (1) exists.

Proof. From Theorem 3.1, if $\overline{u}$ and $\underline{u}$ are super- and sub-solutions respectively, one has $\overline{u}(x,0)\geq u_0(x)\geq\underline{u}(x,0)$. From the Definition 2.1, one may choose $0$ as the sub-solution of problem (1). Next, we construct the super-solution as follows.

Since $u_0(x)$ is nonnegative bounded function, then, we choose $M>0$ sufficiently enough such that

We know that $0$ and $M$ is a pair of the super- and sub-solutions. Therefore, we obtain our conclusion from Theorem 3.1. This completes the proof.

Moreover, the uniqueness of the solutions about the problem (1) will be given as follows.

Theorem 3.3. The problem (1) admits a unique bounded solution for $(x,t)\in B_T$.

Proof. According to Theorem 3.1, there exist solutions for the problem (1). To get our conclusion, we suppose that $u_{1}$ and $u_{2}$ are bounded solutions of (1) in $I_T$. By direct computations, $u_{i}$ (i = 1, 2) satisfies

where $\Phi(x,t)$ is the fundamental solution of the heat equation. Let $\widetilde{u} = u_{1}-u_{2}$, one has

where $\check{\theta}\in(u_{1}, u_{2})$.

From Lemma 3.2, we know that $u_1,\; u_2$ are all nonnegative and bounded functions in $B_T$. Therefore, there exists $N>0$ such that

Define

Furthermore, from (15), we deduce

By Gronwall's inequality, we obtain

Since $\widetilde{u}$ is continuous, we have $\widetilde{u}\equiv0$, that is, $u_1\equiv u_2$ in $I_T$. This completes the proof.

Finally, we prove the uniform boundedness of the solution of problem (1).

Theorem 3.4. The nonnegative solution of the problem (1) is uniformly bounded, i.e. there exists a positive constant $M>0$, such that

Proof. Since $u$ satisfies

we can get $u$ less than the solution of the ODE problem

By a simple computation, we have

where $C$ is a constant, then, there exists a positive constant $M>0$ such that $u(x,t)\leq M$. From the conclusion which implies that the solution of problem (1) blow-up is impossible. This completes the proof.

4.   The asymptotic behavior of Cauchy problem (1)

In this section, we establish the following result for the solution of the Cauchy problem (1) with compactly supported initial data.

Theorem 4.1. Let $u$ be the solution of the Cauchy problem (1) with a non-negative initial condition $u_{0}(x)\in L^{\infty}(\mathbb{R})$, such that $u_{0}(x)\not\equiv 0$, then

Further, if $u_{0}(x)$ is compactly supported, then

where $M$ is an upper bound of $u$.

Proof. We first consider (17), since $u_{0}(x)$ is compactly supported. Hence, there exists $R>0$ such that $u_{0}(x) = 0$ for $|x|\geq R$. Since $u(x,t)\geq0$ for all $t\geq0$ and $x\in\mathbb{R}$, one has

Let $v$ denote the solution of the following Cauchy problem

It follows from the comparison principle that $0\leq u(x,t)\leq v(x,t)$ for all $t>0$ and $x\in\mathbb{R}$. By an elementary calculation, we have

Choose $c>0$ such that $c\geq2\sqrt{kM}$. For $t\geq \frac{R}{c}$ and $|x|\geq ct$, one has

which immediately yields (17).

We now verify (16). Since $u$ is non-negative, for contradiction we assume that there exists $\tilde{c}_{1}\geq0$ and two sequences $(t_{n})_{n\in\mathbb{N}}$ in $(0,+\infty)$ and $(x_{n})_{n\in\mathbb{N}}$ in $\mathbb{R}$ such that

and

We introduce

Up to extraction of a subsequence, one can assume that $c_{n}\rightarrow c_{\infty}\in[-\tilde{c}_{1},\tilde{c}_{1}]\; \mbox{as}\; n\rightarrow+\infty$.

For every $n\in\mathbb{N}\; \mbox{and}\; (x,t)\in\mathbb{R}\times(-t_{n},\infty)$, we define the shifted function

From Theorem 3.4, we know that $\left(\|u_{n}\|_{L^{\infty}(\mathbb{R}\times(-t_{n},+\infty))}\right)_{n\in\mathbb{N}}$ is bounded. Therefore, from the standard parabolic estimate which implies that the function $u_{n}$ converge in $C_{loc}^{2,1}(\mathbb{R}\times\mathbb{R})$, Up to extraction of a subsequence, to a classical bounded solution $u_{\infty}$ of

such that $u_{\infty}\geq0\; \mbox{and}\; u_{\infty}(0,0) = 0$. By viewing $\left(u_{\infty}-u_{\infty}(\phi*u_{\infty})-\frac{b}{k}\right)$ as a coefficient in $L^{\infty}(\mathbb{R}\times\mathbb{R})$, thanks to the strong parabolic maximum principle and uniqueness of the solutions of the Cauchy problem which implies $u_{\infty}(x,t) = 0\; \mbox{for all}\; (x,t)\in\mathbb{R}\times\mathbb{R}$ (since the limit $u_{\infty}$ is unique, one can deduce that the sequence $(u_{n})_{n\in\mathbb{N}}$ converges to $0$ in $C_{loc}^{2,1}(\mathbb{R}\times\mathbb{R}))$. One can define the functions $\tilde{u}_{n}$ in $\mathbb{R}\times(-t_{n},+\infty)$ by

and $\tilde u_{n}(x,t)$ converge to $0$ locally and uniformly in $\mathbb{R}\times\mathbb{R}$, due to the fact that the boundedness of $c_{n}$ defined in (18). Hence the non-negative functions $\phi*\tilde {u}_{n}$ also converge to $0$ locally uniformly in $\mathbb{R}\times\mathbb{R}$ since the sequence $\left(\|\tilde {u}_{n}\|_{L^{\infty}(\mathbb{R}\times(-t_{n},+\infty))}\right)_{n\in\mathbb{N}}$ is bounded.

We now fix some parameters which are independent of $n$. First, we choose $\delta>0$ satisfying

where $M$ is the upper bound of $u$ and $v$, also let $R>0$ such that

Since $u(.,1)$ is continuous from parabolic regularity and is positive in $\mathbb{R}$ from the strong parabolic maximum principle, one can take $\eta>0$ such that

Without loss of generality, we assume that $t_{n}>1\; \mbox{for every}\; n\in\mathbb{N}$. Since $\phi*\tilde{u}_{n}\rightarrow0$ locally uniformly in $\mathbb{R}\times\mathbb{R}\; \mbox{as}\; n\rightarrow+\infty$, we may define

where $\delta$ and $R$ are as in (19) and (20), and assume that $t_{n}^*<0$. Furthermore, for every $n\geq N$, by the continuity of $\phi*\tilde{u}_{n}\; \mbox{in}\; \mathbb{R}\times(-t_{n},\infty)$, we then conclude that the infimum is the minimum in the definition of $t_{n}^*$ and

On the other hand, we have

for all $n\in\mathbb{N}$. According to the above analysis $\phi*\tilde{u}_{n}\; \mbox{in}\; \mathbb{R}\times(-t_{n},+\infty)$ and by minimality of $t_{n}^*$, for each $n\geq N$, we have the following dichotomy:

Next, we assert that there exists $\rho>0$ such that

This assertion is clearly if the second assertion of (22) always holds. Notice that

For contradiction, we assume that (23) is not hold, then up to extraction of a subsequence, there exists a sequence of $(y_{n})_{n\geq N}$ in $[-R,R]$ such that

We use the transformation defined by

for all $n\geq N$ and $(x,t)\in\mathbb{R}\times(-t_{n}-t_{n}^*,+\infty)$. Observe that $\tilde{u}_{n}$ satisfies

then the functions $w_{n}$ also satisfies (24) in $\mathbb{R}\times(-t_{n}-t_{n}^*,+\infty)$. Note that $-t_{n}-t_{n^*}\leq-1$ for all $n\geq N$, that $c_{n}\rightarrow c_{\infty}$ as $n\rightarrow+\infty$, that the functions $w_{n}$ are all non-negative and that the sequence $\left(\|w_{n}\|_{L^\infty(\mathbb{R}\times(-t_{n}-t_{n}^*,+\infty))}\right)_{n\geq N}$ is bounded. Using the standard parabolic estimates, we obtain $w_{n}$ converge in $C_{loc}^{2,1}(\mathbb{R}\times(-1,+\infty))$, up to extraction of a subsequence, to a classical bounded solution $w_{\infty}$ of the following equation

such that

and

Thanks to the strong maximum principle and the uniqueness of the solution for the Cauchy problem, we see that

one has that $w_{n}\rightarrow0$ as $n\rightarrow+\infty$ (at least) are uniform in $\mathbb{R}\times(-1,+\infty)$, it is clearly that

locally uniform in $\mathbb{R}\times(-1,+\infty)$ by boundedness of the sequence $\left(\|w_{n}\|_{L^\infty(\mathbb{R}\times(-1,+\infty))}\right)_{n\geq N}$. We then have

locally uniform in $\mathbb{R}\; \mbox{as}\; n\rightarrow+\infty$. This is a contradiction to (22) due to the fact that both $\delta$ and $\eta$ are positive. Therefore, (23) holds.

Now, in view of (21), (23) and (24), we have the following situation: for each $n\geq N$, one has $-t_{n}+1\leq t_{n}^*<0$ and in the box $[-R,R]\times[t_{n}^*,0]$, the non-negative function $\tilde{u}_n$ satisfies

On the other hand, for every $n\geq N$, the function $\psi_{n}$ defined in $[-R,R]$ by

Then one has $0\leq\psi_{n}\leq\rho$ in $[-R,R]$ from $\psi_{n}(\pm R) = 0$ and

Notice that the time-independent function $\psi_{n}$ is a sub-solution for the problem (25). It follows from the parabolic maximum principle that

for all $n\geq N$. In particular

However, the assumption means that

Since $\rho e^{-cR/2}>0$ from the positivity of $\rho$ in (23) which reaches a contradiction. This completes the proof.

Acknowledgments

The authors would like to thank reviewers for their precious suggestions.