Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics

TL;DR

This paper proves the global existence, boundedness, and stabilization of solutions for a three-dimensional two-species chemotaxis-Stokes system with competitive kinetics, extending previous 2D results to 3D.

Contribution

It provides the first comprehensive results on global solutions and their asymptotic behavior for the 3D coupled chemotaxis-fluid system with competition.

Findings

Global existence of solutions in 3D

Solutions are bounded over time

Solutions stabilize asymptotically

Abstract

This paper considers the two-species chemotaxis-Stokes system with competitive kinetics under homogeneous Neumann boundary conditions in a three-dimensional bounded domain with smooth boundary. Both chemotaxis-fluid systems and two-species chemotaxis systems with competitive terms are studied by many mathematicians. However, there has not been rich results on coupled two-species-fluid systems. Recently, global existence and asymptotic stability in this problem with convection term in the fluid equation of the above system were established in the 2-dimensional case. The purpose of this paper is to give results for global existence, boundedness and stabilization of solutions to this system in the 3-dimensional case.

Equations

$$\{\begin{array}{ll}(n_1{)}_t+u\cdot \nabla n_1=\Delta n_1-{\chi }_1\nabla \cdot (n_1\nabla c)+{\mu }_1{n}_1(1-n_1-a_1{n}_2),& x\in \Omega ,t>0,\\ (n_2{)}_t+u\cdot \nabla n_2=\Delta n_2-{\chi }_2\nabla \cdot (n_2\nabla c)+{\mu }_2{n}_2(1-a_2{n}_1-n_2),& x\in \Omega ,t>0,\\ c_t+u\cdot \nabla c=\Delta c-(\alpha n_1+\beta n_2)c,& x\in \Omega ,t>0,\\ u_t=\Delta u+\nabla P+(\gamma n_1+\delta n_2)\nabla \varphi ,\nabla \cdot u=0,& x\in \Omega ,t>0\end{array}$$

$$\{\begin{array}{ll}(n_1{)}_t+u\cdot \nabla n_1=\Delta n_1-{\chi }_1\nabla \cdot (n_1\nabla c)+{\mu }_1{n}_1(1-n_1-a_1{n}_2),& x\in \Omega ,t>0,\\ (n_2{)}_t+u\cdot \nabla n_2=\Delta n_2-{\chi }_2\nabla \cdot (n_2\nabla c)+{\mu }_2{n}_2(1-a_2{n}_1-n_2),& x\in \Omega ,t>0,\\ c_t+u\cdot \nabla c=\Delta c-(\alpha n_1+\beta n_2)c,& x\in \Omega ,t>0,\\ u_t+\kappa (u\cdot \nabla )u=\Delta u+\nabla P+(\gamma n_1+\delta n_2)\nabla \varphi ,\nabla \cdot u=0,& x\in \Omega ,t>0,\\ {\partial }_{\nu }{n}_1={\partial }_{\nu }{n}_2={\partial }_{\nu }c=0,u=0,& x\in \partial \Omega ,t>0,\\ n_i(x,0)=n_{i,0}(x),c(x,0)=c_0(x),u(x,0)=u_0(x),& x\in \Omega ,i=1,2,\end{array}$$

$$0<n_{1,0},n_{2,0}\in C(\overline{\Omega }),0<c_0\in W^{1,q}(\Omega ),u_0\in D(A^{\vartheta }),$$

$$\varphi \in C^{1+\eta }(\overline{\Omega })$$

$$\chi :=\max \{{\chi }_1,{\chi }_2\}\text{mbox}and\mu :=\min \{{\mu }_1,{\mu }_2\}$$

$$n_1,n_2\in C(\overline{\Omega }\times [0,\infty ))\cap C^{2,1}(\overline{\Omega }\times (0,\infty )),$$

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

$$u\in C(\overline{\Omega }\times [0,\infty ))\cap C^{2,1}(\overline{\Omega }\times (0,\infty ))\cap L_{\mathrm{loc}}^{\infty }([0,\infty );D(A^{\vartheta })),$$

$$P\in C^{1,0}(\overline{\Omega }\times (0,\infty )).$$

$$\Vert n_1(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert n_2(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert c(\cdot ,t){\Vert }_{W^{1,q}(\Omega )}+\Vert u(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}\le C\text{mbox}forallt\in (0,\infty ).$$

$$n_1(\cdot ,t)\to N_1,n_2(\cdot ,t)\to N_2,c(\cdot ,t)\to 0,u(\cdot ,t)\to 0\text{mbox}in{L}^{\infty }(\Omega )\text{mbox}ast\to \infty ,$$

$$N_1:=\frac{1-a_1}{1-a_1{a}_2},N_2:=\frac{1-a_2}{1-a_1{a}_2}.$$

$$n_1(\cdot ,t)\to 0,n_2(\cdot ,t)\to 1,c(\cdot ,t)\to 0,u(\cdot ,t)\to 0\text{mbox}in{L}^{\infty }(\Omega )\text{mbox}ast\to \infty .$$

$$\displaystyle\int_{\Omega}n_{1}^{p}+\int_{\Omega}n_{2}^{p}\leq C\int_{s_{0}}^{t}e^{-(p+1)(t-s)}\int_{\Omega}|\Delta c|^{p+1}-C\int_{s_{0}}^{t}e^{-(p+1)(t-s)}\Bigl{(}\int_{\Omega}n_{1}^{p+1}+\int_{\Omega}n_{2}^{p+1}\Bigr{)}$$

$${\int }_0^{\infty }{\int }_{\Omega }(n_1-N_1{)}^2+{\int }_0^{\infty }{\int }_{\Omega }(n_2-N_2{)}^2\le C$$

$$E:={\int }_{\Omega }\left(n_1-N_1-N_1\log \frac{n_1}{N_1}\right)+b_1{\int }_{\Omega }\left(n_2-N_2-N_2\log \frac{n_2}{N_2}\right)+\frac{b_2}{2}{\int }_{\Omega }{c}^2$$

$$\frac{d}{dt}E(t)\le -\epsilon {\int }_{\Omega }\left[(n_1-N_1{)}^2+(n_2-N_2{)}^2\right]$$

$$n_1,n_2\in C(\overline{\Omega }\times [0,T_{\max }))\cap C^{2,1}(\overline{\Omega }\times (0,T_{\max })),$$

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

$$u\in C(\overline{\Omega }\times [0,T_{\max }))\cap C^{2,1}(\overline{\Omega }\times (0,T_{\max }))\cap L_{\mathrm{loc}}^{\infty }([0,T_{\max });D(A^{\vartheta })),$$

$$n_1,n_2>0,c>0\text{mbox}in\Omega \times (0,T_{\max }).$$

$$\underset{t\nearrow T_{\max }}{\mathrm{limsup}}(\Vert n_1(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert n_2(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert c(\cdot ,t){\Vert }_{W^{1,q}(\Omega )}+\Vert A^{\vartheta }u(\cdot ,t){\Vert }_{L^2(\Omega )})=\infty .$$

$$c(\cdot ,s_0)\in C^2(\overline{\Omega })\text{mbox}with{\partial }_{\nu }c(\cdot ,s_0)=0\text{mbox}on\partial \Omega .$$

$$\Vert c(\cdot ,s_0){\Vert }_{W^{2,\infty }(\Omega )}\le M$$

$${\int }_{s_0}^t{\int }_{\Omega }{e}^{ps}|\Delta c{|}^p$$

$$+C_1{e}^{p{s}_0}(\Vert c(\cdot ,s_0){\Vert }_{L^p(\Omega )}^p+\Vert \Delta c(\cdot ,s_0){\Vert }_{L^p(\Omega )}^p)$$

$$c_{t}=(\Delta-1)c-c+\bigl{(}2c-(\alpha n_{1}+\beta n_{2})c\bigr{)}-u\cdot\nabla c,$$

$$\{\begin{array}{ll}{\tilde{c}}_t=(\Delta -1)\tilde{c}+f,& x\in \Omega ,s\in (s_0,t),\\ {\partial }_{\nu }\tilde{c}=0,& x\in \partial \Omega ,s\in (s_0,t),\\ \tilde{c}(\cdot ,s_0)=e^{s_0}c(\cdot ,s_0)\in W^{2,p}(\Omega ),& x\in \Omega ,\end{array}$$

$$f:=e^s(2c-(\alpha n_1+\beta n_2)c-u\cdot \nabla c)\in L^p(s_0,t;L^p(\Omega )).$$

$${\int }_{\Omega }{n}_i(\cdot ,t)\le C_2$$

$$\Vert c(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}\le \Vert c_0{\Vert }_{L^{\infty }(\Omega )}$$

$$\Vert u(\cdot ,t){\Vert }_{L^r(\Omega )}\le C_3$$

$$\frac{1}{p}{\int }_{\Omega }{n}_1^p(\cdot ,t)+\frac{1}{p}{\int }_{\Omega }{n}_2^p(\cdot ,t)$$

$$\displaystyle\leq-(\mu-\varepsilon-\ell)e^{-(p+1)t}\int_{s_{0}}^{t}e^{(p+1)s}\Bigl{(}\int_{\Omega}n_{1}^{p+1}+\int_{\Omega}n_{2}^{p+1}\Bigr{)}$$

$$+C_4{\ell }^{-p}{\chi }^{p+1}{e}^{-(p+1)t}{\int }_{s_0}^t{e}^{(p+1)s}{\int }_{\Omega }|\Delta c{|}^{p+1}+C_4$$

Full text

0002010Mathematics Subject Classification. Primary: 35K45; Secondary: 92C17; 35Q35. 000Key words and phrases: chemotaxis-Stokes; global existence; asymptotic stability.

**Global existence and asymptotic behavior

of classical solutions for a 3D two-species

chemotaxis-Stokes system

with competitive kinetics **

Xinru Cao

Institut für Mathematik, Universität Paderborn

Warburger Str.100, 33098 Paderborn, Germany

Shunsuke Kurima

Department of Mathematics, Tokyo University of Science

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

Masaaki Mizukami***Corresponding author

Department of Mathematics, Tokyo University of Science

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

masaaki.mizukami.math@gmail.com

• Abstract. This paper considers the two-species chemotaxis-Stokes system with competitive kinetics

[TABLE]

under homogeneous Neumann boundary conditions in a three-dimensional bounded domain $\Omega\subset\mathbb{R}^{3}$ with smooth boundary. Both chemotaxis-fluid systems and two-species chemotaxis systems with competitive terms are studied by many mathematicians. However, there has not been rich results on coupled two-species-fluid systems. Recently, global existence and asymptotic stability in the above problem with $(u\cdot\nabla)u$ in the fluid equation of the above system were established in the 2-dimensional case ([14]). The purpose of this paper is to give results for global existence, boundedness and stabilization of solutions to the above system in the 3-dimensional case when $\frac{\mu_{i}}{\chi_{i}}$ $(i=1,2)$ is sufficiently large.

1 Introduction and results

We consider the following two-species chemotaxis-fluid system with competitive terms:

[TABLE]

where $\Omega$ is a bounded domain in $\mathbb{R}^{3}$ with smooth boundary $\partial\Omega$ and $\partial_{\nu}$ denotes differentiation with respect to the outward normal of $\partial\Omega$; $\kappa\in\{0,1\}$ (in this paper we will deal with the case that $\kappa=0$), $\chi_{1},\chi_{2},a_{1},a_{2}\geq 0$ and $\mu_{1},\mu_{2},\alpha,\beta,\gamma,\delta>0$ are constants; $n_{1,0},n_{2,0},c_{0},u_{0},\phi$ are known functions satisfying

[TABLE]

for some $q>3$, $\vartheta\in\left(\frac{3}{4},1\right)$, $\eta>0$ and $A$ is the Stokes operator.

The problem (1.1) is a generalized system to the chemotaxis-fluid system which is proposed by Tuval et al. [33]. This system describes the evolution of two competing species which react on a single chemoattractant in a liquid surrounding environment. Here $n_{1},n_{2}$ represent the population densities of species, $c$ stands for the concentration of chemoattractant, $u$ shows the fluid velocity field and $P$ represents the pressure of the fluid. The problem (1.1) comes from a problem on account of the influence of chemotaxis, the Lotka–Volterra competitive kinetics and the fluid. In the mathematical point of view, the chemotaxis term: $\nabla\cdot(n_{1}\nabla c)$, the competition term: $n_{1}(1-n_{1}-a_{1}n_{2})$ and the Stokes equation give difficulties in mathematical analysis.

The one-species system (1.1) with $n_{2}=0$ has been studied in some literature. It is known that there exist global classical solutions in the 2-dimensional setting; however, in the 3-dimensional setting, only global weak solutions exist. In this one-species system with $\mu_{1}=0$, Winkler first attained global existence of classical solutions to (1.1), $\kappa=0$ in the 3-dimensional setting and $\kappa=1$ in the 2-dimensional setting ([37]), and also established asymptotic stability of solutions to (1.1) ([38]). Moreover, the convergence rate has been already studied ([43]). Recently, Winkler [41] attained global existence and eventual smoothness of weak solutions and their asymptotic behavior for the 3-dimensional chemotaxis-Navier–Stokes system.

In the analysis of the one-species case the logistic source can enhance the possibility of global existence of solutions. In the 3-dimensional setting, Lankeit [19] obtained global existence of weak solutions in (1.1) with $n_{2}=0$, $\kappa=1$ and with additional external force $f$ in the fourth equation, and also derived eventual smoothness and asymptotic behavior. Even for more complicated problems, Keller–Segel-fluid systems where $-(\alpha n_{1}+\beta n_{2})c$ is replaced with $-c+\alpha n_{1}$ in (1.1) with $n_{2}=0$, logistic source is shown to be helpful for establishing classical bounded solutions. In the 3-dimensional setting, Tao and Winkler [30] established global existence and boundedness of classical solutions by assuming that $\mu_{1}>23$. In the 2-dimensional case, Tao and Winkler [31] also showed global existence of bounded classical solutions in the Keller–Segel-Navier–Stokes system with logistic source with $+rn_{1}-\mu_{1}n_{1}^{2}$ for any $\mu_{1}>0$, and their asymptotic behavior were obtained when $r=0$. For more related works we refer to Ishida [16], Wang and the first author [34], Wang and Xiang [35], Black [3], the first author [5], the first author and Lankeit [6], Kozono, Miura and Sugiyama [17]. These results fully parallel to those for the fluid free model; we can find counterpart in [20, 26, 29].

On the other hand, the study on two-species competitive chemotaxis systems with signal consumption seems pending. We can only find related research with signal production in which the asymptotic behavior of solutions usually relies on some smallness assumption for the chemotaxis sensitivities (e.g., for the noncompetitive case ($a_{1}=a_{2}=0$), see Negreanu and Tello [24, 25], the third author and Yokota [23], the third author [21]; for the competitive case see Tello and Winkler [32], Stinner, Tello and Winkler [28], Bai and Winkler [1], Black, Lankeit and the third author [4], the third author [22]).

As mentioned above, the chemotaxis-fluid systems ($n_{2}=0$ in (1.1)) and the chemotaxis systems with competitive terms ($u=0$ in (1.1)) were studied by many mathematicians. However, the problem (1.1), which is the combination of chemotaxis-fluid systems and chemotaxis-competition systems, had not been studied. Recently, global existence, boundedness of classical solutions and their asymptotic behavior were showed only in the 2-dimensional setting ([14]).

The purpose of the present article is to obtain global existence and boundedness of classical solutions, and their asymptotic stability in the 3-dimensional setting. The main results read as follows. The first theorem gives global existence and boundedness in (1.1). In view of known results on logistic chemotaxis-systems, it is no wonder that an assumption on smallness of $\chi_{1}$ and $\chi_{2}$ related to $\mu_{1}$ and $\mu_{2}$ will be necessary in the considered 3-dimensional case.

Theorem 1.1**.**

Let $\Omega\subset\mathbb{R}^{3}$ be a bounded domain with smooth boundary and let $\kappa=0$, $\chi_{1},\chi_{2},a_{1},a_{2}\geq 0$, $\mu_{1},\mu_{2},\alpha,\beta,\gamma,\delta>0$. Suppose that (1.2) and (1.3) hold. Then there exists a constant $\xi_{0}>0$ such that whenever

[TABLE]

satisfy $\frac{\chi}{\mu}<\xi_{0}$, the problem (1.1) possesses a classical solution $(n_{1},n_{2},c,u,P)$ such that

[TABLE]

Also, the solution is unique in the sense that it allows up to addition of spatially constants to the pressure $P$. Moreover, there exists a constant $C>0$ such that

[TABLE]

The second theorem is concerned with asymptotic stability in (1.1).

Theorem 1.2**.**

Let the assumption of Theorem 1.1 holds. Then the solution of (1.1) has the following properties:**

1. (i)

Assume that $a_{1},a_{2}\in(0,1)$. Then

[TABLE]

where

[TABLE] 2. (ii)

Assume that $a_{1}\geq 1>a_{2}$. Then

[TABLE]

The strategy for the proof of Theorem 1.1 is to derive the $L^{p}$-estimate for $n_{i}$ with $p>\frac{3}{2}$. By using the differential inequality we can see

[TABLE]

with some $C>0$ and $s_{0}>0$. The maximal Sobolev regularity (see Lemma 2.2) will be used to control $\int_{s_{0}}^{t}e^{-(p+1)(t-s)}\int_{\Omega}|\Delta c|^{p+1}$. Combining the maximal Sobolev regularity with some estimate for $|Au|^{2}$, we can obtain the $L^{p}$-estimate for $n_{i}$. On the other hand, the strategy for the proof of Theorem 1.2 is to derive the following inequality:

[TABLE]

with some $C>0$, where $(N_{1},N_{2},0,0)$ is a constant solution to (1.1). In order to obtain this estimate we will use the energy function

[TABLE]

with some $b_{1},b_{2}>0$, and show

[TABLE]

with some $\varepsilon>0$. This estimate and the positivity of $E(t)$ lead to (1.4).

This paper is organized as follows. In Section 2 we collect basic facts which will be used later. In Section 3 we prove global existence and boundedness (Theorem 1.1). Sections 4 is devoted to showing asymptotic stability (Theorem 1.2).

2 Preliminaries

In this section we will provide some results which will be used later. The following lemma gives local existence of solutions to (1.1).

Lemma 2.1**.**

Let $\Omega\subset\mathbb{R}^{3}$ be a bounded domain with smooth boundary. Suppose that (1.2) and (1.3) hold. Then there exists $T_{\rm max}\in(0,\infty]$ such that the problem (1.1) possesses a classical solution $(n_{1},n_{2},c,u,P)$ fulfilling

[TABLE]

Also, the solution is unique up to addition of spatially constants to the pressure $P$. Moreover, either $T_{\rm max}=\infty$ or

[TABLE]

Proof.

The proof of local existence of classical solutions to (1.1) is based on a standard contraction mapping argument, which can be found in [37]. Accordingly, the maximum principle is applied to yield $n_{1},n_{2}>0$ and $c>0$ in $\Omega\times(0,T_{\rm max})$. ∎

Given all $s_{0}\in(0,T_{\rm max})$, from the regularity properties we see that

[TABLE]

In particular, there exists $M=M(s_{0})>0$ such that

[TABLE]

(see e.g., [42]).

The following lemma is referred to as a variation of the maximal Sobolev regularity (see [13, Theorem 3.1]), which is important to prove Theorem 1.1.

Lemma 2.2**.**

Let $s_{0}\in(0,T_{\rm max})$. Then for all $p>1$ there exists a constant $C_{1}=C_{1}(p)>0$ such that

[TABLE]

holds for all $t\in(s_{0},T_{\rm max})$.

Proof.

Let $s_{0}\in(0,T_{\rm max})$ and let $t\in(s_{0},T_{\rm max})$. We rewrite the third equation as

[TABLE]

and use the transformation $\widetilde{c}(\cdot,s)=e^{s}c(\cdot,s)$, $s\in(s_{0},t)$. Then $\widetilde{c}$ satisfies

[TABLE]

where

[TABLE]

Therefore an application of the maximal Sobolev regularity [13, Theorem 3.1] to $\widetilde{c}$ implies this lemma. ∎

3 Boundedness. Proof of Theorem 1.1

In this section we will prove Theorem 1.1 by preparing a series of lemmas.

Lemma 3.1**.**

There exists a constant $C_{2}>0$ such that

[TABLE]

for all $t\in(0,T_{\rm max})$ for $i=1,2$.

Proof.

The same argument as in the proof of [14, Lemma 3.1] implies this lemma. ∎

Lemma 3.2**.**

The function $t\mapsto\|c(\cdot,t)\|_{L^{\infty}(\Omega)}$ is nonincreasing. In particular,

[TABLE]

holds for all $t\in(0,T_{\rm max})$.

Proof.

We can prove this lemma by applying the maximum principle to the third equation in (1.1). ∎

Lemma 3.3**.**

For $r\in(1,3)$ there exists a constant $C_{3}=C_{3}(r)>0$ such that

[TABLE]

for all $t\in(0,T_{\rm max})$.

Proof.

From the well-known Neumann heat semigroup estimates together with Lemma 3.1 we can obtain the $L^{r}$-estimate for $u$ with $r\in(1,3)$ (for more details, see [39, Corollary 3.4]). ∎

Now we fix $s_{0}\in(0,T_{\rm max})\cap(0,1]$. The proofs of the following two lemmas are based on the methods in [42, Lemma 3.1].

Lemma 3.4**.**

For all $p>1$, $\varepsilon>0$ and $\ell>0$ there exists a constant $C_{4}=C_{4}(p)>0$ such that

[TABLE]

for all $t\in(s_{0},T_{\rm max})$, where $\mu=\min\{\mu_{1},\mu_{2}\}$, $\chi=\max\{\chi_{1},\chi_{2}\}$.

Proof.

Let $p>1$. Multiplying the first equation in (1.1) by $n_{1}^{p-1}$ and integrating it over $\Omega$, we see that

[TABLE]

Noting from $\nabla\cdot u=0$ in $\Omega\times(0,T_{\rm max})$ that $\int_{\Omega}u\cdot\nabla n_{1}^{p}=-\int_{\Omega}(\nabla\cdot u)n_{1}^{p}=0$, we obtain from integration by parts and nonnegativity of $n_{1}$, $n_{2}$ that

[TABLE]

Now we let $\varepsilon>0$ and $\ell>0$. By the Young inequality there exists a constant $C_{5}=C_{5}(\mu_{1},\varepsilon,p)>0$ such that

[TABLE]

Moreover, the second term on the right-hand side of (3.1) can be estimated as

[TABLE]

with some $C_{6}=C_{6}(p)>0$. Hence we derive from (3.1), (3.2) and (3.3) that

[TABLE]

Therefore there exists $C_{7}=C_{7}(\mu_{1},\varepsilon,p,|\Omega|,s_{0})>0$ such that

[TABLE]

for each $t\in(s_{0},T_{\rm max})$. Similarly, we see that

[TABLE]

with some $C_{8}=C_{8}(\mu_{2},\varepsilon,p,|\Omega|,s_{0})>0$. Thus from (3.4) and (3.5) we have that there exists $C_{9}=C_{9}(\mu,\varepsilon,p,|\Omega|,s_{0})>0$ such that

[TABLE]

where $\mu=\min\{\mu_{1},\mu_{2}\}$ and $\chi=\max\{\chi_{1},\chi_{2}\}$. ∎

Lemma 3.5**.**

For all $p\in(1,2)$ there exists a constant $C_{10}=C_{10}(p)>0$ such that

[TABLE]

for all $t\in(s_{0},T_{\rm max})$.

Proof.

Fix $\theta\in(1,2)$ and put $\theta^{\prime}=\frac{\theta}{\theta-1}$. We derive from Lemma 2.2 that

[TABLE]

holds with some $C_{11}=C_{11}(p)>0$. Lemma 3.2 and the Hölder inequality imply

[TABLE]

with some $C_{12}=C_{12}(p)>0$. Here we see from the Gagliardo–Nirenberg inequality and Lemma 3.2 that there exist constants $C_{13}=C_{13}(p),C_{14}=C_{14}(p)>0$ such that

[TABLE]

with $a:=\frac{1-\frac{3}{(p+1)\theta^{\prime}}}{2-\frac{3}{p+1}}\in(\frac{1}{2},1)$. By (3.6), (3.7), (3.8) and the Young inequality it holds that

[TABLE]

with some $C_{15}=C_{15}(p)>0$. Here we use $p<2$, which namely enable us to pick $r\in(1,3)$ such that

[TABLE]

holds. Therefore we can obtain that

[TABLE]

Now we see from the Gagliardo–Nirenberg inequality, Lemma 3.3 and the Young inequality that there exists a constant $C_{16}=C_{16}(p)>0$ such that

[TABLE]

with $b:=\frac{\frac{3}{r}-\frac{3}{(p+1)\theta}}{\frac{1}{2}+\frac{3}{r}}\in(0,1)$, since $\frac{p+1}{1-a}b<2$ from (3.9). Similarly, there exists a constant $C_{17}=C_{17}(p)>0$ such that

[TABLE]

for all $s\in(s_{0},T_{\rm max})$. Therefore combination of (3.10) with (3.11) and (3.12) yields that there exists a constant $C_{18}=C_{18}(p)>0$ such that

[TABLE]

for all $t\in(s_{0},T_{\rm max})$, which means the end of the proof. ∎

In order to control $\int_{s_{0}}^{t}e^{(p+1)s}\int_{\Omega}|\Delta c|^{p+1}$ we provide the following lemma.

Lemma 3.6**.**

For all $p>1$ there exists a constant $C_{19}=C_{19}(p)>0$ such that

[TABLE]

for all $t\in(s_{0},T_{\rm max})$.

Proof.

It follows from the fourth equation in (1.1), the Young inequality and the continuity of the Helmholtz projection on $L^{2}(\Omega;\mathbb{R}^{3})$ ([9, Theorem 1]) that

[TABLE]

and hence there exists a constant $C_{20}>0$ such that

[TABLE]

and we derive from [7, Part2, Theorem 14.1], Lemma 3.3 and the Young inequality that

[TABLE]

with some constants $C_{21},C_{22}>0$. By (3.13) and (3.14) we obtain

[TABLE]

with some constant $C_{23}=C_{23}(p)>0$, and hence we have

[TABLE]

with some constant $C_{24}=C_{24}(p)>0$, which concludes the proof. ∎

Lemma 3.7**.**

For all $p\in(1,2)$ and for all $\ell>0$ there exist positive constants $K(p)>0$ and $C_{25}=C_{25}(p,\ell)>0$ such that if $\mu>\mu_{p,\ell}:=\ell+K(p)\ell^{-p}\chi^{p+1}$, then

[TABLE]

for $i=1,2$, where $\mu=\min\{\mu_{1},\mu_{2}\}$, $\chi=\max\{\chi_{1},\chi_{2}\}$.

Proof.

It follows from Lemmas 3.4, 3.5 and 3.6 that there exists a constant $K(p)>0$ such that

[TABLE]

for all $t\in(s_{0},T_{\rm max})$. We assume that $\mu>\mu_{p,\ell}$. Then there exists $\varepsilon\in(0,\mu-\mu_{p,\ell})$ such that

[TABLE]

Thus we derive that

[TABLE]

holds for all $t\in(s_{0},T_{\rm max})$, which concludes the proof of Lemma 3.7. ∎

The proof of the following lemma is based on the method in [42, Proof of Theorem 1].

Lemma 3.8**.**

For all $p\in(\frac{3}{2},2)$ there exists $\xi_{0}>0$ such that if $\frac{\chi}{\mu}<\xi_{0}$, then there exists a constant $C_{26}>0$ such that

[TABLE]

for $i=1,2$.

Proof.

Let $p\in(\frac{3}{2},2)$ and let $\xi_{0}=\xi_{0}(p)>0$ satisfing

[TABLE]

Then we can see that $\frac{\chi}{\mu}<\xi_{0}$ implies $\mu>\mu_{p,\ell}$ with some $\ell>0$. Therefore Lemma 3.7 implies that there exists a constant $C_{27}=C_{27}(p)>0$ such that

[TABLE]

for each $i=1,2$, which implies the end of the proof. ∎

Lemma 3.9**.**

Assume $\frac{\chi}{\mu}<\xi_{0}$. Then there exists a constant $C_{28}>0$ such that

[TABLE]

for all $t\in(0,T_{\rm max})$.

Proof.

Noting that $\frac{1}{2}+\frac{2}{3}(1-\vartheta)\in(\frac{1}{2},\frac{2}{3})$, we can fix p\in\Bigl{(}\frac{1}{\frac{1}{2}+\frac{2}{3}(1-\vartheta)},2\Bigr{)}. It follows from Lemma 3.8, the well-known regularization estimates for Stokes semigroup [11, 27] and the continuity of the Helmholtz projection on $L^{r}(\Omega;\mathbb{R}^{3})$ (see e.g., [9, Theorem 1]) that there exist constants $C_{29},C_{30},C_{31},C_{32}>0$ and $\lambda>0$ such that

[TABLE]

for all $t\in(0,T_{\rm max})$ since $\vartheta+\frac{3}{2}(\frac{1}{p}-\frac{1}{2})<1$. Moreover, the properties of $D(A^{\vartheta})$ ([10, Theorem 3] and [12, Theorem 1.6.1]) imply that there exists $C_{33}>0$ such that

[TABLE]

for all $t\in(0,T_{\rm max})$, which concludes the proof. ∎

Lemma 3.10**.**

Assume $\frac{\chi}{\mu}<\xi_{0}$. Then there exist $r\in(3,6)\cap(1,q]$ and $C_{34}=C_{34}(r)>0$ such that

[TABLE]

Proof.

Let $r\in(3,6)\cap(1,q]$ and fix $p\in(\frac{3q}{q+3},2)$. An application of the variation of constants formula for $c$ leads to

[TABLE]

We first obtain the estimate for the first term on the right-hand side of (3.15). Noting that $q>3$, we derive from the Hölder inequality and [36, Lemma 1.3 (iii)] that there exist constants $C_{35},C_{36}>0$ such that

[TABLE]

We next establish the estimate for the second term on the right-hand side of (3.15). Lemmas 3.2 and 3.8 yield that there exist constants $C_{37},C_{38}>0$ such that

[TABLE]

Here, since $\frac{1}{2}+\frac{3}{2}(\frac{1}{p}-\frac{1}{r})<1$, we have

[TABLE]

Combination of (3.17) and (3.18) derives that

[TABLE]

with some constant $C_{39}>0$. Finally we will deal with the third term on the right-hand side of (3.15). Now we put $\theta\geq r$, $0<\ell<\frac{1}{2}$ satisfying $\frac{1}{2}+\frac{3}{2}(\frac{1}{\theta}-\frac{1}{r})<\ell$ and $a\in(0,\frac{1}{2}-\ell)$. Then we derive from [15, Section 2] and Lemmas 3.2 and 3.9 that there exist constants $C_{40},C_{41},C_{42}>0$ and $\lambda>0$ such that

[TABLE]

Noting that $\ell+\frac{1}{2}+a<1$, we infer that there exists a constant $C_{43}>0$ such that

[TABLE]

From (3.20) and (3.21) we have

[TABLE]

with some constant $C_{44}>0$. Therefore in light of (3.15), (3.16), (3.19) and (3.22) there exists a constant $C_{45}>0$ that

[TABLE]

for all $t\in(0,T_{\rm max})$. ∎

Then we will derive the $L^{\infty}$-estimate for $n_{i}$ by using the well-known semigroup estimates (see [2]).

Lemma 3.11**.**

Assume $\frac{\chi}{\mu}<\xi_{0}$. Then there exists a constant $C_{46}>0$ such that

[TABLE]

for $i=1,2$.

Proof.

We let $q>3$ and let $p\in(\frac{3}{2},2)$ with $\frac{3p}{3-p}<q$. Then thanks to Lemma 3.8, we obtain

[TABLE]

for all $t\in(0,T_{\rm max})$ with some $C_{47}>0$. Now we can choose $r\in(3,q)$ such that $p>\frac{3r}{3+r}$ and $\theta>1$ satisfying

[TABLE]

and put $\theta^{\prime}:=\frac{\theta}{\theta-1}$, and then

[TABLE]

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

[TABLE]

is finite. In order to obtain the estimate for $M(T^{\prime})$ for all $t\in(0,T^{\prime})$ we put $t_{0}:=(t-1)_{+}$ and represent $n_{1}$ according to

[TABLE]

In the case that $t\leq 1$, from the order preserving property of the Neumann heat semigroup we know that

[TABLE]

In the case that $t>1$, by using the $L^{p}$-$L^{q}$ estimate for $(e^{\tau\Delta})_{\tau>0}$ (see [36, Lemma 1.3 (i)]) and Lemma 3.8 we can see that there exists a constant $C_{48}>0$ such that

[TABLE]

Thanks to the elementary inequality

[TABLE]

together with the maximum principle, we see that there exists a constant $C_{49}>0$ such that

[TABLE]

Next we obtain from the known smoothing property of $(e^{\tau\Delta})_{\tau\geq 0}$ (see [8]) that

[TABLE]

for all $t\in(0,T^{\prime})$ with some $C_{50}>0$. Here we note from $\frac{1}{3}+\frac{3}{2r}<1$ that $\int_{0}^{1}\sigma^{-\frac{1}{3}-\frac{3}{2r}}\,d\sigma$ is finite. Then we can obtain that

[TABLE]

for all $s\in(0,T^{\prime})$. Noting from $r\theta^{\prime}<\frac{3p}{3-p}$ that

[TABLE]

we have from Lemma 3.10 that there exists $C_{51}>0$ such that

[TABLE]

Therefore we can find $C_{52}>0$ satisfying

[TABLE]

Similarly, from Lemma 3.9 there exists a constant $C_{53}>0$ such that

[TABLE]

Therefore, Lemma 3.1 leads to the existence of $C_{54},C_{55}>0$ such that

[TABLE]

which implies from the positivity of $n_{1}$ that

[TABLE]

Noting that $r\theta>1$, we derive that there exists $C_{56}>0$ such that

[TABLE]

Similarly we prove that there exists a constant $C_{57}>0$ such that $\|n_{2}(\cdot,t)\|_{L^{\infty}(\Omega)}\leq C_{57}$ for all $t\in(0,T_{\rm max})$. Therefore we can attain the conclusion of the proof. ∎

Proof of Theorem 1.1**.**

Combination of Lemmas 2.1, 3.9, 3.10 and 3.11 directly leads to Theorem 1.1. ∎

4 Stabilization. Proof of Theorem 1.2

4.1 Case 1: $a_{1},a_{2}\in(0,1)$

Now we assume that $\frac{\chi}{\mu}<\xi_{0}$. In this section we will show stabilization in (1.1) in the case $a_{1},a_{2}\in(0,1)$. We will prove the key estimate for the proof of Theorem 1.2. The proof is same as that of [14, Lemma 4.1].

Lemma 4.1**.**

Let $a_{1},a_{2}\in(0,1)$ and let $(n_{1},n_{2},c,u)$ be a solution to (1.1). Under the assumption of Theorem 1.1, there exist $k_{1},\ell_{1}>0$ and $\varepsilon_{1}>0$ such that the nonnegative functions $E_{1}$ and $F_{1}$ defined by

[TABLE]

and

[TABLE]

satisfy

[TABLE]

where

[TABLE]

By using Lemma 4.1 we can show stabilization of $n_{1},n_{2}$.

Lemma 4.2**.**

Let $a_{1},a_{2}\in(0,1)$ and let $(n_{1},n_{2},c,u)$ be a solution to (1.1). Under the assumption of Theorem 1.1, the solution of (1.1) satisfies the following properties:**

[TABLE]

Proof.

Firstly we can see from Lemmas 3.9, 3.10, 3.11 and [18] that there exist constants $C_{58}>0$ and $\alpha_{0}\in(0,1)$ such that

[TABLE]

for all $t\geq 1$. Now we set

[TABLE]

Then the function $f_{1}$ is nonnegative and uniformly continuous. We see from Lemma 4.1 that

[TABLE]

Thus the compactness method ([14, Lemma 4.6]) concludes the proof. ∎

4.2 Case 2: $a_{1}\geq 1>a_{2}$

In this section we assume that $\frac{\chi}{\mu}<\xi_{0}$. This section is devoted to obtaining stabilization in (1.1) in the case $a_{1}\geq 1>a_{2}$. We will give the following lemma for obtaining it. The proof is same as that of [14, Lemma 4.3].

Lemma 4.3**.**

Let $a_{1}\geq 1>a_{2}$ and let $(n_{1},n_{2},c,u)$ be a solution to (1.1). Under the assumption of Theorem 1.1, there exist $k_{2},\ell_{2}>0$ and $\varepsilon_{2}>0$ such that the nonnegative functions $E_{2}$ and $F_{2}$ defined by

[TABLE]

and

[TABLE]

satisfy

[TABLE]

By using a similar argument, Lemma 4.3 leads to stabilization of $n_{1},n_{2}$.

Lemma 4.4**.**

Let $a_{1}\geq 1>a_{2}$ and let $(n_{1},n_{2},c,u)$ be a solution to (1.1). Under the assumption of Theorem 1.1, it holds that

[TABLE]

Proof.

Noting from Lemma 4.3 that

[TABLE]

we can prove this lemma by the same argument as in the proof of Lemma 4.2. ∎

4.3 Convergence for $c$ and $u$

Finally we give the following lemma to establish the decay properties of $c$ and $u$. We first show the lower estimate for $n_{2}$.

Lemma 4.5**.**

Let $a_{2}\in(0,1)$. Under the assumption of Theorem 1.1, there exist constants $C_{59}>0$ and $T^{*}>0$ such that

[TABLE]

Proof.

We first deal with the case that $a_{1},a_{2}\in(0,1)$. Now, we assume that this lemma does not hold. Then there exist $\{x_{j}\}_{j\in\mathbb{N}}\subset\Omega$ and $\{t_{j}\}_{j\in\mathbb{N}}\subset[0,\infty)$ such that $t_{j}\to\infty$ as $j\to\infty$ satisfying

[TABLE]

Thus we have

[TABLE]

which means that $n_{2}(\cdot,t_{j})$ does not converge to $N_{2}$ as $j\to\infty$. However, Lemma 4.2 asserts that

[TABLE]

which contradicts (4.5). In the case that $a_{1}\geq 1>a_{2}$ a similar argument leads to the lower estimate for $n_{2}$. Therefore we can conclude the proof. ∎

Lemma 4.6**.**

Under the assumption of Theorem 1.1, the solution of (1.1) satisfies

[TABLE]

Proof.

Noting from Lemmas 4.1 and 4.3 that $\int_{T^{*}}^{\infty}\int_{\Omega}(\alpha n_{1}+\beta n_{2})c^{2}<\infty$ and using Lemma 4.5, we can establish that

[TABLE]

which entails $\|c(\cdot,t)\|_{L^{\infty}(\Omega)}\to 0$ as $t\to\infty$. Next we will show that $\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\to 0$ as $t\to\infty$. Let $\theta\in(\frac{3}{4},\vartheta)$ and $a=\frac{\theta}{\vartheta}\in(0,1)$. It follows from combination of [10, Theorem 3], [12, Theorem 1.6.1], [7, Part 2, Theorem 14.1] and Lemma 3.9 that there exist constants $C_{60},C_{61},C_{62}>0$ such that

[TABLE]

which means that it is sufficient to show that

[TABLE]

We first note from the Poincaré inequality that there exists a constant $C_{63}>0$ such that

[TABLE]

for all $s\in(0,\infty)$. Put $(n_{1,\infty},n_{2,\infty}):=(N_{1},N_{2})$ if $a_{1},a_{2}\in(0,1)$ or $(n_{1,\infty},n_{2,\infty}):=(0,1)$ if $a_{1}\geq 1>a_{2}$. We infer from the fourth equation in (1.1) and the Young inequality that

[TABLE]

for all $t\in(0,\infty)$ with some constant $C_{64}>0$. Since $\nabla\cdot u=0$ in $\Omega\times(0,\infty)$, the functions

[TABLE]

satisfy

[TABLE]

with some $C_{65}>0$. Hence it holds that

[TABLE]

Here we see from Lemma 3.11 that there exists a constant $C_{66}>0$ such that $h(s)\leq C_{66}$ for all $s>0$, and hence we have

[TABLE]

with some $C_{67}>0$. On the other hand, noting from (4.2) and (4.4) that $\int_{0}^{\infty}h(s)\,ds<\infty$, we can see that

[TABLE]

Therefore combination of (4.6) with (4.7) and (4.8) leads to

[TABLE]

which means the end of the proof. ∎

4.4 Proof of Theorem 1.2

Proof of Theorem 1.2**.**

Lemmas 4.2, 4.4, and 4.6 directly show Theorem 1.2. ∎