Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type

TL;DR

This paper establishes conditions for global bounded solutions and convergence rates in a two-species chemotaxis-competition system, extending previous results to cases where parameters are greater than or equal to one.

Contribution

It introduces new conditions ensuring global existence and stability for all positive parameters, including the previously unaddressed cases where parameters are at least one.

Findings

Global existence of bounded solutions for all positive parameters.

Convergence rates for solutions when parameters are in specific ranges.

Extended stability analysis beyond previous parameter restrictions.

Abstract

This paper deals with the two-species chemotaxis-competition system $u_t = d_1 \Delta u - \chi_1 \nabla \cdot (u \nabla w) + \mu_1 u(1 - u - a_1 v)$, $v_t = d_2 \Delta v - \chi_2 \nabla \cdot (v \nabla w) + \mu_2 v(1 - a_2 u - v)$, $0 = d_3 \Delta w + \alpha u + \beta v - \gamma w$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $n\ge 2$; $\chi_i$ and $\mu_i$ are constants satisfying some conditions. The above system was studied in the cases that $a_1,a_2\in (0,1)$ and $a_1>1>a_2$, and it was proved that global existence and asymptotic stability hold when $\frac{\chi_i}{\mu_i}$ are small. However, the conditions in the above two cases strongly depend on $a_1,a_2$, and have not been obtained in the case that $a_1,a_2\ge 1$. Moreover, convergence rates in the cases that $a_1,a_2\in (0,1)$ and $a_1 > 1 > a_2$ have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all $a_1,a_2>0$ which covers the case that $a_1,a_2 \ge 1$, and lead to convergence rates for solutions of the above system in the cases that $a_1,a_2\in (0,1)$ and $a_1\ge 1 >a_2$.

Equations

$$\{\begin{array}{ll}{u}_t=d_1\Delta u-{\chi }_1\nabla \cdot (u\nabla w)+{\mu }_{1}u(1-u-a_{1}v)& \mathrm{in}\Omega \times (0,\infty ),\\ v_t=d_2\Delta v-{\chi }_2\nabla \cdot (v\nabla w)+{\mu }_{2}v(1-a_{2}u-v)& \mathrm{in}\Omega \times (0,\infty ),\\ 0=d_3\Delta w+\alpha u+\beta v-\gamma w& \mathrm{in}\Omega \times (0,\infty ),\end{array}$$

$$u_t=\Delta u-\chi \nabla \cdot (u\nabla v),\tau v_t=\Delta v+u-v,$$

$$u_t=\Delta u+\chi \nabla \cdot (u\nabla v)+\kappa u-\mu u^2,\tau v_t=\Delta v+u-v$$

$$u_t=\Delta u-{\chi }_1\nabla \cdot (u\nabla w)+{\mu }_{1}u(1-u-a_{1}v),$$

$$v_t=\Delta v-{\chi }_2\nabla \cdot (v\nabla w)+{\mu }_{2}v(1-a_{2}u-v),$$

$$\tau w_t=\Delta w+\alpha u+\beta v-\gamma w$$

$$\{\begin{array}{ll}{u}_t=d_1\Delta u-{\chi }_1\nabla \cdot (u\nabla w)+{\mu }_{1}u(1-u-a_{1}v),& x\in \Omega ,t>0,\\ v_t=d_2\Delta v-{\chi }_2\nabla \cdot (v\nabla w)+{\mu }_{2}v(1-a_{2}u-v),& x\in \Omega ,t>0,\\ 0=d_3\Delta w+\alpha u+\beta v-\gamma w,& x\in \Omega ,t>0,\\ \nabla u\cdot \nu =\nabla v\cdot \nu =\nabla w\cdot \nu =0,& x\in \partial \Omega ,t>0,\\ u(x,0)=u_0(x),v(x,0)=v_0(x),& x\in \Omega ,\end{array}$$

$$u_t=u(1-u-a_{1}v),v_t=v(1-a_{2}u-v).$$

$$2({\chi }_1+{\chi }_2)+a_2{\mu }_1<{\mu }_2\text{mbox}and2({\chi }_1+{\chi }_2)+a_1{\mu }_2<{\mu }_1.$$

$$\frac{{\chi }_1}{{\mu }_1}\in \left[0,\frac{d_3}{2\alpha }\right)\cap \left[0,\frac{a_1{d}_3}{\beta }\right),\frac{{\chi }_2}{{\mu }_2}\in \left[0,\frac{d_3}{2\beta }\right)\cap \left[0,\frac{a_2{d}_3}{\alpha }\right),$$

$$a_1{a}_2{d}_3^2<\left(d_3-\frac{2\alpha {\chi }_1}{{\mu }_1}\right)\left(d_3-\frac{2\beta {\chi }_2}{{\mu }_2}\right)$$

$$\frac{{\chi }_1}{{\mu }_1}\le a_1,\frac{{\chi }_2}{{\mu }_2}<\frac{1}{2}\text{mbox}and$$

$$\frac{\alpha {\chi }_1}{{\mu }_1}+\max \left\{\frac{{\chi }_2}{{\mu }_2},\frac{a_2({\mu }_2-{\chi }_2)}{{\mu }_2-2{\chi }_2},\frac{(\alpha -a_2){\chi }_2}{{\mu }_2-2{\chi }_2}\right\}<1$$

$$\frac{{\chi }_1}{{\mu }_1}<\frac{n{d}_3}{n-2}\min \left\{\frac{1}{\alpha },\frac{a_1}{\beta }\right\}\text{mbox}and\frac{{\chi }_2}{{\mu }_2}<\frac{n{d}_3}{n-2}\min \left\{\frac{1}{\beta },\frac{a_2}{\alpha }\right\}.$$

$$0\le u_0\in C(\overline{\Omega })\setminus \{0\},0\le v_0\in C(\overline{\Omega })\setminus \{0\}.$$

$$u,v\in C(\overline{\Omega }\times [0,\infty ))\cap C^{2,1}(\overline{\Omega }\times (0,\infty )),$$

$$w\in C(\overline{\Omega }\times [0,\infty ))\cap C^{2,1}(\overline{\Omega }\times (0,\infty ))\cap L_{\mathrm{loc}}^{\infty }([0,\infty );W^{1,q}(\Omega )),$$

$$\Vert u(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert v(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert w(\cdot ,t){\Vert }_{W^{1,q}(\Omega )}\le C\text{mbox}forallt\ge 0,$$

$$\Vert u{\Vert }_{C^{\theta ,\frac{\theta }{2}}(\overline{\Omega }\times [t,t+1])}+\Vert v{\Vert }_{C^{\theta ,\frac{\theta }{2}}(\overline{\Omega }\times [t,t+1])}+\Vert w{\Vert }_{C^{\theta ,\frac{\theta }{2}}(\overline{\Omega }\times [t,t+1])}\le M\text{mbox}forallt\ge 1.$$

$${\chi }_1<\frac{2n{a}_1({\chi }_1+{\chi }_2)(1+a_1)}{(n-2)(1-a_1{a}_2)}\text{mbox}and{\chi }_2<\frac{2n{a}_2({\chi }_1+{\chi }_2)(1+a_2)}{(n-2)(1-a_1{a}_2)},$$

$$\frac{\alpha {\chi }_1}{{\mu }_1}+\frac{{\chi }_2}{{\mu }_2}<1,$$

$$\frac{\alpha (n-2)}{n}<\min \{1,a_1\alpha ,a_2\}$$

$$4{\delta }_1-a_1{a}_2(1+{\delta }_1{)}^2>0$$

$${\mu }_1>\frac{{\chi }_1^2(1+{\delta }_1)(1-a_1)({\alpha }^2{a}_1{\delta }_1+{\beta }^2{a}_2-\alpha \beta a_1{a}_2(1+{\delta }_1))}{4a_1{d}_1{d}_3\gamma (1-a_1{a}_2)(4{\delta }_1-a_1{a}_2(1+{\delta }_1{)}^2)},$$

$${\mu }_2>\frac{{\chi }_2^2(1+{\delta }_1)(1-a_2)({\alpha }^2{a}_1{\delta }_1+{\beta }^2{a}_2-\alpha \beta a_1{a}_2(1+{\delta }_1))}{4a_2{d}_2{d}_3\gamma (1-a_1{a}_2)(4{\delta }_1-a_1{a}_2(1+{\delta }_1{)}^2)}.$$

$$\Vert u{\Vert }_{C^{\theta ,\frac{\theta }{2}}(\overline{\Omega }\times [t,t+1])}+\Vert v{\Vert }_{C^{\theta ,\frac{\theta }{2}}(\overline{\Omega }\times [t,t+1])}+\Vert w{\Vert }_{C^{\theta ,\frac{\theta }{2}}(\overline{\Omega }\times [t,t+1])}\le M\text{mbox}forallt>0$$

$$\Vert u(\cdot ,t)-u^{\ast }{\Vert }_{L^{\infty }(\Omega )}+\Vert v(\cdot ,t)-v^{\ast }{\Vert }_{L^{\infty }(\Omega )}+\Vert w(\cdot ,t)-w^{\ast }{\Vert }_{L^{\infty }(\Omega )}\le C{e}^{-\ell t}\text{mbox}forallt>0,$$

$$u^{\ast }:=\frac{1-a_1}{1-a_1{a}_2},v^{\ast }:=\frac{1-a_2}{1-a_1{a}_2}{w}^{\ast }:=\frac{\alpha u^{\ast }+\beta v^{\ast }}{\gamma }.$$

$$4{\delta }_1-a_1^{\prime }{a}_2(1+{\delta }_1{)}^2>0,$$

$${\mu }_2>\frac{{\chi }_2^2{\delta }_1({\alpha }^2{a}_1^{\prime }{\delta }_1+{\beta }^2{a}_2-\alpha \beta a_1^{\prime }{a}_2(1+{\delta }_1))}{4a_2{d}_2{d}_3\gamma (4{\delta }_1-a_1^{\prime }{a}_2(1+{\delta }_1{)}^2)}.$$

$$\Vert u{\Vert }_{C^{\theta ,\frac{\theta }{2}}(\overline{\Omega }\times [t,t+1])}+\Vert v{\Vert }_{C^{\theta ,\frac{\theta }{2}}(\overline{\Omega }\times [t,t+1])}+\Vert w{\Vert }_{C^{\theta ,\frac{\theta }{2}}(\overline{\Omega }\times [t,t+1])}\le M\text{mbox}forallt>0$$

$$\Vert u(t){\Vert }_{L^{\infty }(\Omega )}+\Vert v(t)-1{\Vert }_{L^{\infty }(\Omega )}+{\Vert w(t)-\frac{\beta }{\gamma }\Vert }_{L^{\infty }(\Omega )}\le C{e}^{-\ell t}\text{mbox}forallt>0.$$

$$\Vert u(t){\Vert }_{L^{\infty }(\Omega )}+\Vert v(t)-1{\Vert }_{L^{\infty }(\Omega )}+{\Vert w(t)-\frac{\beta }{\gamma }\Vert }_{L^{\infty }(\Omega )}\le C(t+1{)}^{-\ell }\text{mbox}forallt>0.$$

$$\frac{1}{p}\frac{d}{dt}{\int }_{\Omega }{u}^p\le (p-1){\chi }_1{\int }_{\Omega }{u}^{p-1}\nabla u\cdot \nabla v+{\mu }_1{\int }_{\Omega }{u}^p(1-u-a_{1}v).$$

Full text

0002010Mathematics Subject Classification. Primary: 35K51; Secondary: 92C17, 35B40. 000*Key words and phrases: chemotaxis; Lotka–Volterra; global existence; stabilization. *

**Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type **

Masaaki Mizukami 000 E-mail: masaaki.mizukami.math@gmail.com

Department of Mathematics, Tokyo University of Science

1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

• Abstract. This paper deals with the two-species chemotaxis-competition system

[TABLE]

where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$, $n\geq 2$; $\chi_{i}$ and $\mu_{i}$ are constants satisfying some conditions. The above system was studied in the cases that $a_{1},a_{2}\in(0,1)$ and $a_{1}>1>a_{2}$, and it was proved that global existence and asymptotic stability hold when $\frac{\chi_{i}}{\mu_{i}}$ are small ([5, 32, 34]). However, the conditions in the above two cases strongly depend on $a_{1},a_{2}$, and have not been obtained in the case that $a_{1},a_{2}\geq 1$. Moreover, convergence rates in the cases that $a_{1},a_{2}\in(0,1)$ and $a_{1}>1>a_{2}$ have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all $a_{1},a_{2}>0$ which covers the case that $a_{1},a_{2}\geq 1$, and lead to convergence rates for solutions of the above system in the cases that $a_{1},a_{2}\in(0,1)$ and $a_{1}\geq 1>a_{2}$.

1 Introduction

Many phenomena, which appear in natural science, especially, biology, chemistry and physics, support animals’ lives. In this paper we focus on chemotaxis which is one of the important properties and is related to e.g., movement of sperm, migration of neurons or lymphocytes and tumor invasion. Chemotaxis is the property such that species move towards higher concentration of the chemical substance when they plunge into hunger.

A mathematical problem which describes a part of the life cycle of cellular slime molds with chemotaxis is called the Keller–Segel system:

[TABLE]

where $\chi>0$ and $\tau\in\{0,1\}$. Moreover, the chemotaxis system with growth terms

[TABLE]

was proposed by [25, 31], where $\chi,\kappa,\mu>0$ and $\tau\in\{0,1\}$. After the pioneering work of Keller–Segel [19], the Keller–Segel system and the chemotaxis system are intensively studied (see e.g., [2, 13, 15]). A generalized problem of Keller–Segel systems, which means a two-species chemotaxis system, was proposed in [36] and also has studied (see e.g., [3, 4, 7, 8, 10, 21, 37]; global existence was proved in [7, 8, 37]; and thier asymptotic stability was shown in [37]; related works which deal with blow-up of solutions can be seen in [3, 4, 7, 8, 10, 21]). Recently, a two-species chemotaxis system with competitive kinetics

[TABLE]

with some $\chi_{1},\chi_{2},\mu_{1},\mu_{2},a_{1},a_{2}>0$ and $\tau\in\{0,1\}$, which describes the evolution of two competing species which react on a single chemoattractant, was proposed by Tello–Winkler [34] and was studied (see [1, 22, 26, 27, 28, 29, 30, 38]). About this problem with $\tau=1$, global existence and boundedness was obtained in the 2-dimensional case ([1]) and the $n$-dimensional setting ([22]); moreover, asymptotic behavior of solutions was established in [1, 27]. Related works which dealt with global existence and boundedness in this two-species problem with sensitivity functions can be found in [27, 38]; and related works which treated the non-competition case are in [26, 28, 29, 30]. These results in the case $\tau=1$ are motivated by the results ([5, 32, 34]) in the case $\tau=0$. Therefore the parabolic-parabolic-elliptic problem reduced by letting $\tau=0$ seems to be helpful to analyze the fully parabolic case.

In this paper we consider the two-species chemotaxis system with competitive kinetics of parabolic-parabolic-elliptic type

[TABLE]

where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ ($n\geq 2$) with smooth boundary $\partial\Omega$ and $\nu$ is the outward normal vector to $\partial\Omega$. The constants $d_{1},d_{2},d_{3},\chi_{1},\chi_{2},\mu_{1},\mu_{2},a_{1},a_{2}$ and $\alpha,\beta,\gamma$ are positive. The initial data $u_{0},v_{0}$ are assumed to be nonnegative functions. The unknown functions $u(x,t)$ and $v(x,t)$ represent the population densities of two species and $w(x,t)$ shows the concentration of the chemical substance at place $x$ and time $t$.

The problem (1.1) is a problem on account of the influence of chemotaxis, diffusion, and the Lotka–Volterra competitive kinetics, i.e., with coupling coefficients $a_{1},a_{2}>0$ in

[TABLE]

The mathematical difficulties of the problem (1.1) are to deal with the chemotaxis term $\nabla\cdot(u\nabla w)$ and the competition term $u(1-u-a_{1}v)$. To overcome these difficulties, in the case that $a_{1},a_{2}\in(0,1)$ and $d_{3}=\alpha=\beta=1$ in (1.1), Tello–Winkler [34] applied comparison methods to this problem and obtained global existence of classical bounded solutions and their asymptotic behavior under the conditions that

[TABLE]

However, if $\chi_{1}\to 0$ or $\mu_{1}\to\infty$, then these conditions break down. Recently, it was shown that the conditions

[TABLE]

lead to global existence and asymptotic stability in (1.1) in the case that $a_{1},a_{2}\in(0,1)$ ([5]). The conditions (1.4)–(1.5) partially relax (1.3) in view of the point mentioned above. On the other hand, in the case that $a_{1}>1>a_{2}$ and $d_{3}=\beta=1$ in (1.1) Stinner–Tello–Winkler [32] established global existence and stabilization of global classical solutions when

[TABLE]

are satisfied. In summary the two-species chemotaxis-competition model (1.1) were studied in the cases that $a_{1},a_{2}\in(0,1)$ and $a_{1}>1>a_{2}$, and it was proved that global existence and same asymptotic behavior as solutions to the Lotka–Volterra competition model (1.2) hold when $\frac{\chi_{i}}{\mu_{i}}$ are small. However, the conditions in the above two cases strongly depend on $a_{1},a_{2}$, and have not been obtained in the case that $a_{1},a_{2}\geq 1$. Moreover, convergence rates in the cases that $a_{1},a_{2}\in(0,1)$ and $a_{1}>1>a_{2}$ have not been studied.

The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all $a_{1},a_{2}>0$ which covers the case that $a_{1},a_{2}\geq 1$, and lead to convergence rates for solutions of (1.1) in the cases that $a_{1},a_{2}\in(0,1)$ and $a_{1}>1>a_{2}$.

For establishing global existence and boundedness we shall suppose that $\chi_{1},\chi_{2}$ and $\mu_{1},\mu_{2}$ satisfy the following conditions:

[TABLE]

We assume that the initial data $u_{0},v_{0}$ satisfy

[TABLE]

Now the main results read as follows. The first one is concerned with global existence and boundedness in (1.1).

Theorem 1.1**.**

Let $d_{1},d_{2},d_{3}>0$, $\mu_{1},\mu_{2}>0$, $a_{1},a_{2}>0$, $\chi_{1},\chi_{2}>0$, $\alpha,\beta,\gamma>0$ and let $\Omega\subset\mathbb{R}^{n}$ $(n\geq 2)$ be a bounded domain with smooth boundary. Assume that (1.6) are satisfied. Then for any $u_{0},v_{0}$ satisfying (1.7) with some $q>n$, there exists an exactly one pair $(u,v,w)$ of nonnegative functions

[TABLE]

which satisfy (1.1). Moreover, the solutions $u,v,w$ are uniformly bounded, i.e., there exists a constant $C>0$ such that

[TABLE]

and the solutions $u,v,w$ are the Hölder continuous functions, i.e., there exist $\theta\in(0,1)$ and $M>0$ such that

[TABLE]

Remark 1.1**.**

This result give the existence of global classical bounded solutions in the case that $a_{1},a_{2}\geq 1$. Moreover, the condition (1.6) relaxes (1.4) which assumed for global existence of solutions in [5]. Indeed, if $\chi_{1},\chi_{2}$ and $\mu_{1},\mu_{2}$ satisfy the condition (1.4), then $\chi_{1},\chi_{2}$ and $\mu_{1},\mu_{2}$ satisfy the condition (1.6). However, the condition (1.6) does not always relax those assumed in [32] and [34]; in the case that $a_{1},a_{2}\in(0,1)$ the condition (1.6) relaxes (1.3) under the condition

[TABLE]

and in the case that $a_{1}>1>a_{2}$ the condition (1.6) relaxes the condition

[TABLE]

which was used to obtain global existence in [32], when

[TABLE]

hold.

The main theorem tells us the following result in the 2-dimensional case.

Corollary 1.2**.**

Let $d_{1},d_{2},d_{3}>0$, $\mu_{1},\mu_{2}>0$, $a_{1},a_{2}>0$, $\chi_{1},\chi_{2}>0$, $\alpha,\beta,\gamma>0$ and let $\Omega\subset\mathbb{R}^{2}$ be a bounded domain with smooth boundary. Then for any $u_{0},v_{0}$ satisfying (1.7) with some $q>n$, (1.1) possesses a unique global bounded classical solution.

In the case $a_{1},a_{2}\in(0,1)$ asymptotic behavior of solutions to (1.1) will be discussed under the following additional conditions: there exists $\delta_{1}>0$ such that

[TABLE]

and

[TABLE]

The second theorem gives asymptotic behavior in (1.1) in the case $a_{1},a_{2}\in(0,1)$.

Theorem 1.3**.**

Let $d_{1},d_{2},d_{3}>0$, $\mu_{1},\mu_{2}>0$, $a_{1},a_{2}\in(0,1)$, $\chi_{1},\chi_{2}>0$, $\alpha,\beta,\gamma>0$ and let $\Omega\subset\mathbb{R}^{n}$ $(n\geq 2)$ be a bounded domain with smooth boundary. Assume that there exists a unique global classical solution $(u,v,w)$ of (1.1) satisfying

[TABLE]

with some $M>0$. Then under the conditions (1.8)–*(1.10), $(u,v,w)$ satisfies that there exist $C>0$ and $\ell>0$ such that *

[TABLE]

where

[TABLE]

Remark 1.2**.**

If the assumption of Theorem 1.1 and (1.8)–(1.10) are satisfied, then Theorem 1.3 gives the convergence rates for solutions of (1.1) in the case that $a_{1},a_{2}\in(0,1)$. Moreover, the conditions (1.8)–(1.10) are the same conditions as that assumed in [27] in the case that $a_{1},a_{2}\in(0,1)$ and $h(u,v,w)=\alpha u+\beta v-\gamma w$.

In the case $a_{1}\geq 1>a_{2}$ asymptotic behavior of solutions to (1.1) will be discussed under the following additional conditions: there exist $\delta_{1}>0$ and $a^{\prime}_{1}\in[1,a_{1}]$ such that

[TABLE]

The third one gives asymptotic behavior in (1.1) in the case $a_{1}\geq 1>a_{2}$.

Theorem 1.4**.**

Let $d_{1},d_{2},d_{3}>0$, $\mu_{1},\mu_{2}>0$, $a_{1}\geq 1$, $a_{2}\in(0,1)$, $\chi_{1},\chi_{2}>0$, $\alpha,\beta,\gamma>0$ and let $\Omega\subset\mathbb{R}^{n}$ $(n\geq 2)$ be a bounded domain with smooth boundary. Assume that there exists a unique global classical solution $(u,v,w)$ of (1.1) such that

[TABLE]

with some $M>0$. Then under the conditions (1.11)–(1.12), $(u,v,w)$ has the following properties:**

1. (i)

If $a_{1}>1$ and take $a^{\prime}_{1}\in(1,a_{1}]$ in (1.11)–(1.12), then there exist $C>0$ and $\ell>0$ satisfying

[TABLE] 2. (ii)

If $a_{1}=1$, then there exist $C>0$ and $\ell>0$ satisfying

[TABLE]

Remark 1.3**.**

If the assumption of Theorem 1.1 and (1.11)–(1.12) are satisfied, then Theorem 1.4 gives the convergence rates for solutions in the cases that $a_{1}>1>a_{2}$ and $a_{1}=1>a_{2}$. Moreover, the conditions (1.11)–(1.12) are the same conditions as that assumed in [27] in the case that $a_{1}\geq 1>a_{2}$ and $h(u,v,w)=\alpha u+\beta v-\gamma w$.

Remark 1.4**.**

Stabilization in the case that $a_{1},a_{2}\geq 1$ is a still open question. In the case that $a_{1},a_{2}>1$ a Lotka–Volterra competition model with diffusion term was studied; however, its analysis is difficult and it is known that solutions have complicated structures (see cf. [9, 16, 17, 18, 20, 23, 24]).

The strategy of the proof of Theorem 1.1 is to extend a method in [33] to a two-species case. We first aim to establish the $L^{p}$-estimate for $u$ with some $p>\frac{n}{2}$ from the following derivative of $\int_{\Omega}u^{p}$:

[TABLE]

Since the third equation in (1.1) derives that

[TABLE]

we shall show that a combination of (1.13) and (1.14), along with the condition (1.6) implies

[TABLE]

which leads to $L^{p}$-estimate for $u$. Then aided by standard semigroup estimates, we can obtain the $L^{\infty}$-estimate for $u$. On the other hand, one of the keys for the proof of Theorems 1.3 and 1.4 is to derive the following energy estimate:

[TABLE]

for all $t>0$ with some positive function $E$ and some constant $\varepsilon>0$, where $(\overline{u},\overline{v},\overline{w})\in\mathbb{R}^{3}$ is a solution of (1.1). Thanks to (1.15), we can obtain that there exists $C>0$ such that

[TABLE]

which together with the regularity of the solution leads to Theorems 1.3 and 1.4.

This paper is organized as follows. In Section 2 we prove global existence and boundedness (Theorem 1.1) through a series of lemmas. Section 3 is devoted to the proof of asymptotic stability (Theorems 1.3 and 1.4); we first provide some lemmas which will be used later, and we next devide the section into Sections 3.1 and 3.2 according to the proof of Theorem 1.3 and that of Theorem 1.4, respectivly.

2 Global existence and boundedness

In this section we shall show global existence and boundedness in (1.1). First we will recall the known result about local existence of solutions to (1.1) ([5, Lemma 2.1], [32, Lemma 2.1]).

Lemma 2.1**.**

Let $d_{1},d_{2},d_{3}>0$, $\mu_{1},\mu_{2}>0$, $a_{1},a_{2}>0$, $\chi_{1},\chi_{2}>0$, $\alpha,\beta,\gamma>0$ and let $\Omega\subset\mathbb{R}^{n}$ $(n\in\mathbb{N})$ be a bounded domain with smooth boundary. Then for any $u_{0},v_{0}$ satisfying (1.7) for some $q>n$, there exist $T_{\rm max}\in$ $(0,\infty]$ and an exactly one pair $(u,v,w)$ of nonnegative functions

[TABLE]

*which satisfy (1.1). Moreover, *

[TABLE]

We next give the $L^{p}$-estimate for $u$ with some $p>\frac{n}{2}$ which plays an important role in deriving $L^{\infty}$-estimate for $u$. The proof is based on the proof of [33, Lemma 2.2].

Lemma 2.2**.**

Assume that (1.6)–(1.7) are satisfied. Then for all $p\in I_{1}$, there exists $C(p)>0$ such that

[TABLE]

for all $t>0$, where

[TABLE]

Proof.

We fix $p\in I_{1}$. Here we note from the condition (1.6) that $I_{1}\neq\emptyset$. Multiplying the first equation in (1.1) by $u^{p-1}$ and integrating it over $\Omega$, we obtain that

[TABLE]

Then integration by parts and the third equation in (1.1) imply that

[TABLE]

Therefore a combination of (2.1) with (2.2) yields that

[TABLE]

Recalling $p\in I_{1}=\left(\frac{n}{2},\min\left\{\frac{\alpha\chi_{1}}{(\alpha\chi_{1}-d_{3}\mu_{1})_{+}},\frac{\beta\chi_{1}}{(\beta\chi_{1}-a_{1}d_{3}\mu_{1})_{+}}\right\}\right)$ that

[TABLE]

we establish from the Hölder inequality

[TABLE]

that there exists $\varepsilon>0$ satisfying

[TABLE]

which implies that

[TABLE]

Thus we can attain the conclusion of this lemma. ∎

Similarly, we can confirm the following $L^{p}$-estimate for $v$ with some $p>\frac{n}{2}$.

Lemma 2.3**.**

Assume that (1.6)–(1.7) are satisfied. Then for all $p\in I_{2}$, there exists $C(p)>0$ such that

[TABLE]

where $I_{2}:=\left(\frac{n}{2},\min\left\{\frac{\beta\chi_{2}}{(\beta\chi_{2}-d_{3}\mu_{2})_{+}},\frac{\alpha\chi_{2}}{(\alpha\chi_{2}-a_{2}d_{3}\mu_{2})_{+}}\right\}\right)$.

Proof.

A similar argument as in the proof of Lemma 2.3 derives this lemma. ∎

Now we could construct all estimates which will enable us to obtain the estimate for the solution; Lemmas 2.2 and 2.3 lead to the following lemma. The proof is based on a known argument involving semigroup estimates which derive the $L^{\infty}$-estimate for $u$ from $L^{p}$-estimate with $p>\frac{n}{2}$ (see e.g., [2]).

Lemma 2.4**.**

Assume that (1.6)–(1.7) are satisfied. Then there exists $C>0$ such that

[TABLE]

Moreover, there exist $M>0$ and $\theta\in(0,1)$ such that

[TABLE]

Proof.

We fix $p\in I_{1}\cap I_{2}\cap(0,n)$, where $I_{1}$ and $I_{2}$ are the intervals defined in Lemmas 2.2 and 2.3. Then thanks to Lemmas 2.2 and 2.3, we can find $C_{1}>0$ such that

[TABLE]

We first verify the $L^{\frac{np}{n-p}}$-estimate for $\nabla v$. Here for all $q\in(1,\infty)$, the standard elliptic regularity argument (see e.g., [11, Theorem 19.1]) leads to the existence of a constant $C_{E}(q)>0$ satisfying

[TABLE]

Therefore a combination of (2.5) with (2.4) yields from the Sobolev embedding theorem that there exists $C_{2}>0$ such that

[TABLE]

since $p<n$. We next establish the $L^{\infty}$-estimate for $u$. Since $p>\frac{n}{2}$, we can take $r\in(n,q)$ such that

[TABLE]

We take $\vartheta>1$ satisfying

[TABLE]

Then $\vartheta^{\prime}:=\frac{\vartheta}{\vartheta-1}$ satisfies

[TABLE]

Now for all $T\in(0,T_{\rm max})$ we note that

[TABLE]

is finite. To obtain the estimate for $A(T)$ we put $t_{0}:=(t-1)_{+}$ and represent $u$ according to

[TABLE]

for $t\in(0,T_{\rm max})$. In the case that $t\leq 1$, i.e., $t_{0}=0$, from the order preserving property of the Neumann heat semigroup we see that

[TABLE]

In the case that $t>1$ using the $L^{p}$-$L^{q}$ estimate for $(e^{\tau\Delta})_{\tau\geq 0}$ (see [35, Lemma 1.3]) yields that there is $C_{3}>0$ such that

[TABLE]

Next due to a known smoothing property of $(e^{\tau\Delta})_{\tau\geq 0}$ (see [12, Lemma 3.3]), we can find $C_{4}>0$ such that

[TABLE]

Noting from $r\vartheta^{\prime}<\frac{np}{n-p}$ and (2.4) that

[TABLE]

with some $C_{5}>0$, we establish that there exists $C_{6}>0$ such that

[TABLE]

Finally, the maximum principle together with the elementary inequality

[TABLE]

implies that there exists $C_{7}>0$ such that

[TABLE]

Therefore a combination of (2.6), the nonnegativity of $u$ with (2.7), (2.8), (2.9), (2.10) tells us that there exist $C_{8},C_{9}>0$ such that

[TABLE]

which implies from $p<r\vartheta$ that

[TABLE]

with some $C_{10}>0$. Thus we obtain the $L^{\infty}$-estimate for $u$. Similarly, we can verify the $L^{\infty}$-estimate for $v$. Then invoking (2.5), we see that there exists $C_{11}>0$ such that

[TABLE]

which implies (2.3). Moreover, known regularity arguments (see [6, Proposition 2.3]) enable us to find $C_{12}>0$ and $\theta\in(0,1)$ satisfying

[TABLE]

which implies the end of the proof. ∎

Proof of Theorem 1.1.

Lemma 2.4 directly shows Theorem 1.1. ∎

3 Stabilization

In this section we will establish stabilization of solutions to (1.1). Here we assume that there exists a unique global classical solution $(u,v,w)$ of (1.1) satisfying

[TABLE]

with some $M>0$. we first recall a important lemma for the proof of Theorems 1.3 and 1.4 (see [14, Lemma 4.6]).

Lemma 3.1**.**

Let $n\in C^{0}(\overline{\Omega}\times[0,\infty))$ satisfy that there exist constants $C^{*}>0$ and $\theta^{*}>0$ such that

[TABLE]

Assume that

[TABLE]

with some constant $N^{\ast}>0$. Then

[TABLE]

We next provide the following lemma which will be used to confirm that the assumption of Lemma 3.1 is satisfied.

Lemma 3.2**.**

Let $a,b,c,d,e,f\in\mathbb{R}$. Suppose that

[TABLE]

Then

[TABLE]

holds for all $x,y,z\in\mathbb{R}$.

Proof.

Straightforward calculations lead to the conclusion of this lemma (for more details, see [27, Lemma 3.2]). ∎

Finally, we give the following lemma which enables us to upgrade the $L^{2}$-convergence rate to $L^{\infty}$-convergence rate.

Lemma 3.3**.**

Let $(\overline{u},\overline{v},\overline{w})\in\mathbb{R}^{3}$ be a solution to (1.1). Assume that there exists a decreasing function $h:[0,\infty)\to\mathbb{R}$ satisfying

[TABLE]

Then there exists $C>0$ such that

[TABLE]

for all $t>1$.

Proof.

For all $p>2$ we first obtain from the Hölder inequality that

[TABLE]

holds for all $f\in L^{\infty}(\Omega)$, which means from the boundedness of $u,v$ that

[TABLE]

with some $C_{1}(p)>0$. Here (2.5) enables us to see that

[TABLE]

Thus we have that there is $C_{2}>0$ such that

[TABLE]

Then by using a similar argument as in the proof of [1, Lemma 3.6] we infer that there exists $C_{3}>0$ such that

[TABLE]

Finally, since $(\overline{u},\overline{v},\overline{w})$ satisfies

[TABLE]

we can apply the maximum principle to

[TABLE]

and hence obtain the existence of a constant $C_{4}>0$ such that

[TABLE]

which concludes the proof of this lemma. ∎

3.1 Convergence. Case 1: $a_{1},a_{2}\in(0,1)$

In this subsection we establish stabilization in the case that $a_{1},a_{2}\in(0,1)$. We first confirm that the assumption of Lemma 3.1 are satisfied.

Lemma 3.4**.**

Assume that (1.8)–(1.10) are satisfied. Then there exist a nonnegative function $E_{1}:(0,\infty)\to\mathbb{R}$ and a constant $\varepsilon>0$ such that

[TABLE]

holds for all $t>0$. Moreover, there exists $C>0$ satisfying

[TABLE]

Proof.

Let $\delta_{1}>0$ be a constant defined in (1.8)–(1.10). First we shall show that the function $E_{1}:(0,\infty)\to\mathbb{R}$ defined as

[TABLE]

satisfies that (3.1) holds for all $t>0$ with some $\varepsilon>0$. From straightforward calculations we infer that

[TABLE]

Here in light of (1.8)–(1.10) we can take $\delta_{2}>0$ satisfying

[TABLE]

and

[TABLE]

Invoking the Young inequality, we obtain that

[TABLE]

and

[TABLE]

Therefore since the definition of $\delta_{2}$ yields

[TABLE]

a combination of (3.3) with (3.4) and (3.5) implies

[TABLE]

Noting from the third equation in (1.1) that

[TABLE]

we establish that

[TABLE]

where

[TABLE]

for all $t>0$. In order to see (3.1) we will show that

[TABLE]

with some $\varepsilon>0$ by using Lemma 3.2. To confirm that the assumption of Lemma 3.2 is satisfied we put

[TABLE]

for $\varepsilon>0$, and shall see that there exists $\varepsilon_{1}>0$ such that $g_{i}(\varepsilon_{1})>0$ for $i=1,2,3$. Here $g_{1}(0)=\mu_{1}>0$ obviously holds, and the condition (1.8) implies that

[TABLE]

Moreover, aided by the definition of $\delta_{2}$, we can obtain that

[TABLE]

Therefore a combination of the above inequalities and the continuity argument implies that there exists $\varepsilon_{1}>0$ such that $g_{i}(\varepsilon_{1})>0$ hold for $i=1,2,3$. Thus Lemma 3.2 derives that

[TABLE]

which yields that (3.1) holds for all $t>0$. Then since from the Taylor formula $E_{1}$ is a nonnegative function for $t>0$ (more details, see [1, Lemma 3.2]), integrating (3.1) over $(0,\infty)$ concludes the proof of this lemma. ∎

Lemma 3.5**.**

Assume that (1.8)–(1.10) are satisfied. Then

[TABLE]

Proof.

A combination of Lemmas 3.1 and 3.4 implies this lemma. ∎

Next we desire to establish convergence rates for the solution of (1.1). We note that in view of Lemma 3.3 it is sufficient to confirm the $L^{2}$-convergence rates for the solution.

Lemma 3.6**.**

Assume that (1.8)–(1.10) are satisfied. Then there exist $C>0$ and $\ell>0$ such that

[TABLE]

Proof.

Aided by Lemma 3.5 and the L’Hôpital theorem, a similar argument as in the proof of [1, Lemma 3.7] (or [27, Proof of Theorem 1.2]) derives that there exist $C_{1},C_{2}>0$ and $t_{0}>0$ such that for all $t>t_{0}$,

[TABLE]

where $E_{1}$ is the function defined as (3.2). Therefore we obtain from (3.1) that

[TABLE]

with some $C_{3}>0$, which implies that there exists $C_{4}>0$ such that

[TABLE]

Thus a combination of (3.6) and (3.7) yields that

[TABLE]

which concludes the proof of this lemma. ∎

3.2 Convergence. Case 2: $a_{1}\geq 1>a_{2}$

In this subsection we will obtain stabilization in the case that $a_{1}\geq 1>a_{2}$. In this case we also have to confirm that the assumption of Lemma 3.1 is satisfied.

Lemma 3.7**.**

Assume that (1.11)–(1.12) are satisfied. Then there exist a nonnegative function $E_{2}:(0,\infty)\to\mathbb{R}$ and constants $\varepsilon_{1}\geq 0$ and $\varepsilon_{2}>0$ such that

[TABLE]

holds for all $t>0$. Moreover, there exists $C>0$ satisfying

[TABLE]

Proof.

Let $\delta_{1}>0$ and $a_{1}^{\prime}\in[1,a_{1}]$ be a constant defined in (1.11)–(1.12). We first show that the function $E_{2}:(0,\infty)\to\mathbb{R}$ defined as

[TABLE]

fulfils (3.8) for all $t>0$ with $\varepsilon_{1}:=a_{1}^{\prime}-1$ and some $\varepsilon_{2}>0$. Noting from the relation $a_{1}^{\prime}\leq a_{1}$ that

[TABLE]

from straightforward calculations we derive that

[TABLE]

for all $t>0$. Here thanks to (1.11)–(1.12), we can take $\delta_{2}>0$ such that

[TABLE]

Invoking the Young inequality, we obtain that

[TABLE]

Therefore since the definition of $\delta_{2}$ yields

[TABLE]

the equation (3.10) implies that

[TABLE]

for all $t>0$, where $\varepsilon_{1}=a_{1}^{\prime}-1$. Noting from the third equation in (1.1) that

[TABLE]

we establish that for all $t>0$,

[TABLE]

where

[TABLE]

Then by using the same argument as in the proof of Lemma 3.4 we can see that

[TABLE]

with some $\varepsilon_{2}>0$, which means that (3.8) holds with $\varepsilon_{1}=a_{1}^{\prime}-1\geq 0$ and $\varepsilon_{2}>0$. ∎

Then we will establish the convergence result for the solution to (1.1) in the case that $a_{1}\geq 1>a_{2}$.

Lemma 3.8**.**

Assume that (1.11)–(1.12) are satisfied. Then we have

[TABLE]

Proof.

A combination of Lemmas 3.1 and 3.7 implies this lemma. ∎

Finally, we shall show two lemmas which give asymptotic behavior in the case that $a_{1}>1>a_{2}$.

Lemma 3.9**.**

Let $a_{1}>1$ and $a_{2}\in(0,1)$. Assume that (1.11)–(1.12) are satisfied with $\delta_{1}>0$ and $a_{1}^{\prime}>0$. Then there exist $C>0$ and $\ell>0$ satisfying

[TABLE]

Proof.

In the case that $a_{1}\geq 1$ and $a_{2}\in(0,1)$ a similar argument as in the proof of [27, Lemmas 4.3] enables us to see that there exist $C_{1},C_{2}>0$ and $t_{0}>0$ such that

[TABLE]

where $E_{2}$ is the function defined as (3.9) and

[TABLE]

Thus a combination of the above inequality and (3.8) means that

[TABLE]

holds for all $t>t_{0}$, which together with the same argument as in the proof of Lemma 3.6 leads to the conclusion of this lemma in the case that $a_{1}>1$ and $a_{2}\in(0,1)$. ∎

Lemma 3.10**.**

Let $a_{1}=1$ and $a_{2}\in(0,1)$. Assume that (1.11)–(1.12) are satisfied. Then there exist $C>0$ and $\ell>0$ satisfying

[TABLE]

Proof.

First we can verify from the same argument as in the proof of [27, Lemma 3.7] that there exist $C_{4},C_{5}>0$ and $t_{1}>0$ such that

[TABLE]

for all $t>t_{1}$. Hence it follows from the Cauchy–Schwarz inequality and the boundedness of $v$ that

[TABLE]

which implies from (3.8) that

[TABLE]

Therefore we can find $C_{8}>0$ such that

[TABLE]

Therefore thanks to the boundedness of $u$ and (3.11), we obtain that

[TABLE]

which proves this lemma. ∎

Proof of Theorems 1.3 and 1.4.

A combination of Lemmas 3.6, 3.9, 3.10 and 3.3 immediately leads to the conclusions of these theorems. ∎