On a Two-Species Attraction–Repulsion Chemotaxis System with Nonlocal Terms

Abstract

This paper deals with a two-species attraction–repulsion chemotaxis system

$$\begin{aligned} \left\{ \begin{aligned}{}&u_t=d_{1}\Delta u-\xi _{1}\nabla \cdot (u\nabla v)+\chi _{1}\nabla \cdot (u\nabla z)+g_{1}(u,w),&(x,t)\in \Omega \times (0,\infty ), \\&\tau v_{t}=d_{2}\Delta v+w-v,&(x,t)\in \Omega \times (0,\infty ),\\&w_t=d_{3}\Delta w-\xi _{2}\nabla \cdot (w\nabla z)+\chi _{2}\nabla \cdot (w\nabla v)+g_{2}(u,w),&(x,t)\in \Omega \times (0,\infty ), \\&\tau z_{t}=d_{4}\Delta z+u-z,&(x,t)\in \Omega \times (0,\infty ) \end{aligned} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega \subset {\mathbb {R}}^{n}$ for $n\ge 1$, where $\tau \in \{0,1\}$, the parameters $d_{i}(i=1,2,3,4),\xi _{j},\chi _{j}(j=1,2)$ are positive and the kinetic terms $g_{1}(u,w),g_{2}(u,w)$ satisfy

$$\begin{aligned} \left\{ \begin{aligned}{}&g_{1}(u,w)=u\bigg (a_{0}-a_{1}u-a_{2}w-a_{3}\int _{\Omega }u{\text {d}}x-a_{4}\int _{\Omega }w{\text {d}}x\bigg ),\\&g_{2}(u,w)=w\bigg (b_{0}-b_{1}u-b_{2}w-b_{3}\int _{\Omega }u{\text {d}}x-b_{4}\int _{\Omega }w{\text {d}}x\bigg )\\ \end{aligned} \right. \end{aligned}$$

with $a_{0},a_{1},b_{0},b_{2}>0,a_{2},a_{3},a_{4},b_{1},b_{3},b_{4}\in {\mathbb {R}}$. It is shown that under some suitable parameter conditions, the above system possesses a unique global and uniformly bounded solution in any spatial dimension. Moreover, we investigate the asymptotic stability of solutions under the locally intraspecific competition and globally interspecific cooperation. Finally, we present some numerical simulations, which not only support our analytically theoretical results, but also find some new and interesting phenomena.

Appendix A. Proof of Lemma 2.1

Proof

The ideas of proof are similar to (Winkler 2010a, Lemma 1.1) and (Stinner et al. 2014b, Lemma 2.1). For reader’s convenience, we give the sketch of the proof.

(i) Existence. Under the assumptions of Lemma 2.1, we claim that for all $L>0$ there exists $T=T(L)>0$ such that $\Vert u_{0}\Vert _{L^{\infty }(\Omega )}\le L,\Vert w_{0}\Vert _{L^{\infty }(\Omega )}\le L,\Vert v_{0}\Vert _{W^{1,q}(\Omega )}\le L$ and $\Vert z_{0}\Vert _{W^{1,q}(\Omega )}\le L$, then system (1.1) is classically solvable in $\Omega \times (0,T)$. As a consequence of a standard extension argument, this will imply the existence of a maximal existence time $T_{\max }$ satisfying (2.1).

Now, we prove the local existence of solutions for system (1.1) when $\tau =1$ and $\tau =0$, respectively.

When $\tau =1$, according to the well-known Neumann heat semigroup $\big (e^{t\Delta }\big )_{t\ge 0}$ in (Winkler 2010b, Lemma 3.1), we can pick $K>0$ such that $\Vert e^{t\Delta }v\Vert _{W^{1,q}(\Omega )}\le K\Vert v\Vert _{W^{1,q}(\Omega )}$ and $\Vert e^{t\Delta }z\Vert _{W^{1,q}(\Omega )}\le K\Vert z\Vert _{W^{1,q}(\Omega )}$ for all $v,z\in W^{1,q}(\Omega )$. For small $T\in (0, 1)$ to be fixed below, we introduce the Banach space

$$\begin{aligned} X:= & {} C^{0}\bigg ([0,T];C^{0}({\overline{\Omega }})\bigg )\times C^{0}\bigg ([0,T];W^{1,q}(\Omega )\bigg )\times C^{0}\bigg ([0,T];C^{0}({\overline{\Omega }})\bigg )\\{} & {} \times C^{0}\bigg ([0,T];W^{1,q}(\Omega )\bigg ), \end{aligned}$$

and the close subset

$$\begin{aligned} \begin{aligned} F:=\big \{&(u,v,w,z)\in X\mid \Vert u\Vert _{L^{\infty }((0,T);L^{\infty }(\Omega ))}\le L+1,\Vert v\Vert _{L^{\infty }((0,T);W^{1,q}(\Omega ))}\le KL+1,\\&\Vert w\Vert _{L^{\infty }((0,T);L^{\infty }(\Omega ))}\le L+1,\Vert z\Vert _{L^{\infty }((0,T);W^{1,q}(\Omega ))}\le KL+1\big \}. \end{aligned} \end{aligned}$$

For $(u,v,w,z)\in F$ and $t\in (0,T)$, we define the mapping

$$\begin{aligned} \begin{gathered} \Psi _{1}(u,v,w,z)(t):=\left( \begin{array}{c} \Psi _{11}(u,v, w,z)(t) \\ \Psi _{12}(u,v,w,z)(t) \\ \Psi _{13}(u,v,w,z)(t) \\ \Psi _{14}(u,v,w,z)(t) \end{array}\right) :=\\ \left( \begin{array}{c} e^{td_{1}\Delta }u_{0}-\xi _{1}\int _{0}^{t} e^{(t-s)d_{1}\Delta }\nabla \cdot \big (u(s)\nabla v(s)\big ) ds+\chi _{1}\int _{0}^{t}e^{(t-s)d_{1}\Delta }\nabla \cdot \big (u(s)\nabla z(s)\big )ds+\int _{0}^{t} e^{(t-s)d_{1}\Delta }g_{1}(u,w)ds\\ e^{t(d_{2}\Delta -1)}v_{0}+\int _{0}^{t} e^{(t-s)(d_{2}\Delta -1)}w(s)ds\\ e^{td_{3}\Delta }w_{0}-\xi _{2}\int _{0}^{t} e^{(t-s)d_{3}\Delta }\nabla \cdot \big (w(s)\nabla z(s)\big ) ds+\chi _{2}\int _{0}^{t}e^{(t-s)d_{3}\Delta }\nabla \cdot \big (w(s)\nabla v(s)\big )ds+\int _{0}^{t} e^{(t-s)d_{3}\Delta }g_{2}(u,w)ds\\ e^{t(d_{4}\Delta -1)}z_{0}+\int _{0}^{t} e^{(t-s)(d_{4}\Delta -1)}u(s)ds \end{array}\right) . \end{gathered} \end{aligned}$$

Then, we have

$$\begin{aligned} \begin{aligned} \Vert \Psi _{11}(u,v,w,z)(t)\Vert _{L^{\infty }(\Omega )}\le&\Vert e^{t d_{1}\Delta } u_{0}\Vert _{L^{\infty }(\Omega )}+\xi _{1} \int _{0}^{t}\bigg \Vert e^{(t-s)d_{1}\Delta }\nabla \cdot \bigg (u(s)\nabla v(s)\bigg )\bigg \Vert _{L^{\infty }(\Omega )}ds\\&+\chi _{1} \int _{0}^{t}\bigg \Vert e^{(t-s)d_{1}\Delta }\nabla \cdot \bigg (u(s) \nabla z(s)\bigg )\bigg \Vert _{L^{\infty }(\Omega )}ds\\&+\int _{0}^{t}\bigg \Vert e^{(t-s)d_{1}\Delta }g_{1}(u,w)\bigg \Vert _{L^{\infty }(\Omega )}ds\\ \le&\xi _{1} \int _{0}^{t}\bigg \Vert e^{(t-s)d_{1}\Delta }\nabla \cdot \bigg (u(s) \nabla v(s)\bigg )\bigg \Vert _{L^{\infty }(\Omega )}ds\\&+\chi _{1} \int _{0}^{t}\bigg \Vert e^{(t-s)d_{1}\Delta }\nabla \cdot \bigg (u(s) \nabla z(s)\bigg )\bigg \Vert _{L^{\infty }(\Omega )}ds\\&+L+T\cdot \Vert g_{1}(u,w)\Vert _{L^{\infty }((-R-1,R+1))}, \end{aligned} \end{aligned}$$

(7.1)

where by the maximum principle

$$\begin{aligned} \begin{aligned} \Vert e^{td_{1}\Delta }u_{0}\Vert _{L^{\infty }(\Omega )} \le \Vert u_{0}\Vert _{L^{\infty }(\Omega )}\le L \end{aligned} \end{aligned}$$

(7.2)

and

$$\begin{aligned} \begin{aligned} \int _{0}^{t}\Vert e^{td_{1}\Delta }g_{1}(u,w)\Vert _{L^{\infty }(\Omega )}ds&\le \int _{0}^{t}\Vert g_{1}(u,w)\Vert _{L^{\infty }(\Omega )}ds\\&\le T\cdot \Vert g_{1}(u,w)\Vert _{L^{\infty }((-R-1,R+1))}, \end{aligned} \end{aligned}$$

(7.3)

for all $t\in (0,T)$. Furthermore, by picking any $p>\frac{nq}{q-n}$ and then $\alpha \in \big (\frac{n}{p}, \frac{1}{2}-\frac{n}{2}(\frac{1}{q}-\frac{1}{p})\big )$, we obtain $p\alpha >n$ and the fractional power $A^{\alpha }$ of the sectorial operator $A:=-d_{1}\Delta +1$ with Neumann data in $L^{p}(\Omega )$ satisfies $\Vert \phi \Vert _{L^{\infty }(\Omega )}\le C\Vert A^{\alpha } \phi \Vert _{L^{p}(\Omega )}$ as well as $\Vert A^{\alpha } e^{\rho d_{1}\Delta }\phi \Vert _{L^{p}(\Omega )}\le C \rho ^{-\alpha }\Vert \phi \Vert _{L^{p}(\Omega )}$ for all $\phi \in C^{\infty }_{0}(\Omega )$ (cf. Henry (1981)). Here and below, $C_{i}(i=1,2,\cdots ,23)$ denote generic positive constants. Therefore, by $T<1,\alpha \in \bigg (\frac{n}{p}, \frac{1}{2}-\frac{n}{2}(\frac{1}{q}-\frac{1}{p})\bigg )$ and $\Vert e^{\rho d_{1}\Delta }\nabla \cdot \psi \Vert _{L^{p}(\Omega )}\le C \rho ^{-\frac{1}{2}-\frac{n}{2}(\frac{1}{q}-\frac{1}{p})}\Vert \psi \Vert _{L^{q}(\Omega )}$ for $\rho <1$ and all ${\mathbb {R}}-$valued $\psi \in C^{\infty }_{0}(\Omega )$ (cf. Weinberger (1982)), we have

$$\begin{aligned} \begin{aligned}&\xi _{1} \int _{0}^{t}\bigg \Vert e^{(t-s)d_{1}\Delta }\nabla \cdot \bigg (u(s) \nabla v(s)\bigg )\bigg \Vert _{L^{\infty }(\Omega )}ds\\&\le C_{1}\int _{0}^{t}\bigg \Vert A^{\alpha }e^{(t-s)d_{1}\Delta }\nabla \cdot \bigg (u(s)\nabla v(s)\bigg )\bigg \Vert _{L^{p}(\Omega )}ds\\&\le C_{2}\int _{0}^{t}(t-s)^{-\alpha }\bigg \Vert e^{\frac{t-s}{2}d_{1}\Delta }\nabla \cdot \bigg (u(s)\nabla v(s)\bigg )\bigg \Vert _{L^{p}(\Omega )}ds\\&\le C_{3}\int _{0}^{t}(t-s)^{-\alpha } \cdot (t-s)^{-\frac{1}{2}-\frac{n}{2}(\frac{1}{q}-\frac{1}{p})}\bigg \Vert u(s)\nabla v(s)\bigg \Vert _{L^{q}(\Omega )}ds\\&\le C_{4}T^{\frac{1}{2}-\alpha -\frac{n}{2}(\frac{1}{q}-\frac{1}{p})}\cdot (L+1)\cdot (KL+1) \end{aligned} \end{aligned}$$

(7.4)

for all $t\in (0,T)$. Similarly, we obtain

$$\begin{aligned} \begin{aligned}&\chi _{1} \int _{0}^{t}\bigg \Vert e^{(t-s)d_{1}\Delta }\nabla \cdot \bigg (u(s) \nabla z(s)\bigg )\bigg \Vert _{L^{\infty }(\Omega )}ds\\&\quad \le C_{5}T^{\frac{1}{2}-\alpha -\frac{n}{2}(\frac{1}{q}-\frac{1}{p})}\cdot (L+1)\cdot (KL+1) \end{aligned} \end{aligned}$$

(7.5)

for all $t\in (0,T)$. For the term $\Vert \Psi _{13}(u,v,w,z)(t)\Vert _{L^{\infty }(\Omega )}$, we can use the similar way.

For $\Vert \Psi _{12}(u,v,w,z)(t)\Vert _{L^{\infty }(\Omega )}$, we have

$$\begin{aligned} \begin{aligned} \Vert \Psi _{12}(u,v,w,z)(t)\Vert _{W^{1,q}(\Omega )}&\le e^{-t}\Vert e^{td_{1}\Delta }v_{0}\Vert _{W^{1,q}(\Omega )}+C_{6}\int _{0}^{t}(t-s)^{-\frac{1}{2}}\Vert w(s)\Vert _{L^{q}(\Omega )}ds\\&\le K\Vert v_{0}\Vert _{W^{1,q}(\Omega )}+C_{7}\int _{0}^{t}(t-s)^{-\frac{1}{2}}\Vert w(s)\Vert _{L^{\infty }(\Omega )}ds\\&\le KL+C_{8}T^{\frac{1}{2}}\cdot (L+1) \end{aligned} \end{aligned}$$

(7.6)

for all $t\in (0,T)$. Similarly, we can estimate the term $\Vert \Psi _{14}(u,v,w,z)(t)\Vert _{L^{\infty }(\Omega )}$. Then, it follows from (7.1)–(7.6) that if we fix $T_{0} \in (0, 1)$ small enough such that $T\in (0, T_{0})$, then $\Psi _{1}$ maps F into itself.

Moreover, using the same ideas with (7.4), for $(u,v,w,z)\in F$ and $({\bar{u}},{\bar{v}},{\bar{w}},{\bar{z}})\in F$, we get

$$\begin{aligned}{} & {} \left\| \Psi _{11}(u, v, w, z)(t)-\Psi _{11}({\bar{u}}, {\bar{v}}, {\bar{w}}, {\bar{z}})(t)\right\| _{L^{\infty }(\Omega )}\nonumber \\{} & {} \le C_{9} \int _{0}^{t}\left\| A^{\alpha } e^{(t-l)d_{1}\Delta }\nabla \cdot \bigg (u(s)\nabla v(s)-{\bar{u}}(s)\nabla {\bar{v}}(s)\bigg )\right\| _{L^{p}(\Omega )}ds\nonumber \\{} & {} +C_{9}\int _{0}^{t}\left\| A^{\alpha } e^{(t-s)d_{1}\Delta }\nabla \cdot \bigg (u(s)\nabla z(s)-{\bar{u}}(s)\nabla {\bar{z}}(s)\bigg )\right\| _{L^{p}(\Omega )}ds\nonumber \\{} & {} +\int _{0}^{t}\left\| e^{(t-s) d_{1}\Delta }\bigg (g_{1}(u, w)-g_{1}({\bar{u}}, {\bar{w}})\bigg )\right\| _{L^{\infty }(\Omega )}ds\nonumber \\{} & {} \le C_{10} \int _{0}^{t}(t-s)^{-\alpha -\frac{1}{2}-\frac{n}{2}\left( \frac{1}{q}-\frac{1}{p}\right) }\Vert u(s)\nabla v(s)-{\bar{u}}(s)\nabla {\bar{v}}(s)\Vert _{L^{q}(\Omega )}ds\nonumber \\{} & {} +C_{10} \int _{0}^{t}(t-s)^{-\alpha -\frac{1}{2}-\frac{n}{2}\left( \frac{1}{q}-\frac{1}{p}\right) }\Vert u(s)\nabla z(s)-{\bar{u}}(s)\nabla {\bar{z}}(s)\Vert _{L^{q}(\Omega )}ds\nonumber \\{} & {} +\int _{0}^{t} \Vert g_{1}(u,w)-g_{1}({\bar{u}}, {\bar{w}})\Vert _{L^{\infty }(\Omega )}ds\nonumber \\{} & {} \le C_{11} T^{\frac{1}{2}-\alpha -\frac{n}{2}\left( \frac{1}{q}-\frac{1}{p}\right) }\bigg ((L+1)+(KL+1)\bigg )\cdot \Vert (u, v, w, z)-({\bar{u}}, {\bar{v}}, {\bar{w}}, {\bar{z}})\Vert _{X} \nonumber \\{} & {} +T\cdot \left\| g_{1}^{\prime }\right\| _{L^{\infty }((-L-1, L+1))}\cdot \Vert (u, v, w, z)-({\bar{u}}, {\bar{v}}, {\bar{w}}, {\bar{z}})\Vert _{X}. \end{aligned}$$

(7.7)

Similarly, we have

$$\begin{aligned} \begin{aligned}&\left\| \Psi _{13}(u, v, w, z)(t)-\Psi _{13}({\bar{u}}, {\bar{v}}, {\bar{w}}, {\bar{z}})(t)\right\| _{L^{\infty }(\Omega )}\\ \le&C_{12} T^{\frac{1}{2}-\alpha -\frac{n}{2}\left( \frac{1}{q}-\frac{1}{p}\right) }\bigg ((L+1)+(KL+1)\bigg )\cdot \Vert (u, v, w, z)-({\bar{u}}, {\bar{v}}, {\bar{w}}, {\bar{z}})\Vert _{X} \\&+T\cdot \left\| g_{2}^{\prime }\right\| _{L^{\infty }((-L-1, L+1))}\cdot \Vert (u, v, w, z)-({\bar{u}}, {\bar{v}}, {\bar{w}}, {\bar{z}})\Vert _{X} \end{aligned} \end{aligned}$$

(7.8)

and

$$\begin{aligned} \begin{aligned}&\left\| \Psi _{12}(u, v, w, z)(t)-\Psi _{12}({\bar{u}}, {\bar{v}}, {\bar{w}}, {\bar{z}})(t)\right\| _{L^{\infty }(\Omega )}\\&\le C_{13} \int _{0}^{t}(t-s)^{-\frac{1}{2}}\Vert w(s)-{\bar{w}}(s)\Vert _{L^{q}(\Omega )}ds \\&\le C_{14} T^{\frac{1}{2}}\cdot \Vert (u, v, w, z)-({\bar{u}},{\bar{v}},{\bar{w}},{\bar{z}})\Vert _{X} \end{aligned} \end{aligned}$$

(7.9)

as well as

$$\begin{aligned} \begin{aligned}&\left\| \Psi _{14}(u,v,w,z)(t)-\Psi _{14}({\bar{u}}, {\bar{v}}, {\bar{w}}, {\bar{z}})(t)\right\| _{L^{\infty }(\Omega )} \\ {}&\le C_{15} T^{\frac{1}{2}}\cdot \Vert (u, v, w, z)-({\bar{u}},{\bar{v}},{\bar{w}},{\bar{z}})\Vert _{X} \end{aligned} \end{aligned}$$

(7.10)

for all $t\in (0,T)$, which shows that $\Psi$ is a contraction mapping if $T\in (0, T_{0})$ is small enough. Then, by using the Banach fixed point theorem, we know that the existence of some $(u,v,w,z)\in F$ such that $\Psi _{1}(u,v,w,z)=(u,v,w,z)$. Once again using standard arguments involving semigroup estimates, it can easily be checked that in fact (u, v, w, z) lies in the asserted regularity class and is a classical solution of (1.1) in $\Omega \times (0,T)$. Since $g_{1}(0,0)\ge 0$ and $g_{2}(0,0)\ge 0$ hold, the maximum principle moreover ensures that u, w, v, z are nonnegative.

When $\tau =0$, we introduce the Banach space

$$\begin{aligned} {\bar{X}}:=C^{0}\big ([0,T];C^{0}({\overline{\Omega }})\big )\times C^{0}\big ([0,T];C^{0}({\overline{\Omega }})\big ), \end{aligned}$$

and consider the close subset

$$\begin{aligned} \begin{aligned} {\bar{F}}:=\bigg \{(u,w)\in X\mid \Vert u\Vert _{L^{\infty }((0,T);L^{\infty }(\Omega ))}\le L+1,\Vert w\Vert _{L^{\infty }((0,T);L^{\infty }(\Omega ))}\le L+1\bigg \}, \end{aligned} \end{aligned}$$

where $T\in (0,1)$ is small. Similarly, we define the mapping

$$\begin{aligned} \begin{gathered} \Psi _{2}(u,v,w,z)(t):=\left( \begin{array}{c} \Psi _{21}(u,v, w,z)(t) \\ \Psi _{22}(u,v,w,z)(t) \\ \end{array}\right) \\:=\left( \begin{array}{c} e^{td_{1}\Delta }u_{0}+\int _{0}^{t} e^{(t-s)d_{1}\Delta }\nabla \cdot \bigg (\chi _{1}u(s)\nabla z(s)-\xi _{1}u(s)\nabla v(s)\bigg )ds+\int _{0}^{t} e^{(t-s)d_{1}\Delta }g_{1}(u,w)ds\\ e^{td_{3}\Delta }w_{0}+\int _{0}^{t} e^{(t-s)d_{3}\Delta }\nabla \cdot \bigg (\chi _{2}w(s)\nabla v(s)-\xi _{2}w(l)\nabla z(s)\bigg )ds+\int _{0}^{t} e^{(t-s)d_{3}\Delta }g_{2}(u,w)ds \end{array}\right) . \end{gathered} \end{aligned}$$

for $(u,w)\in {\bar{F}}$ and $t\in (0,T)$, where $\big (e^{td_{i}\Delta }\big )_{t\ge 0}$ denotes the Neumann heat semigroup. From the second and fourth equation in (1.1), we have $-d_{1}\Delta v+v=w$ and $-d_{3}\Delta z+z=u$ under homogeneous Neumann boundary conditions. According to the same methods in case of $\tau =1$, we get that $\Psi _{2}$ is a contraction mapping on ${\bar{F}}$ if $T\in (0, T_{0})$ is sufficiently small. Hence, the Banach fixed point theorem implies the existence of some $(u,w)\in {\bar{F}}$ such that $\Psi _{2}(u,w)=(u,w)$. Moreover, by applying the similar arguments and the strong maximum principle, we deduce that (u, w) is nonnegative. And by the strong elliptic maximum principle applied to the second and fourth equation in (1.1), we also obtain the nonnegativity of (v, z).

(ii) Uniqueness. Proceeding as in Gajewski and Zacharias (1998), for given $T>0$ and two solutions $(u,v,w,z),({\bar{u}},{\bar{v}},{\bar{w}},{\bar{z}})$ in $\Omega \times (0,T)$, we fix $T_{1}\in (0,T)$ and set $U:=u-{\bar{u}},V:=v-{\bar{v}},W:=w-{\bar{w}},Z:=z-{\bar{z}}$. By applying straightforward testing procedures to (1.1), we have

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}}{{\text {d}}t}\int _{\Omega }U^{2}+2d_{1}\int _{\Omega }|\nabla U|^{2}&=2\xi _{1}\int _{\Omega }U\nabla v\cdot \nabla U+2\xi _{1}\int _{\Omega }{\bar{u}}\nabla U\cdot \nabla V\\&\quad -2\chi _{1}\int _{\Omega }U\nabla z\cdot \nabla U -2\chi _{1}\int _{\Omega }{\bar{u}}\nabla U\cdot \nabla Z \\&\quad +2\int _{\Omega }\bigg (g_{1}(u,w)-g_{1}({\bar{u}}, {\bar{w}})\bigg )U \end{aligned} \end{aligned}$$

(7.11)

and

$$\begin{aligned} \begin{aligned} \int _{\Omega }W^{2}+2d_{3}\int _{\Omega }|\nabla W|^{2}&=2\xi _{2}\int _{\Omega }W\nabla z\cdot \nabla W+2\xi _{2}\int _{\Omega }{\bar{w}}\nabla W\cdot \nabla Z\\&\quad -2\chi _{2}\int _{\Omega }W\nabla v\cdot \nabla W -2\chi _{2}\int _{\Omega }{\bar{w}}\nabla W\cdot \nabla V \\&\quad +2\int _{\Omega }\bigg (g_{2}(u,w)-g_{2}({\bar{u}},{\bar{w}})\bigg )W \end{aligned} \end{aligned}$$

(7.12)

for all $t\in (0,T_{1})$.

When $\tau =1$, by the second and fourth equations in (1.1), we obtain

$$\begin{aligned} \frac{{\text {d}}}{{\text {d}}t}\int _{\Omega }|\nabla V|^{2}+2d_{2}\int _{\Omega }|\Delta V|^{2}+2\int _{\Omega }|\nabla V|^{2}=-2\int _{\Omega }W\Delta V \end{aligned}$$

(7.13)

and

$$\begin{aligned} \frac{{\text {d}}}{{\text {d}}t}\int _{\Omega }|\nabla Z|^{2}+2d_{4}\int _{\Omega }|\Delta Z|^{2}+2\int _{\Omega }|\nabla Z|^{2}=-2\int _{\Omega }U\Delta Z \end{aligned}$$

(7.14)

for all $t\in (0,T_{1})$. By the Hölder, Young and Gagliardo–Nirenberg inequalities, we get

$$\begin{aligned} \begin{aligned} 2\xi _{1}\int _{\Omega }U\nabla v\cdot \nabla U&\le 2\xi _{1}\bigg (\int _{\Omega }|\nabla U|^{2}\bigg )^{\frac{1}{2}}\cdot \bigg (\int _{\Omega }|\nabla v|^{q}\bigg )^{\frac{1}{q}}\cdot \bigg (\int _{\Omega } U^{\frac{2 q}{q-2}}\bigg )^{\frac{q-2}{2 q}}\\&\le C_{22}\bigg (\int _{\Omega }|\nabla U|^{2}\bigg )^{\frac{1}{2}+\frac{n}{2 q}}\cdot \bigg (\int _{\Omega }|\nabla v|^{q}\bigg )^{\frac{1}{q}}\cdot \bigg (\int _{\Omega } U^{2}\bigg )^{\frac{q-n}{2q}}\\&\le \frac{d_{1}}{2}\int _{\Omega }|\nabla U|^{2}+C_{16}\int _{\Omega } U^{2}, \end{aligned} \end{aligned}$$

(7.15)

where we have used the fact that $\int _{\Omega }U=0$ by a simple integration of (1.1), and $\Vert \nabla v\Vert _{L^{q}(\Omega )}\le C_{17}$ for $t\in (0,T_{1})$ as well as $q>n\ge 2$. By using the same method with (7.15), we have

$$\begin{aligned} \begin{aligned} -2\chi _{1}\int _{\Omega }U\nabla z\cdot \nabla U\le \frac{d_{1}}{2}\int _{\Omega }|\nabla U|^{2}+C_{18}\int _{\Omega }U^{2}. \end{aligned} \end{aligned}$$

(7.16)

Furthermore, we have

$$\begin{aligned} \begin{aligned} 2\xi _{1}\int _{\Omega }{\bar{u}}\nabla U\cdot \nabla V\le \frac{d_{1}}{2}\int _{\Omega }|\nabla U|^{2}+C_{17}\int _{\Omega }|\nabla V|^{2} \end{aligned} \end{aligned}$$

(7.17)

and

$$\begin{aligned} \begin{aligned} -2\chi _{1}\int _{\Omega }{\bar{u}}\nabla U\cdot \nabla Z\le \frac{d_{1}}{2}\int _{\Omega }|\nabla U|^{2}+C_{19}\int _{\Omega }|\nabla Z|^{2} \end{aligned} \end{aligned}$$

(7.18)

as well as

$$\begin{aligned} \begin{aligned} 2\int _{\Omega }\bigg (g_{1}(u,w)-g_{1}({\bar{u}}, {\bar{w}})\bigg )U\le C_{20}\int _{\Omega }U^{2}, \end{aligned} \end{aligned}$$

(7.19)

in view of the boundedness of u and ${\bar{u}}$ in $\Omega \times (0,T_{1})$ and the local Lipschitz continuity of $g_{1}$. Then, by substituting (7.15)–(7.19) into (7.11), we derive

$$\begin{aligned} \frac{{\text {d}}}{{\text {d}}t}\int _{\Omega }U^{2}\le (C_{16}+C_{18}+C_{20})\int _{\Omega }U^{2}+C_{17}\int _{\Omega }|\nabla V|^{2}+C_{19}\int _{\Omega }|\nabla Z|^{2}. \end{aligned}$$

(7.20)

By using the same method to (7.12), we have

$$\begin{aligned} \frac{{\text {d}}}{{\text {d}}t}\int _{\Omega }W^{2}\le C_{21}\int _{\Omega }W^{2}+C_{21}\int _{\Omega }|\nabla V|^{2}+C_{21}\int _{\Omega }|\nabla Z|^{2}. \end{aligned}$$

(7.21)

By Young’s inequality, we obtain from (7.13) and (7.14) that

$$\begin{aligned} \frac{{\text {d}}}{{\text {d}}t}\int _{\Omega }|\nabla V|^{2}+2\int _{\Omega }|\nabla V|^{2}\le \frac{1}{2d_{2}}\int _{\Omega }W^{2} \end{aligned}$$

(7.22)

and

$$\begin{aligned} \frac{{\text {d}}}{{\text {d}}t}\int _{\Omega }|\nabla Z|^{2}+2\int _{\Omega }|\nabla Z|^{2}\le \frac{1}{2d_{4}}\int _{\Omega }U^{2}. \end{aligned}$$

(7.23)

By combining (7.20)–(7.23), one can find a positive constant $C_{22}$ such that

$$\begin{aligned} \begin{aligned}&\frac{{\text {d}}}{{\text {d}}t}\bigg (\int _{\Omega }U^{2}+\int _{\Omega }W^{2}+\int _{\Omega }|\nabla V|^{2}+\int _{\Omega }|\nabla Z|^{2}\bigg ) \\&\le C_{22}\bigg (\int _{\Omega }U^{2}+\int _{\Omega } W^{2}+\int _{\Omega }|\nabla V|^{2}+\int _{\Omega }|\nabla Z|^{2}\bigg ). \end{aligned} \end{aligned}$$

(7.24)

When $\tau =0$, by a straightforward computation, we deduce

$$\begin{aligned} \frac{{\text {d}}}{{\text {d}}t}\int _{\Omega }U^{2}\le C_{23}\bigg (\int _{\Omega } U^{2}+\int _{\Omega }W^{2}+\int _{\Omega }|\nabla V|^{2}+\int _{\Omega }|\nabla Z|^{2}\bigg ) \end{aligned}$$

(7.25)

and

$$\begin{aligned} \frac{{\text {d}}}{{\text {d}}t}\int _{\Omega }W^{2}\le C_{23}\bigg (\int _{\Omega } U^{2}+\int _{\Omega }W^{2}+\int _{\Omega }|\nabla V|^{2}+\int _{\Omega }|\nabla Z|^{2}\bigg ) \end{aligned}$$

(7.26)

as well as

$$\begin{aligned} \int _{\Omega }|\nabla V|^{2}\le \int _{\Omega }W^{2} \end{aligned}$$

(7.27)

and

$$\begin{aligned} \int _{\Omega }|\nabla Z|^{2}\le \int _{\Omega }U^{2}. \end{aligned}$$

(7.28)

Then, by combining (7.25)–(7.28), we have

$$\begin{aligned} \begin{aligned} \frac{{\text {d}}}{{\text {d}}t}\bigg (\int _{\Omega }U^{2}+\int _{\Omega }W^{2}\bigg )\le C_{23}\bigg (\int _{\Omega }U^{2}+\int _{\Omega }W^{2}\bigg ). \end{aligned} \end{aligned}$$

(7.29)

Now with the aid of Grönwall’s lemma, we obtain that $U\equiv 0,V\equiv 0,W\equiv 0,Z\equiv 0$ in $\Omega \times (0,T_{1})$. Hence, we obtain $(u,v,w,z)\equiv ({\bar{u}},{\bar{v}},{\bar{w}},{\bar{z}})$ in $\Omega \times (0,T)$, because $T_{1}\in (0,T)$ is arbitrary. The proof of Lemma 2.1 is complete. $\square$