1 Introduction and main results

Chemotaxis is one of the most important components in the process of reproduction and migration, which describes the partial movement of biological species or cells to the gradient of chemotactic substances. The classic chemotactic model was proposed by Keller and Segel [10] as the following

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}=\nabla \cdot (D(u) \nabla u-S(u) \nabla v)+f(u), & x \in \Omega , \quad t>0, \\ v_{t}=\Delta v-v+ g(u), & x \in \Omega , \quad t>0, \end{array}\right. } \end{aligned}$$
(1.1)

where \(\Omega \subset {\mathbb {R}}^{n}\). In the system (1.1), u(x, t) and v(x, t) represent the density of the population and the concentration of the chemical substance at space x and time t, respectively. D(u) and S(u) are the density-dependent diffusion function and the density-dependent sensitivity function, respectively. The function f(u) is the logistic source and g(u) is the production or consumption of chemical substances. When \(f(u)=g(u)=0\), the asymptotic property of \(\frac{S(u)}{D(u)} \simeq u^{\frac{2}{n}}\) is the critical condition for blow-up and global boundedness (see [27, 40]). Further studies even have provided further information in which the respective blow-up phenomenon either can occur within finite time (see [4,5,6]), or only arises in the sense of an infinite-time grow-up (see [4, 6, 41]). Moreover, some rigorous results of (1.1) are shown that the sub-logistic source f(u) can prevent the blow-up of solutions (see [39, 42]). In addition, more fruitful results have been obtained for the classical chemotaxis model (1.1) and its variants forms, we refer [3, 8, 11, 22, 28, 37] for further reading.

One of the well-known variants form of (1.1) is the two-species and one-stimuli type, that is, two species respond to the same chemical signaling substance produced by themselves. To describe the movement of two species, the following chemotaxis system

$$\begin{aligned} \left\{ \begin{array}{lll} u_t=\nabla \cdot \left( D_1(u) \nabla u-S_1(u) \nabla w\right) +f_{1}(u,v), & \quad x \in \Omega , & \quad t>0, \\ v_t=\nabla \cdot \left( D_2(v) \nabla v-S_2(v) \nabla w\right) +f_{2}(u,v), & \quad x \in \Omega , & \quad t>0, \\ \tau w_t=\Delta w-w+g(u, v), & \quad x \in \Omega , & \quad t>0, \end{array}\right. \end{aligned}$$
(1.2)

was proposed by Tello and Winkler in [29], where \(\tau \in \left\{ 0, 1\right\} \). In the system (1.2), u(x, t) and v(x, t) represent the density of different species respectively, and w(x, t) denotes the concentrations of chemical substances. The functions \(f_{i} (i=1,2)\) contain the logistic source and interaction between species, and g(u, v) is the production or consumption of chemical substances. So far, most of the conclusions are focused on the Lotka–Volterra case that \(f_{1}(u,v)=\mu _{1}u(1-u-a_{1}v)\), \(f_{2}(u,v)=\mu _{2}v(1-v-a_{2}u)\) and \(g(u,v)=u+v\). Specifically, when the linear case \(D_{i}(s)=1\) and \(S_{i}(s)=s\) for \(i=1,2\), in the case of \(\tau =0\), Tello and Winkler obtained the global existence and asymptotic behavior of solutions when \(a_1, a_2 \in [0,1)\) (see [29]). When \(a_1>1>a_2 \ge 0\), Stinner et al. proved that the semi-trivial steady state is asymptotically stable (see [25]). In case \(\tau =1\), Bai and Winkler derived the global existence of classical solutions when \(n\le 2\) and asymptotical behavior when the damping terms are suitably strong (i.e., \(\mu _{1}\) and \(\mu _{2}\) are large enough) in [2]; the bounded and asymptotic results are optimized in [16]. Moreover, in the quasilinear case, when \(f_{i}=0\), it has been proved in [31] that if \(D_i(s)\) and \(S_i(s)\) satisfy \(K_{0 i}(s+1)^{l_i-1} \le D_i(s) \le K_{1 i}(s+1)^{L_i-1}\) and \(\frac{S_i(s)}{D_i(s)} \le K_i(s+1)^{\alpha _i},(i=1,2)\), then the solutions are globally bounded under the conditions that \(0<\alpha _i<\frac{2}{n}\); and the finite-time blow-up of solution was also obtained. When the functions \(D_{i}, S_{i}, f_{i} (i=1,2)\) and g satisfy some conditions, Pan and Wang proved that this system possesses a global bounded smooth solution under some specific conditions with or without the logistic functions \(f_i(s)\) and further obtain the asymptotic stability for the solutions of system (1.2) (see [21]). More related interesting results can be found in [17, 18, 20, 46]. Furthermore, for results of the system with two-species and two signaling substances, we can refer to [32, 43, 44].

Recently, inspired by the work [42], Li found several explicit conditions involving the kinetic functions f, g, the parameters \(\chi , \lambda \) and the initial mass \(\left\| u_{0}\right\| _{L^{1}(\Omega )}\) to ensure the global-in-time existence and uniform boundedness for a chemotaxis model with indirect signal production and general kinetic function (see [12]). Subsequently, Shan and Zheng applied this idea to a chemotactic model describing the immune system, and obtained some global boundedness results (see [24, 47]).

Motivated by the above works, this paper is concerned with the following two-species chemotaxis system with generalized volume-filling effect and general kinetic functions

$$\begin{aligned} \left\{ \begin{array}{ll} u_t=\nabla \cdot (D_{1}(u)\nabla u)- \nabla \cdot ( \chi _{1}(u)\nabla w) + f_{1}(u)-\mu _{1}a_{1}uv, & \quad (x,t)\in \Omega \times (0,\infty ), \\ v_t=\nabla \cdot (D_{2}(v)\nabla v)- \nabla \cdot ( \chi _{2}(v)\nabla w) + f_{2}(v)-\mu _{2}a_{2}uv, & \quad (x,t)\in \Omega \times (0,\infty ), \\ \tau w_t=\Delta w - w + g_{1}(u) + g_{2}(v), & \quad (x,t)\in \Omega \times (0,\infty ),\\ \dfrac{\partial u}{\partial \nu }=\dfrac{\partial v}{\partial \nu }=\dfrac{\partial w}{\partial \nu } =0, & \quad (x,t)\in \partial \Omega \times (0,\infty ), \\ u(x,0)=u_{0}(x), v(x,0)=v_{0}(x), \tau w(x,0)=w_{0}(x),& \quad x\in \Omega , \end{array} \right. \end{aligned}$$
(1.3)

where \(\tau \in \left\{ 0, 1\right\} \), \(\Omega \subset {\mathbb {R}}^{n}\) \((n\ge 1)\) is a smoothly bounded domain and u, v, w have the same meanings as in (1.2). The parameters \(\mu _{i}, a_{i}>0 (i=1,2)\) and nonnegative initial data satisfies

$$\begin{aligned} u_0 \in C^0({\bar{\Omega }}), v_0 \in C^0({\bar{\Omega }}), \tau w_0 \in C^1({\bar{\Omega }}). \end{aligned}$$
(1.4)

Firstly, we consider the global boundedness of solutions for (1.3) under the following hypotheses with \(i=1,2\),

$$\begin{aligned}&(S_{1})\;D_{i}(s) \ge (s+1)^{-\alpha _{i}} \quad \text{ with } \alpha _{i}\in {\mathbb {R}};&\\&(S_{2})\;\chi _{i}(s) \le C_{\chi _{i}}s(s+1)^{\beta _{i}-1} \quad \text {with }C_{\chi _{i}}\ge 0, \beta _{i}\in {\mathbb {R}};&\\&(S_{3})\;f_{i}(0)\ge 0, \chi _{i}(0)=g_{i}(0)=0\text { and } g_{i}(s), \chi _{i}(s)>0 \text{ for } \text{ all } s>0;\\&(S_{4})\;g_{i}(s)\le s^{\gamma _{i}} \quad \text{ with } \gamma _{i}>0;\\&(S_{5})\;D_{i},\chi _{i}, f_{i}, g_{i} \text { belong to } C^{2}. \end{aligned}$$

Based on \((S_{1})-(S_{5})\), our main results of global existence and boundedness are stated as follows.

Theorem 1.1

Let \(\tau =1\), \(\Omega \subset {\mathbb {R}}^{n} (n\ge 1)\) be a bounded domain with smooth boundary. Assume \((S_{1})-(S_{5})\) hold. For \(i=1,2\), if one of the following conditions hold:

$$\begin{aligned} \begin{aligned}&\mathrm{(i)}\;\;f_{i} \equiv 0 \text { and } \alpha _i<\min \left\{ \frac{2}{n}, 1+\frac{2}{n}-\gamma _{i} -\beta _i\right\} ;\\&{\mathrm{(ii)} \;\; \exists \iota _{i}^\star =\iota _{i}^\star (\beta _{1}, \beta _{2}, \gamma _{1}, \gamma _{2}, C_{\chi _{1}}, C_{\chi _{2}})>0 \text { such that }\lim _{s \rightarrow \infty } \inf \left\{ -\frac{f_{i}(s)}{s^{\max \left\{ 1, \beta _{i}+\gamma _{i}\right\} }}\right\} =: \iota _{i} \in \left( \iota _{i}^{\star }, \infty \right] ,} \end{aligned} \end{aligned}$$

then system (1.3) possesses a classical solution (u, v, w), which is uniformly bounded in time.

Remark 1.1

Our results extend the results in [7, 9, 13, 26, 47] from one species to two species. Specifically, when system (1.3) meets linear diffusion and chemosensitivity (i.e., \(\alpha _{i}=0\) and \(\beta _{i}=1\) for \(i=1,2\)), the condition (i) of Theorem 1.1 indicates that suitable sublinear signal production (\(\gamma _{i}<\frac{2}{n}\) with \(n\ge 2\) and \(i=1,2\)) can ensure the uniform boundedness, which is similar to the results for simpler (single-species) Keller-Segel system (see Theorem 1.1 in [13]). When system (1.3) meets linear diffusion and signal production (i.e., \(\alpha _{i}=0\) and \(\gamma _{i}=1\) for \(i=1,2\)), the condition (i) indicates that suitable sublinear chemoattractant (\(\beta _{i}<\frac{2}{n}\) with \(n\ge 2\) and \(i=1,2\)) can ensure the uniform boundedness (see Theorem 4.1 in [9]). Furthermore, when \(f_{i}(s)=S_i-\mu _{i} s^{k_{i}}\) with \(S_i, \mu _{i}, k_{i}>0\) for \(i=1,2\), the condition (ii) of Theorem 1.1 indicates that if \(\beta _{i}+\gamma _{i}<k_{i}\) or \(\beta _{i}+\gamma _{i}=k_{i}\) with \(\mu _{i}\) large enough for \(i=1,2\), then problem (1.3) possesses a unique global classical solution that is bounded in \(\Omega \times (0, \infty )\) (see Theorem 1.2 in [26] or Theorem 1.1 in [7]). Moreover, when system (1.3) meets linear chemosensitivity and signal production, the condition (ii) implies that the superquadratic degradation mechanisms or quadratic degradation mechanisms with sufficiently large damping coefficients of species can ensure the uniform boundedness of the solution, while the subquadratic case is still a problem that needs to be developed.

In addition, we get rid of the constraint of \(\gamma _{1}=\gamma _{2}\le 1\) relative to [21], which relies on the Sobolev regularity estimate (Lemma 3.1) and condition \(\alpha _{i}<\frac{2}{n}\) for \(i=1,2\).

Theorem 1.2

Let \(\tau =0\), \(\Omega \subset {\mathbb {R}}^{n} (n\ge 1)\) be a bounded domain with smooth boundary, \(\gamma =\max \left\{ \gamma _{1},\gamma _{2}\right\} \). Assume \((S_{1})-(S_{5})\) hold. For \(i=1,2\), if one of the following conditions hold:

$$\begin{aligned} \begin{aligned}&\mathrm{(i)}\;\;f_{i} \equiv 0 \text { and } \alpha _i<1+\frac{2}{n}-\gamma -\beta _i;\\&{\mathrm{(ii)}\;\; \exists \tilde{\iota _{i}}=\tilde{\iota _{i}}(\beta _{1}, \beta _{2}, \gamma _{1}, \gamma _{2}, C_{\chi _{1}}, C_{\chi _{2}})>0 \text { such that } \lim _{s \rightarrow \infty } \inf \left\{ -\frac{f_{i}(s)}{s^{ \beta _{i}+\gamma }}\right\} =: \iota _{i} \in \left( \tilde{\iota _{i}}, \infty \right] ,} \end{aligned} \end{aligned}$$

then system (1.3) possesses a classical solution (u, v, w), which is uniformly bounded in time.

Remark 1.2

Compared with Theorem 1.1, the conditions of Theorem 1.2 have been relatively simplified, thanks to the elliptic properties of the third equation in (1.3). Moreover, Our results extend the results from one species to two species. Specifically, when system (1.3) meets linear diffusion and chemosensitivity, the condition (i) of Theorem 1.2 indicates that suitable sublinear signal production (\(\gamma <\frac{2}{n}\) with \(n\ge 2\) and \(i=1,2\)) can ensure the uniform boundedness, which is similar to the results for single-species Keller-Segel system (see Proposition 1.3 in [36] and Theorem 1.1 in [33]). When \(f_{i}(s)=S_i-\mu _{i} s^{k_{i}}\) with \(S_i, \mu _{i}, k_{i}>0\) for \(i=1,2\), the condition (ii) of Theorem 1.2 indicates that when \(\beta _{i}+\gamma _{i}<k_{i}\) or \(\beta _{i}+\gamma _{i}=k_{i}\) with \(\mu _{i}\) large enough for \(i=1,2\), problem (1.3) possesses a uniformly bounded solution (see [35, 45]). In addition, our results highlight the independence of \(\beta _{i}\) and \(\alpha _{i}\) for \(i=1,2\) respectively, compared with [23].

After the globally bounded solutions obtained, we next consider the large time behavior of solutions to the system (1.3). First, we taking into account the effects of Lotka–Volterra competitive kinetic functional response term and linear signal generations, i.e., \(f_{i}, g_{i}\) \((i=1,2)\) satisfy

$$\begin{aligned}&(S_{6})\;f_{i}(s)=\mu _{i}s(1-s) \text { and } g_{i}(s)=s \text { for all } s>0.&\end{aligned}$$

Then, it is easy to obtain that the system (1.3) has four possible constant steady states \(\left( u_\star , v_\star , w_\star \right) \):

$$\begin{aligned} {\left\{ \begin{array}{ll}P_1 \text{ or } P_2 \text{ or } P_3 \text{ or } P_*, & \text{ if } a_1, a_2<1 \text{ or } a_1, a_2>1, \\ P_1 \text{ or } P_2 \text{ or } P_3, & \text{ other } \text{ cases, } \end{array}\right. } \end{aligned}$$

where \(P_1\) is the extinction state, \(P_2\) and \(P_3\) are two semi trivial steady states, and \(P_*\) is the coexistence steady states with

$$\begin{aligned} \left\{ \begin{array}{l} P_1:=(0,0,0), \quad P_2:=(1,0,1), \quad P_3:=(0,1,1) \\ P_\star =\left( u_\star , v_\star , w_\star \right) :=\left( \frac{1-a_1}{1-a_1 a_2}, \frac{1-a_2}{1-a_1 a_2}, \frac{2-a_1-a_2}{1-a_1 a_2}\right) . \end{array}\right. \end{aligned}$$

In view of the existing works [2, 14,15,16], the weakly competitive case \(\left( a_1, a_2 \in [0,1)\right) \) and the strongly-weakly competitive case \(\left( a_1 \ge 1>a_2 \ge 0\right) \) have been concerned. Moreover, the case of strong competition \(\left( a_1, a_2 \ge 1\right) \) have been partly given in [20]. Along with the previous work, we further consider weakly competitive and the strongly-weakly competitive cases, and give a more concise result to ensure the asymptotic stability of the solutions in the case of \(\tau =1\) or \(\tau =0\).

Moreover, it follows from Theorems 1.1 and 1.2 that we can find positive constants \(k_1\) and \(k_2\) satisfying

$$\begin{aligned} k_1=\max _{0 \le u \le \Vert u\Vert _{L^{\infty }(\Omega )}}(u+1)^{2 \beta _1+\alpha _1-2}, k_2=\max _{0 \le v \le \Vert v\Vert _{L^{\infty }(\Omega )}}(v+1)^{2 \beta _2+\alpha _2-2}. \end{aligned}$$
(1.5)

Then, we can get the following results about large time behavior of solutions for (1.3).

Theorem 1.3

(Strongly-weakly competitive case) Let \(\tau =\left\{ 0, 1\right\} \) and the assumption \((S_{6})\) hold. Assume that \(a_1\ge 1>a_2>0\). If

$$\begin{aligned} \mu _{2}>\frac{C_{\chi _{2}}^{2}}{8}k_{2}, \end{aligned}$$
(1.6)

then the unique global bounded solution (u, v, w) of (1.3) obtained by Theorem 1.1 or 1.2 satisfies

$$\begin{aligned} \Vert u(\cdot , t)\Vert _{L^{\infty }(\Omega )}+\Vert v(\cdot , t)-1\Vert _{L^{\infty }(\Omega )}+\Vert w(\cdot , t)-1\Vert _{L^{\infty }(\Omega )} \rightarrow 0 \quad \text{ as } t \rightarrow \infty . \end{aligned}$$

Theorem 1.4

(Weakly competitive case) Let \(\tau =\left\{ 0,1\right\} \) and the assumption \((S_{6})\) hold. Assume that \(0<a_1, a_2<1\). If

$$\begin{aligned} \mu _{2}>\frac{C_{\chi _{1}}^{2}u_{\star }\mu _{2}(2-a_{2})}{8a_{1}\mu _{1}}k_{1}+\frac{C_{\chi _{2}}^{2}v_{\star }}{8}k_{2}, \end{aligned}$$
(1.7)

then the unique global bounded solution (u, v, w) of (1.3) obtained by Theorem 1.1 or 1.2 satisfies

$$\begin{aligned} \Vert u(\cdot , t)-u_{\star }\Vert _{L^{\infty }(\Omega )}+\Vert v(\cdot , t)-v_\star \Vert _{L^{\infty }(\Omega )}+\Vert w(\cdot , t)-w_\star \Vert _{L^{\infty }(\Omega )} \rightarrow 0 \quad \text{ as } t \rightarrow \infty . \end{aligned}$$

Remark 1.3

Our results in Theorems 1.3 and 1.4 are more concise than the existing results in [2, 14,15,16], and the condition of Theorem 1.3 is free from the influence of parameters \(a_{1}\) and \(a_{2}\). Specifically, when \(a_1\ge 1>a_2>0\), the conditions in [2] are

$$\begin{aligned} \mu _2>\frac{C_{\chi _2}^2 v_{\star }\left( a_1^{\prime } +a_2 -2 a_1^{\prime } a_2 \right) }{16 a_2 \left( 1-a_1^{\prime } a_2\right) }. \end{aligned}$$
(1.8)

for some \(a_1^{\prime } \in \left( 1, a_1\right] \) such that \(a_1^{\prime } a_2<1\). It is not hard to get, \(\frac{\left( a_1^{\prime } +a_2 -2 a_1^{\prime } a_2 \right) }{2a_2 \left( 1-a_1^{\prime } a_2\right) }>1\) when \(a_{2}>\frac{1}{2}\) for all \(a_1^{\prime }>1\). Therefore, Our results in Theorem 1.3 partly improved the existing work in [2]. For the condition (1.7) in Theorem 1.4, it can be rewritten to \(\mu _{2}>\frac{C_{\chi _{2}}^{2}v_{\star }}{8d_{\star }}k_{2}\), where \(d_{\star }:=1-\frac{C_{\chi _{1}}^{2}u_{\star }(2-a_{2})}{8a_{1}\mu _{1}}k_{1}>0\) when \(\mu _{1}>\frac{C_{\chi _{1}}^{2}u_{\star }(2-a_{2})}{8a_{1}}k_{1}\), which implies that if \(\mu _{1}\) is sufficiently large, then the condition of \(\mu _{2}\) can be relaxed accordingly.

The rest of the article is organized as follows. In Sect. 2, we give some preliminary lemmas and the local existence of solution for system (1.3). In Sects. 3 and 4, we study the global existence and boundedness of solutions for system (1.3), and prove Theorems 1.1 and 1.2. In Sect. 5, we study the asymptotic behavior of global solutions for system (1.3), and prove Theorems 1.3 and 1.4. In the following content, we let \(u(\cdot ,t)=u(x,t)\) and shall use \(K_{i}, C_{i}(i=1,2,\ldots )\) to denote a generic positive constant which may vary in the context. Without confusion, the integration sign dx and dxdt will be omitted.

2 Local existence and preliminaries

The following local existence for solutions of the system (1.3) can be obtained by adapting established techniques. The details can be found in [1, 2, 9, 25, 46] and can therefore be omitted here.

Lemma 2.1

Let \(\tau \in \left\{ 0,1\right\} \) and \(\Omega \subset {\mathbb {R}}^{n}(n\ge 1)\) be a smoothly bounded domain. Assume that the functions \(D_{i},\chi _{i},f_{i}\) and \(g_{i}\) (\(i=1,2\)) satisfy \((S_{3})\) and \((S_{5})\), as well as the nonnegative initial data satisfies (1.4. Then there exist \(T_{\max }\in (0,\infty ]\) and uniquely determined nonnegative functions

$$\begin{aligned} \left\{ \begin{array}{l} u, v \in C^0\left( {\bar{\Omega }} \times \left[ 0, T_{\max }\right) \right) \cap C^{2,1}\left( {\bar{\Omega }} \times \left( 0, T_{\max }\right) \right) , \\ w \in C^0\left( {\bar{\Omega }} \times \left[ 0, T_{\max }\right) \right) \cap C^{2,1}\left( {\bar{\Omega }} \times \left( 0, T_{\max }\right) \right) \cap L_{\text{ loc } }^{\infty }\left( \left[ 0, T_{\max }\right) ; W^{1, q}(\Omega )\right) \end{array}\right. \end{aligned}$$

such that (u, v, w) solves problem (1.3)classically in \(\Omega \times \left( 0, T_{\max }\right) \), where \(q>n\) and \(T_{\max }\) is the maximal existence time. Moreover, if \(T_{\max }<\infty \), then

$$\begin{aligned} \sup \left\{ ||u(\cdot ,t)||_{L^{\infty }(\Omega )}+||v(\cdot ,t)||_{L^{\infty }(\Omega )} +||w(\cdot ,t)||_{W^{1,\infty }(\Omega )}\right\} \rightarrow \infty ,\;\;\;\text {as}\, t\nearrow T_{\max }. \end{aligned}$$
(2.1)

Lemma 2.2

(see [19]) Let \(\Omega \subset {\mathbb {R}}^{n}\), \(n\ge 1\) be a bounded domain with smooth boundary, and let \(p\geqslant 1\), \(q\in (0,p)\). Then there exists a constant \(C_{GN}>0\) such that

$$\begin{aligned} ||u||_{L^{p}(\Omega )}\le C_{GN}(||\nabla u||_{L^{2}(\Omega )}^{\delta }||u||_{L^{q}(\Omega )}^{1-\delta }+||u||_{L^{q}(\Omega )}), \end{aligned}$$

where \(\delta =\frac{\frac{n}{q}-\frac{n}{p}}{1-\frac{n}{2}+\frac{n}{q}}\in (0,1)\).

Lemma 2.3

Let the conditions in Lemma 2.1 hold and \(f_{i}=0 (i=1,2)\), then there exists \(K_{1} > 0\) such that the solution components u, v of (1.3) satisfy

$$\begin{aligned} ||u(\cdot ,t)||_{L^{1}(\Omega )} + ||v(\cdot ,t)||_{L^{1}(\Omega )}\le K_{1} \quad \text{ for } \text{ all } t \in \left( 0, T_{\max }\right) . \end{aligned}$$
(2.2)

Proof

Integrating the first two equations in (1.3) with respect to \(x\in \Omega \), we end up with

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t} \int \limits _{\Omega } u + a_{1}\mu _{1}\int \limits _{\Omega }uv=0=\frac{\textrm{d}}{\textrm{d} t} \int \limits _{\Omega } v + a_{2}\mu _{2}\int \limits _{\Omega }uv, \end{aligned}$$

due to \(f_{i}=0 (i=1,2)\). Then, we have

$$\begin{aligned} \Vert u(\cdot , t)\Vert _{L^1(\Omega )} \le \left\| u_0\right\| _{L^1(\Omega )} \text { and } \Vert v(\cdot , t)\Vert _{L^1(\Omega )} \le \left\| v_0\right\| _{L^1(\Omega )} \end{aligned}$$
(2.3)

for all \(t \in \left( 0, T_{\max }\right) \). \(\square \)

3 Global boundedness for \(\tau =1\)

In this section, with the aid of damping term and diffusion respectively, we study the global boundedness of system (1.3) when \(n\ge 1\) and \(\tau =1\). The ideas come from [11, 21, 47]. Firstly, we need the following estimate.

Lemma 3.1

(see Lemma 2.3 in [34]) Let \(\Omega \subset {\mathbb {R}}^{n}(n\ge 1)\) be a smoothly bounded domain, and let \(0\le t_{0}<T_{\max }\le \infty \) and \(p\in (n,+\infty )\). Assume that \(z_{0}\in W^{2,p}(\Omega )\) with \(\partial _{\nu }z_{0}=0\) on \(\partial \Omega \) and \(h_{1}, h_{2} \in L^{p}\left( [0,T_{\max });L^{p}(\Omega )\right) \). Then the problem

$$\begin{aligned} \left\{ \begin{array}{ll} z_t=\Delta z-z+h_{1}+h_{2}, & \quad (x,t)\in \Omega \times (0,T_{\max }), \\ \dfrac{\partial z}{\partial \nu }=0, & \quad (x,t)\in \partial \Omega \times (0,T_{\max }), \\ z(\cdot ,0)=z_{0}, & \quad x\in \Omega , \end{array} \right. \end{aligned}$$

exists a unique solution \(z\in W^{1,p}\left( [0,T_{\max });L^{p}(\Omega )\right) \bigcap L^{p}\left( [0,T_{\max }); W^{2,p}(\Omega )\right) \) and there exists \(C_{S}(p)>0\) such that

$$\begin{aligned} \int \limits _{t_{0}}^{t}e^{p\tau }\int \limits _{\Omega }|\Delta z(\cdot ,\tau )|^{p}\textrm{d}\tau \le C_{S}(p)\int \limits _{t_{0}}^{t}e^{p\tau }\int \limits _{\Omega }\left( |h_{1}(\cdot ,s)|^{p} +|h_{2}(\cdot ,s)|^{p}\right) \textrm{d}\tau +C_{S}(p)e^{{pt_{0}}}||\Delta z(\cdot ,t_{0})||_{L^{p}(\Omega )}^{p} \end{aligned}$$

for any \(t\in (t_{0},T_{\max })\).

Next, we establish a priori estimates about u, v, which are of great help to get the main result.

Lemma 3.2

Let the conditions in Lemma 2.1 hold and \(\tau =1\). For any \(p_{i}>p^{\star }_{i}:=\max \left\{ 1, \alpha _{i}, 1-\beta _{i}\right\} \) and \(\epsilon _{i}>0\) with \(i=1,2\), the solution components u, v of (1.3) satisfy

$$\begin{aligned} \begin{aligned}&\frac{1}{p_{1}}\frac{\textrm{d}}{\textrm{d} t} \int \limits _{\Omega }(u+1)^{p_{1}} +\frac{4 \left( p_1-1\right) }{\left( p_1-\alpha _1\right) ^2} \int \limits _{\Omega }\left| \nabla \left( u+1\right) ^{\frac{p_1-\alpha _1}{2}}\right| ^2 \\&\quad \le \epsilon _1 \int \limits _{\Omega }(u+1)^{p_{1}+\beta _{1}+\gamma _{1}-1}+M_1\left( \epsilon _1\right) \int \limits _{\Omega }|\Delta w|^{\tilde{p_{1}}} +\int \limits _{\Omega }(u+1)^{p_{1}-1}f_{1}(u), \end{aligned} \end{aligned}$$
(3.1)

and

$$\begin{aligned} \begin{aligned}&\frac{1}{p_{2}}\frac{\textrm{d}}{\textrm{d} t} \int \limits _{\Omega }(v+1)^{p_{2}} +\frac{4 \left( p_2-1\right) }{\left( p_2-\alpha _2\right) ^2} \int \limits _{\Omega }\left| \nabla \left( v+1\right) ^{\frac{p_2-\alpha _2}{2}}\right| ^2\\&\quad \le \epsilon _2 \int \limits _{\Omega }(v+1)^{p_{2}+\beta _{2}+\gamma _{2}-1}+M_2\left( \epsilon _2\right) \int \limits _{\Omega }|\Delta w|^{\tilde{p_{2}}} +\int \limits _{\Omega }(v+1)^{p_{2}-1}f_{2}(v), \end{aligned} \end{aligned}$$
(3.2)

where \(\tilde{p_{i}}=\frac{p_{i}+\beta _{i}+\gamma _{i}-1}{\gamma _{i}}\) and

$$\begin{aligned} \begin{aligned} M_i\left( \epsilon _i\right)&= \frac{\gamma _{i}}{p_{i}+\beta _{i}+\gamma _{i}-1}\left( \epsilon _1 \cdot \frac{p_{i}+\beta _{i}+\gamma _{i}-1}{p_{i}+\beta _{i}-1}\right) ^{\frac{1-p_{i}-\beta _{i}}{\gamma _{i}}} \left( \frac{C_{\chi _{i}}(p_{i}-1)}{p_{i}+\beta _{i}-1}\right) ^{\frac{p_{i}+\beta _{i}+\gamma _{i}-1}{\gamma _{i}}}, \end{aligned} \end{aligned}$$
(3.3)

for \(i=1,2\).

Proof

Testing the first equation in (1.3) by \((u+1)^{p_{1}-1}\) for \(p_{1}>1\), we have

$$\begin{aligned} \begin{aligned}&\frac{1}{p_{1}}\frac{\textrm{d}}{\textrm{d} t} \int \limits _{\Omega }(u+1)^{p_{1}} \\&\quad =-(p_{1}-1) \int \limits _{\Omega }(u+1)^{p_{1}-2} D_{1}(u)|\nabla u|^{2} +(p_{1}-1) \int \limits _{\Omega }(u+1)^{p_{1}-2} \chi _{1}(u) \nabla u \cdot \nabla w\\&\qquad + \int \limits _{\Omega }(u+1)^{p_{1}-1}f_{1}(u). \end{aligned} \end{aligned}$$
(3.4)

Due to \((S_{1})\), \((S_{2})\), \(p_{1}>\alpha _{1}\) and \(p_{1}>1-\beta _{1}\), we get

$$\begin{aligned} \begin{aligned} -(p_{1}-1) \int \limits _{\Omega }(u+1)^{p_{1}-2} D_{1}(u)|\nabla u|^{2}&\le -(p_{1}-1)\int \limits _{\Omega }(u+1)^{p-\alpha _{1}-2}|\nabla u|^2 \\&\le \frac{4 \left( p_1-1\right) }{\left( p_1-\alpha _1\right) ^2} \int \limits _{\Omega }\left| \nabla \left( u+1\right) ^{\frac{p_1-\alpha _1}{2}}\right| ^2, \end{aligned} \end{aligned}$$
(3.5)

and

$$\begin{aligned} \begin{aligned} (p_{1}-1) \int \limits _{\Omega }(u+1)^{p_{1}-2} \chi _{1}(u) \nabla u \cdot \nabla w&=-(p_{1}-1)\int \limits _{\Omega }\left( \int \limits _0^u \Gamma (s) d s\right) \Delta w \\&\le (p_{1}-1) \int \limits _{\Omega }\left| \int \limits _0^u \Gamma (s) d s\right| |\Delta w| \\&\le \frac{C_{\chi _{1}}(p_{1}-1)}{p_{1}+\beta _{1}-1} \int \limits _{\Omega }(u+1)^{p_{1}+\beta _{1}-1}|\Delta w|, \end{aligned} \end{aligned}$$
(3.6)

where

$$\begin{aligned} \Gamma (s)=(s+1)^{p_{1}-2} \chi _{1}(s) \le C_{\chi _{1}}(s+1)^{p_{1}+\beta _{1}-2}. \end{aligned}$$

For the right term of (3.6), since \(\gamma _{1}>0\) and \(p_{1}>1-\beta _{1}\), using the fact \(\frac{\gamma _{1}}{p_{1}+\beta _{1}+\gamma _{1}-1}\in \left( 0,1\right) \) and Young’s inequality that for any \(\epsilon _1>0\), we obtain

$$\begin{aligned} \frac{C_{\chi _{1}}(p_{1}-1)}{p_{1}+\beta _{1}-1} \int \limits _{\Omega }(u+1)^{p_{1}+\beta _{1}-1}|\Delta w|\le \epsilon _1 \int \limits _{\Omega }(u+1)^{p_{1}+\beta _{1}+\gamma _{1}-1}+M_1\left( \epsilon _1\right) \int \limits _{\Omega }|\Delta w|^{\tilde{p_{1}}} \end{aligned}$$
(3.7)

where \(\tilde{p_{1}}=\frac{p_{1}+\beta _{1}+\gamma _{1}-1}{\gamma _{1}}\) and \(M_1\left( \epsilon _1\right) \) is defined in (3.3). Then, collecting (3.4), (3.5) and (3.7) that we prove (3.1). Similarly, (3.2) can be obtained by using the same framework. \(\square \)

Then, based on the above lemmas, we establish the uniform boundedness of \(\left\| u\right\| _{L^{p_1}(\Omega )}\), \(\left\| v\right\| _{L^{p_2}(\Omega )}\) under some appropriate parameter conditions.

Lemma 3.3

Let (u, v, w) be a solution ensured in Lemma 2.1 and the assumptions in Theorem 1.1 hold. Then there exists a positive constant \(K_{2}\) such that for all \(p_{1}, p_{2}>1\)

$$\begin{aligned} \left\| u(\cdot , t)\right\| _{L^{p_1}(\Omega )}+\left\| v(\cdot , t)\right\| _{L^{p_2}(\Omega )} \le K_{2}\;\;\;\text {for all}\; t\in (0,T_{\max }). \end{aligned}$$
(3.8)

Proof

Applying Lemma 3.1 to the third equation in (1.3) and combining \(\left( S_4\right) \), for \(i=1,2\), there exists \(C_S(p_{i})>0\) such that

$$\begin{aligned} \begin{aligned}&\int \limits _{t_0}^t e^{\tilde{p_{i}} \tau } \int \limits _{\Omega }|\Delta w(\cdot , \tau )|^{\tilde{p_{i}}} \textrm{d} \tau \\&\quad \le C_S(p_{i}) \int \limits _{t_0}^t e^{\tilde{p_{i}} \tau } \int \limits _{\Omega }\left( u^{\gamma _{1}}+v^{\gamma _{2}}\right) ^{\tilde{p_{i}}} \textrm{d} \tau +C_S(p_{i}) e^{\tilde{p_{i}} t_0}\left\| \Delta w\left( \cdot , t_0\right) \right\| _{L^{\tilde{p_{i}}}(\Omega )}^{\tilde{p_{i}}} \\&\quad \le C_S(p_{i})2^{\tilde{p_{i}}}\int \limits _{t_0}^t e^{\tilde{p_{i}} \tau } \int \limits _{\Omega }\left\{ (u(\cdot , \tau )+1)^{\tilde{p_{i}} \gamma _{1}}+(v(\cdot , \tau )+1)^{\tilde{p_{i}} \gamma _{2}}\right\} \textrm{d} \tau \\&\qquad + C_S(p_{i}) e^{\tilde{p_{i}} t_0}\left\| \Delta w\left( \cdot , t_0\right) \right\| _{L^{\tilde{p_{i}}}(\Omega )}^{\tilde{p_{i}}}. \\ \end{aligned} \end{aligned}$$
(3.9)

Let \(\tilde{p_{1}}=\tilde{p_{2}}\). Next, we divide our proof into the following two parts.

Part 1 We shall deal with the hypothesis (i) in Theorem 1.1 by using the aid of the diffusion when \(f_{i}=0 (i=1,2)\).

For \(i=1,2\), \(\psi _{i} \in W^{1, 2}(\Omega ) \cap L^{\frac{2}{p_{i}-\alpha _{i}}}(\Omega )\) and the same \(p_{i}\) in Lemma 3.2, combining Lemmas 2.2, 2.3 and Young’s inequality, there exist positive constants \(C^{i}_1, C^{i}_2, C^{i}_3\) such that,

$$\begin{aligned} \begin{aligned} \frac{\tilde{p_{i}}}{p_{i}}\int \limits _{\Omega }(\psi _{i}+1)^{p_{i}}&=\frac{\tilde{p_{i}}}{p_{i}}\left\| (\psi _{i}+1)^{\frac{p_{i} -\alpha _{i}}{2}}\right\| _{L^{\frac{2p_{i}}{p_{i}-\alpha _{i}}}(\Omega )}^{\frac{2p_{i}}{p_{i}-\alpha _{i}}} \\&\le C^{i}_1\left\| \nabla (\psi _{i}+1)^{\frac{p_{i}-\alpha _{i}}{2}}\right\| _{L^2(\Omega )}^{\frac{2p_{i}}{p_{i}-\alpha _{i}} \delta _{1}^{i}}\left\| (\psi _{i}+1)^{\frac{p_{i}-\alpha _{i}}{2}}\right\| _{L^{\frac{2}{p_{i} -\alpha _{i}}}(\Omega )}^{\frac{2p_{i}}{p_{i}-\alpha _{i}}\left( 1-\delta _{1}^{i}\right) } \\&\quad +C^{i}_1\left\| (\psi _{i}+1)^{\frac{p_{i}-\alpha _{i}}{2}}\right\| _{L^{\frac{2}{p_{i} -\alpha _{i}}}(\Omega )}^{\frac{2p_{i}}{p_{i}-\alpha _{i}}} \\&\le C^{i}_2\left\| \nabla (\psi _{i}+1)^{\frac{p_{i}-\alpha _{i}}{2}}\right\| _{L^2(\Omega )}^{\frac{2p_{i}}{p_{i} -\alpha _{i}} \delta _{1}^{i}}+C^{i}_2 \\&\le \frac{2 \left( p_i-1\right) }{\left( p_i-\alpha _i\right) ^2} \int \limits _{\Omega }\left| \nabla \left( \psi _{i}+1\right) ^{\frac{p_i-\alpha _i}{2}}\right| ^2+C^{i}_3, \\ \end{aligned} \end{aligned}$$
(3.10)

where \(\delta _{1}^{i}:=\frac{\frac{n (p_{i}-\alpha _{i})}{2}\left( 1-\frac{1}{p_{i}}\right) }{1-\frac{n}{2}+\frac{n(p_{i}-\alpha _{i})}{2}} \in (0,1)\) and \(\frac{2p_{i}}{p_{i}-\alpha _{i}} \delta _{1}^{i}\in (0,2)\) due to \(p_{i}>1\) and \(\alpha _{i}<\frac{2}{n}\) with \(n \ge 1\).

Moreover, let

$$\begin{aligned} \begin{aligned}&M_{3}^{1}=\epsilon _1+M_{1}(\epsilon _1)C_{S}(p_{1})2^{\tilde{p_{1}}}+M_{2}(\epsilon _2)C_{S}(p_{2})2^{\tilde{p_{2}}},\\&M_{3}^{2}=\epsilon _2+M_{1}(\epsilon _1)C_{S}(p_{1})2^{\tilde{p_{1}}}+M_{2}(\epsilon _2)C_{S}(p_{2})2^{\tilde{p_{2}}}, \end{aligned} \end{aligned}$$
(3.11)

and \(p_{i}>\max \left\{ 2-\beta _{i}-\gamma _{i}, \alpha _{i}+\frac{n-2}{n}, \frac{n\alpha _{i}}{2}+\frac{n-2}{2}(\beta _{i}+\gamma _{i}-1)\right\} \) with \(i=1,2\), combining Lemmas 2.2, 2.3 and Young’s inequality again, for \(i=1,2\), there exist positive constants \(C^{i}_{4}, C^{i}_{5}, C^{i}_{6}\) such that

$$\begin{aligned} \begin{aligned}&M^{i}_{3}\int \limits _{\Omega }(\psi _{i}+1)^{p_{i}+\beta _{i}+\gamma _{i}-1}\\&\qquad =M^{i}_{3}\left\| (\psi _{i}+1)^{\frac{p_{i}-\alpha _{i}}{2}}\right\| _{L^{\frac{2(p_{i} +\beta _{i}+\gamma _{i}-1)}{p_{i}-\alpha _{i}}}(\Omega )}^{\frac{2(p_{i}+\beta _{i}+\gamma _{i}-1)}{p_{i}-\alpha _{i}}} \\&\qquad \le C^{i}_4\left\| \nabla (\psi _{i}+1)^{\frac{p_{i}-\alpha _{i}}{2}}\right\| _{L^2(\Omega )}^{\frac{2(p_{i} +\beta _{i}+\gamma _{i}-1)}{p_{i}-\alpha _{i}} \delta _{2}^{i}} \left\| (\psi _{i}+1)^{\frac{p_{i}-\alpha _{i}}{2}}\right\| _{L^{\frac{2}{p_{i} -\alpha _{i}}}(\Omega )}^{\frac{2(p_{i}+\beta _{i}+\gamma _{i}-1)}{p_{i}-\alpha _{i}}\left( 1-\delta _{2}^{i}\right) } \\&\quad +C^{i}_4\left\| (\psi _{i}+1)^{\frac{p_{i}-\alpha _{i}}{2}}\right\| _{L^{\frac{2}{p_{i} -\alpha _{i}}}(\Omega )}^{\frac{2(p_{i}+\beta _{i}+\gamma _{i}-1) }{p_{i}-\alpha _{i}}} \\&\qquad \le C^{i}_5\left\| \nabla (\psi _{i}+1)^{\frac{p_{i}-\alpha _{i}}{2}}\right\| _{L^2(\Omega )}^{\frac{2(p_{i}+\beta _{i} +\gamma _{i}-1)}{p_{i}-\alpha _{i}} \delta _{2}^{i}}+C^{i}_5 \\&\qquad \le \frac{2 \left( p_i-1\right) }{\left( p_i-\alpha _i\right) ^2} \int \limits _{\Omega }\left| \nabla \left( \psi _{i} +1\right) ^{\frac{p_i-\alpha _i}{2}}\right| ^2+C^{i}_6, \\ \end{aligned} \end{aligned}$$
(3.12)

where \(\delta ^{i}_2:=\frac{\frac{n (p_{i}-\alpha _{i})}{2}\left( 1-\frac{1}{p_{i}+\beta _{i} +\gamma _{i}-1}\right) }{1-\frac{n}{2}+\frac{n(p_{i}-\alpha _{i})}{2}} \in (0,1)\) and \(\frac{2(p_{i}+\beta _{i}+\gamma _{i}-1)}{p_{i}-\alpha _{i}} \delta ^{i}_2\in (0,2)\) due to \(\alpha _{i}+\beta _{i}+\gamma _{i}<1+\frac{2}{n}\) with \(n \ge 1\).

Together (3.10), (3.12) with (3.1) and (3.2), we have

$$\begin{aligned} \begin{aligned}&\frac{1}{p_{1}}\frac{\textrm{d}}{\textrm{d} t} \int \limits _{\Omega }(u+1)^{p_{1}} + \frac{\tilde{p_{1}}}{p_{1}}\int \limits _{\Omega }(u+1)^{p_{1}} +M^{1}_{3}\int \limits _{\Omega }(u+1)^{p_{1}+\beta _{1}+\gamma _{1}-1} \\&\quad \le \epsilon _1 \int \limits _{\Omega }(u+1)^{p_{1}+\beta _{1}+\gamma _{1}-1}+M_1\left( \epsilon _1\right) \int \limits _{\Omega }|\Delta w|^{\tilde{p_{1}}}, \end{aligned} \end{aligned}$$
(3.13)

and

$$\begin{aligned} \begin{aligned}&\frac{1}{p_{2}}\frac{\textrm{d}}{\textrm{d} t} \int \limits _{\Omega }(v+1)^{p_{2}} + \frac{\tilde{p_{2}}}{p_{2}}\int \limits _{\Omega }(v+1)^{p_{2}} +M^{2}_{3}\int \limits _{\Omega }(v+1)^{p_{2}+\beta _{2}+\gamma _{2}-1}\\&\quad \le \epsilon _2 \int \limits _{\Omega }(v+1)^{p_{2}+\beta _{2}+\gamma _{2}-1}+M_2\left( \epsilon _2\right) \int \limits _{\Omega }|\Delta w|^{\tilde{p_{2}}}, \end{aligned} \end{aligned}$$
(3.14)

due to \(f_{i}=0 (i=1,2)\). Integrating (3.13) and (3.14) from \(t_{0}\) to t yields

$$\begin{aligned} \begin{aligned}&\frac{1}{p_{1}}\int \limits _{\Omega }(u+1)^{p_{1}}(\cdot , t)+ \left( M_{3}^{1}-\epsilon _1\right) e^{-\tilde{p_{1}} t} \int \limits _{t_0}^t e^{\tilde{p_{1}} \tau } \int \limits _{\Omega }(u(\cdot , \tau )+1)^{p_{1}+\beta _{1}+\gamma _{1}-1} \\&\quad \le M_{1}(\epsilon _1) e^{-\tilde{p_{1}} t} \int \limits _{t_0}^t e^{\tilde{p_{1}} \tau } \int \limits _{\Omega }|\Delta w(\cdot , \tau )|^{\tilde{p_{1}}}+\frac{1}{p_{1}}e^{\tilde{p_{1}}\left( t_0-t\right) } \int \limits _{\Omega }\left( u\left( \cdot , t_0\right) +1\right) ^{p_{1}} \end{aligned} \end{aligned}$$
(3.15)

and

$$\begin{aligned} \begin{aligned}&\frac{1}{p_{2}}\int \limits _{\Omega }(v+1)^{p_{2}}(\cdot , t)+ \left( M_{3}^{2} -\epsilon _2\right) e^{-\tilde{p_{2}} t} \int \limits _{t_0}^t e^{\tilde{p_{2}} \tau } \int \limits _{\Omega }(v(\cdot , \tau )+1)^{p_{2}+\beta _{2}+\gamma _{2}-1} \\&\quad \le M_{2}(\epsilon _1) e^{-\tilde{p_{2}} t} \int \limits _{t_0}^t e^{\tilde{p_{2}} \tau } \int \limits _{\Omega }|\Delta w(\cdot , \tau )|^{\tilde{p_{2}}}+\frac{1}{p_{2}}e^{\tilde{p_{2}}\left( t_0-t\right) } \int \limits _{\Omega }\left( v\left( \cdot , t_0\right) +1\right) ^{p_{2}}. \end{aligned} \end{aligned}$$
(3.16)

Together (3.15) and (3.16) with (3.9), we obtain that for all \(t\in (0, T_{\max })\),

$$\begin{aligned} \begin{aligned}&\frac{1}{p_{1}}\int \limits _{\Omega }(u+1)^{p_{1}}(\cdot , t)+ \frac{1}{p_{2}}\int \limits _{\Omega }(v+1)^{p_{2}}(\cdot , t)\le C_{7} \end{aligned} \end{aligned}$$
(3.17)

for some constant \(C_{7}>0\) due to \(\tilde{p_{1}}=\tilde{p_{2}}\).

Part 2 We deal with the hypothesis (ii) in Theorem 1.1 by using the effect of damping terms. Firstly, it follows from \(M_1\left( \epsilon _1\right) \) and \(M_2\left( \epsilon _2\right) \) in (3.3) and simple calculations that \(M_{3}^{1}(\epsilon _1)\) and \(M_{3}^{2}(\epsilon _2)\) defined in (3.11) attain the minimum value

$$\begin{aligned} \min _{\epsilon _1>0} M_{3}^{1}\left( \epsilon _1\right) =\frac{2C_{\chi _{1}}(p_{1}-1)C_{S}(p_{1})^{\frac{1}{\tilde{p_{1}}}}}{p_{1} +\beta _{1}-1}+\frac{2\gamma _{2}C_{\chi _{2}}(p_{2}-1)C_{S}(p_{2})^{\frac{1}{\tilde{p_{2}}}}}{(p_{2} +\beta _{2}+\gamma _{2}-1)(p_{2}+\beta _{2}-1)}:=M_{4}^{1} \end{aligned}$$
(3.18)

and

$$\begin{aligned} \min _{\epsilon _2>0} M_{3}^{2}\left( \epsilon _2\right) =\frac{2C_{\chi _{2}}(p_{2}-1)C_{S}(p_{2})^{\frac{1}{\tilde{p_{2}}}}}{p_{2} +\beta _{2}-1}+\frac{2\gamma _{1}C_{\chi _{1}}(p_{1}-1)C_{S}(p_{1})^{\frac{1}{\tilde{p_{1}}}}}{(p_{1} +\beta _{1}+\gamma _{1}-1)(p_{1}+\beta _{1}-1)}:=M_{4}^{2} \end{aligned}$$
(3.19)

respectively, when

$$\begin{aligned} \epsilon _i=\frac{2C_{\chi _{i}}(p_{i}-1)C_{S}(p_{i})^{\frac{1}{\tilde{p_{i}}}}}{p_{i}+\beta _{i}+\gamma _{i}-1} \end{aligned}$$
(3.20)

for \(i=1,2\).

Adding \(\frac{\tilde{p_{1}}}{p_{1}} \int \limits _{\Omega }(u+1)^{p_{1}}\) and \(\frac{\tilde{p_{2}}}{p_{2}} \int \limits _{\Omega }(v+1)^{p_{2}}\) to both sides of (3.1) and (3.2), respectively, we have

$$\begin{aligned} \begin{aligned}&\frac{1}{p_{1}}\frac{\textrm{d}}{\textrm{d} t} \int \limits _{\Omega }(u+1)^{p_{1}}+\frac{\tilde{p_{1}}}{p_{1}} \int \limits _{\Omega }(u+1)^{p_{1}}\\&\quad \le \frac{\tilde{p_{1}}}{p_{1}} \int \limits _{\Omega }(u+1)^{p_{1}} +\epsilon _1 \int \limits _{\Omega }(u+1)^{p_{1}+\beta _{1}+\gamma _{1}-1}+M_1\left( \epsilon _1\right) \int \limits _{\Omega }|\Delta w|^{\tilde{p_{1}}}\\&\qquad +\int \limits _{\Omega }(u+1)^{p_{1}-1}f_{1}(u), \end{aligned} \end{aligned}$$
(3.21)

and

$$\begin{aligned} \begin{aligned}&\frac{1}{p_{2}}\frac{\textrm{d}}{\textrm{d} t} \int \limits _{\Omega }(v+1)^{p_{2}} +\frac{\tilde{p_{2}}}{p_{2}} \int \limits _{\Omega }(v+1)^{p_{2}}\\&\quad \le \frac{\tilde{p_{2}}}{p_{2}} \int \limits _{\Omega }(v+1)^{p_{2}} +\epsilon _2 \int \limits _{\Omega }(v+1)^{p_{2}+\beta _{2}+\gamma _{2}-1} +M_2\left( \epsilon _2\right) \int \limits _{\Omega }|\Delta w|^{\tilde{p_{2}}}\\&\qquad +\int \limits _{\Omega }(v+1)^{p_{2}-1}f_{2}(v). \end{aligned} \end{aligned}$$
(3.22)

Integrating (3.21), (3.22) from \(t_{0}\) to t and using (3.9) yields

$$\begin{aligned} \begin{aligned}&\frac{1}{p_{1}}\int \limits _{\Omega }(u+1)^{p_{1}}(\cdot , t) +\frac{1}{p_{2}}\int \limits _{\Omega }(v+1)^{p_{2}}(\cdot , t) \\&\quad \le \frac{\tilde{p_{1}}}{p_{1}} e^{-\tilde{p_{1}} t} \int \limits _{t_0}^t e^{\tilde{p_{1}} \tau } \int \limits _{\Omega }(u(\cdot , \tau )+1)^{p_{1}} + M^{1}_{3}(\epsilon _1) e^{-\tilde{p_{1}} t} \int \limits _{t_0}^t e^{\tilde{p_{1}} \tau } \int \limits _{\Omega }(u(\cdot , \tau )+1)^{p_{1}+\beta _{1}+\gamma _{1}-1}\\&\qquad +e^{-\tilde{p_{1}} t} \int \limits _{t_0}^t e^{\tilde{p_{1}} \tau } \int \limits _{\Omega }(u+1)^{p_{1}-1}f_{1}(u)(\cdot , \tau )+\frac{1}{p_{1}}e^{\tilde{p_{1}}\left( t_0-t\right) } \int \limits _{\Omega }\left( u\left( \cdot , t_0\right) +1\right) ^{p_{1}} \\&\qquad + \frac{\tilde{p_{2}}}{p_{2}} e^{-\tilde{p_{2}} t} \int \limits _{t_0}^t e^{\tilde{p_{2}} \tau } \int \limits _{\Omega }(v(\cdot , \tau )+1)^{p_{2}} + M^{2}_{3}(\epsilon _2) e^{-\tilde{p_{2}} t} \int \limits _{t_0}^t e^{\tilde{p_{2}} \tau } \int \limits _{\Omega }(v(\cdot , \tau )+1)^{p_{2}+\beta _{2}+\gamma _{2}-1}\\&\qquad +e^{-\tilde{p_{2}} t} \int \limits _{t_0}^t e^{\tilde{p_{2}} \tau } \int \limits _{\Omega }(v+1)^{p_{2}-1}f_{2}(v)(\cdot , \tau )+\frac{1}{p_{2}}e^{\tilde{p_{2}}\left( t_0-t\right) } \int \limits _{\Omega }\left( v\left( \cdot , t_0\right) +1\right) ^{p_{2}} \end{aligned} \end{aligned}$$
(3.23)

due to \(\tilde{p_{1}}=\tilde{p_{2}}\). With the \(\epsilon _i\) in (3.20) for \(i=1,2\), we obtain

$$\begin{aligned} \begin{aligned}&\frac{1}{p_{1}}\int \limits _{\Omega }(u+1)^{p_{1}}(\cdot , t)+\frac{1}{p_{2}}\int \limits _{\Omega }(v+1)^{p_{2}}(\cdot , t) \\&\quad \le e^{-\tilde{p_{1}} t} \int \limits _{t_0}^t e^{\tilde{p_{1}} \tau } \int \limits _{\Omega }F_{1}(u)(\cdot , \tau )+ e^{-\tilde{p_{2}} t} \int \limits _{t_0}^t e^{\tilde{p_{2}} \tau } \int \limits _{\Omega }F_{2}(v)(\cdot , \tau )+C_{8} \end{aligned} \end{aligned}$$
(3.24)

where \(C_{8}=\frac{1}{p_{1}}e^{\tilde{p_{1}}\left( t_0-t\right) } \int \limits _{\Omega }\left( u\left( \cdot , t_0\right) +1\right) ^{p_{1}}+\frac{1}{p_{2}}e^{\tilde{p_{2}}\left( t_0-t\right) } \int \limits _{\Omega }\left( v\left( \cdot , t_0\right) +1\right) ^{p_{2}} \) and

$$\begin{aligned} F_{i}(s):= \frac{\tilde{p_{i}}}{p_{i}} (s+1)^{p_{i}} + M_{4}^{i} (s+1)^{p_{i}+\beta _{i}+\gamma _{i}-1}+(s+1)^{p_{i}-1}f_{i}(s) \end{aligned}$$
(3.25)

for \(i=1,2\). Let

$$\begin{aligned} \iota _{1}^{\star }={\left\{ \begin{array}{ll} \frac{\tilde{p_{1}}}{p_{1}}, & \text{ when } \beta _{1}+\gamma _{1}<1; \\ \frac{\tilde{p_{1}}}{p_{1}}+M_{4}^{1}, & \text{ when } \beta _{1}+\gamma _{1}=1; \\ M_{4}^{1}, & \text{ when } \beta _{1}+\gamma _{1}>1,\end{array}\right. } \end{aligned}$$
(3.26)

and

$$\begin{aligned} \iota _{2}^{\star }={\left\{ \begin{array}{ll} \frac{\tilde{p_{2}}}{p_{2}}, & \text{ when } \beta _{2}+\gamma _{2}<1; \\ \frac{\tilde{p_{2}}}{p_{2}}+M_{4}^{2}, & \text{ when } \beta _{2}+\gamma _{2}=1; \\ M_{4}^{2}, & \text{ when } \beta _{2}+\gamma _{2}>1,\end{array}\right. } \end{aligned}$$
(3.27)

then we can divide the three cases (a) \(\beta _{1}+\gamma _{1}<1\), (b) \(\beta _{1}+\gamma _{1}=1\) and (c) \(\beta _{1}+\gamma _{1}>1\) to deal with \(F_{1}\), and \(F_{2}\) is treated in a similar way. Here we only give the proof of case (a).

If \(\beta _{1}+\gamma _{1}<1\), it follows from \(\lim _{s \rightarrow \infty } \inf \left\{ -\frac{f_{1}(s)}{s}\right\} =: \iota _{1} \in \left( \iota _{1}^{\star }, \infty \right] \) for \(\iota _{1}^{\star }=\frac{\tilde{p_{1}}}{p_{1}}\) that

$$\begin{aligned} \begin{aligned} \lim _{s \rightarrow \infty } \inf \frac{1}{(s+1)^{p_{1}}} F_1(s)&=\frac{\tilde{p_{1}}}{p_{1}}+M_{4}^{1} \lim _{s \rightarrow \infty } \inf (s+1)^{\beta _{1}+\gamma _{1}-1}-\iota _{1} \\&=\frac{\tilde{p_{1}}}{p_{1}}-\iota _{1}<0, \end{aligned} \end{aligned}$$
(3.28)

so that

$$\begin{aligned} \exists {\bar{s}}_1>0 \text{, } \text{ s.t. } F_1(s)<0 \text{ for } \text{ all } s>{\bar{s}}_1 \text{. } \end{aligned}$$

Therefore, we further have

$$\begin{aligned} \int \limits _{\Omega }F_{1}(u)(\cdot , \tau )= \int \limits _{u\le {\bar{s}}_1}F_{1}(u)(\cdot , \tau )+ \int \limits _{u>{\bar{s}}_1}F_{1}(u)(\cdot , \tau )\le \sup _{0<s\le {\bar{s}}_1}F_{1}(s)|\Omega |<\infty . \end{aligned}$$
(3.29)

In this similar way, we can also get

$$\begin{aligned} \int \limits _{\Omega }F_{1}(u)(\cdot , \tau )<\infty \quad \text { when case } (b), (c) \end{aligned}$$
(3.30)

and

$$\begin{aligned} \int \limits _{\Omega }F_{2}(v)(\cdot , \tau )<\infty \quad \text { when case } (a), (b), (c) \end{aligned}$$
(3.31)

under the condition \(\lim _{s \rightarrow \infty } \inf \left\{ -\frac{f_{i}(s)}{s^{\max \left\{ 1, \beta _{i}+\gamma _{i}\right\} }}\right\} =: \iota _{i} \in \left( \iota _{i}^{\star }, \infty \right] \) by the definition of \(\iota _{i}^{\star }\) in (3.26) and (3.27) with \(i=1,2\). Combining (3.24) with (3.30) and (3.31), we obtain the existence of positive constant \(C_{9}\) such that

$$\begin{aligned} \frac{1}{p_{1}}\int \limits _{\Omega }(u+1)^{p_{1}}(\cdot , t)+\frac{1}{p_{2}}\int \limits _{\Omega }(v+1)^{p_{2}}(\cdot , t) \le C_{9} \end{aligned}$$
(3.32)

for all \(t\in (0, T_{\max })\). The proof of Lemma 3.3 is complete. \(\square \)

By Lemma 3.3, we can complete the proof of Theorem 1.1.

Proof of Theorem 1.1

By using the well-known \(L^p-L^q\) estimate in [38] and selecting the appropriate values for \(p_{1}, p_{2}\) in Lemma 3.3, we can obtain the boundedness of \(\Vert w(\cdot , t)\Vert _{W^{1,\infty }}\). Then, it follows form Moser iteration technique (Appendix A of [27]) that there exists a constant \(K_{3}>0\) independent of t such that

$$\begin{aligned} \Vert u(\cdot , t)\Vert _{L^{\infty }}+\Vert v(\cdot , t)\Vert _{L^{\infty }}+\Vert w(\cdot , t)\Vert _{W^{1,\infty }} \le K_{3}, \end{aligned}$$

which together with the extension criterion in Lemma 2.1 proves Theorem 1.1.

4 Global boundedness for \(\tau =0\)

In this section, we shall prove the global boundedness of solutions for (1.3) in any space dimension when \(\tau =0\). Firstly, we give the coupled estimate of \(\int \limits _{\Omega }u^{p_{1}}\) and \(\int \limits _{\Omega }v^{p_{2}}\) for some suitably large \(p_{i}>1 (i=1,2)\).

Lemma 4.1

Let \(\tau =0\) and (u, v, w) be a solution ensured in Lemma 2.1. Then the solution (u, v, w) satisfies

$$\begin{aligned} \begin{aligned}&\frac{1}{p_{1}}\frac{\textrm{d}}{\textrm{d} t} \int \limits _{\Omega }(u+1)^{p_{1}}+\frac{4(p_{1}-1)}{(p_{1}-\alpha _{1})^{2}} \int \limits _{\Omega }\left| \nabla (u+1)^{\frac{p_{1}-\alpha _{1}}{2}}\right| ^{2} \\&\quad \le \frac{C_{\chi _{1}}(p_{1}-1)}{(p_{1}+\beta _{1}-1)}\int \limits _{\Omega }(u+1)^{p_{1}+\beta _{1} +\gamma _{1}-1}+\frac{C_{\chi _{1}}(p_{1}-1)}{p_{1}+\beta _{1}+\gamma _{2}-1}\int \limits _{\Omega }(u+1)^{p_{1}+\beta _{1}+\gamma _{2}-1} \\&\qquad +\frac{C_{\chi _{1}}\gamma _{2}(p_{1}-1)}{(p_{1}+\beta _{1}+\gamma _{2}-1)(p_{1} +\beta _{1}-1)}\int \limits _{\Omega }(v+1)^{p_{1}+\beta _{1}+\gamma _{2}-1}-\int \limits _{\Omega }f_{1}(u)(u+1)^{p_{1}-1} \end{aligned} \end{aligned}$$
(4.1)

and

$$\begin{aligned} \begin{aligned}&\frac{1}{p_{2}}\frac{\textrm{d}}{\textrm{d} t} \int \limits _{\Omega }(v+1)^{p_{2}}+\frac{4(p_{2}-1)}{(p_{2} -\alpha _{2})^{2}} \int \limits _{\Omega }\left| \nabla (v+1)^{\frac{p_{2}-\alpha _{2}}{2}}\right| ^{2} \\&\quad \le \frac{C_{\chi _{2}}(p_{2}-1)}{(p_{2}+\beta _{2}-1)}\int \limits _{\Omega }(v+1)^{p_{2} +\beta _{2}+\gamma _{2}-1}+\frac{C_{\chi _{2}}(p_{2}-1)}{p_{2}+\beta _{2}+\gamma _{1}-1}\int \limits _{\Omega }(v+1)^{p_{2}+\beta _{2}+\gamma _{1}-1} \\&\qquad +\frac{C_{\chi _{2}}\gamma _{1}(p_{2}-1)}{(p_{2}+\beta _{2}+\gamma _{1}-1) (p_{2}+\beta _{2}-1)}\int \limits _{\Omega }(u+1)^{p_{2}+\beta _{2}+\gamma _{1}-1}-\int \limits _{\Omega }f_{2}(v)(v+1)^{p_{2}-1} \end{aligned} \end{aligned}$$
(4.2)

for all \(t\in (0, T_{\max })\), where

$$\begin{aligned} p_{i}>\max \{1, 1-\beta _{i}, \alpha _{i}\}\quad \text { with } i=1,2. \end{aligned}$$
(4.3)

Proof

Testing the first equation in (1.3) by \((u+1)^{p_{1}-1}\), since \((S_{1})\) and \(p_{1}>\alpha _{1}\), we have

$$\begin{aligned} \begin{aligned}&\frac{1}{p_{1}}\frac{\textrm{d}}{\textrm{d} t} \int \limits _{\Omega }(u+1)^{p_{1}} \\&\quad = \int \limits _{\Omega }(u+1)^{p_{1}-1} \nabla \cdot (D_{1}(u) \nabla u) - \int \limits _{\Omega }(u+1)^{p_{1}-1} \nabla \cdot (\chi _{1}(u) \nabla w)-\int \limits _{\Omega }f_{1}(u)(u+1)^{p_{1}-1}\\&\quad \le \frac{-4(p_{1}-1)}{(p_{1}-\alpha _{1})^{2}} \int \limits _{\Omega }\left| \nabla (u+1)^{\frac{p_{1} -\alpha _{1}}{2}}\right| ^{2}+(p_{1}-1) \int \limits _{\Omega }(u+1)^{p_{1}-2} \chi _{1}(u) \nabla u \cdot \nabla w\\&\qquad -\int \limits _{\Omega }f_{1}(u)(u+1)^{p_{1}-1}. \end{aligned} \end{aligned}$$
(4.4)

Due to \((S_{2}), (S_{4})\) and \(p_{1}>1-\beta _{1}\), using Young’s inequality, we obtain

$$\begin{aligned} \begin{aligned}&(p_{1}-1) \int \limits _{\Omega }(u+1)^{p_{1}-2}\chi _{1}(u) \nabla u \cdot \nabla w\\&\quad =(p_{1}-1) \int \limits _{\Omega }\nabla \varpi (u)\cdot \nabla w\\&\quad =-(p_{1}-1) \int \limits _{\Omega }\varpi (u)\cdot \Delta w\\&\quad \le (p_{1}-1) \int \limits _{\Omega }\varpi (u)(u^{\gamma _{1}}+v^{\gamma _{2}}-w)\\&\quad \le \frac{C_{\chi _{1}}(p_{1}-1)}{(p_{1}+\beta _{1}-1)} \int \limits _{\Omega }(u+1)^{p_{1} +\beta _{1}-1}\left( u^{\gamma _{1}}+v^{\gamma _{2}}\right) \\&\quad \le \frac{C_{\chi _{1}}(p_{1}-1)}{(p_{1}+\beta _{1}-1)}\left\{ \int \limits _{\Omega }(u+1)^{p_{1} +\beta _{1}+\gamma _{1}-1}+\frac{p_{1}+\beta _{1}-1}{p_{1}+\beta _{1} +\gamma _{2}-1}\int \limits _{\Omega }(u+1)^{p_{1}+\beta _{1}+\gamma _{2}-1}\right. \\&\qquad \left. +\frac{\gamma _{2}}{p_{1}+\beta _{1}+\gamma _{2}-1}\int \limits _{\Omega }(v+1)^{p_{1}+\beta _{1}+\gamma _{2}-1}\right\} \end{aligned} \end{aligned}$$
(4.5)

where

$$\begin{aligned} \varpi (u)=\int \limits _{0}^{u}(s+1)^{p_{1}-2} \chi _{1}(s)ds\le \int \limits _{0}^{u}(s+1)^{p_{1}+\beta _{1}-2}ds. \end{aligned}$$

Then, collecting (4.4) and (4.5) that we prove (4.1). Similarly, (4.2) can be obtained by using the same framework. \(\square \)

Then, based on the above lemmas, we establish the uniform boundedness of \(\left\| u\right\| _{L^{p_1}(\Omega )}\), \(\left\| v\right\| _{L^{p_2}(\Omega )}\) under some appropriate parameter conditions.

Lemma 4.2

Let (u, v, w) be a solution ensured in Lemma 2.1 and the assumptions in Theorem 1.2 hold. Then, for all \(p_{1}, p_{2}>1\), there exists a positive constant \(K_{4}\) such that

$$\begin{aligned} \left\| u(\cdot , t)\right\| _{L^{p_1}(\Omega )}+\left\| v(\cdot , t)\right\| _{L^{p_2}(\Omega )} \le K_{4}\;\;\;\text {for all}\; t\in (0,T_{\max }). \end{aligned}$$
(4.6)

Proof

Let \(p_{1}+\beta _{1}+\gamma _{2}=p_{2}+\beta _{2}+\gamma _{1}\) in Lemma 4.1. Similar to Lemma 3.3, we divide our proof into the following two parts.

Part 1 We shall deal with the hypothesis (i) in Theorem 1.2 by using the aid of the diffusion when \(f_{i}=0 (i=1,2)\). Combining (4.1)–(4.2) and apply Young’s inequality, there exist some constant \(C_{10}, C_{11}>0\) such that

$$\begin{aligned} \begin{aligned}&\frac{\textrm{d}}{\textrm{d} t} \left\{ \frac{1}{p_{1}}\int \limits _{\Omega }(u+1)^{p_{1}}+\frac{1}{p_{2}} \int \limits _{\Omega }(v+1)^{p_{2}}\right\} \\&\qquad +\frac{4(p_{1}-1)}{(p_{1}-\alpha _{1})^{2}} \int \limits _{\Omega } \left| \nabla (u+1)^{\frac{p_{1}-\alpha _{1}}{2}}\right| ^{2} +\frac{4(p_{2}-1)}{(p_{2} -\alpha _{2})^{2}} \int \limits _{\Omega }\left| \nabla (v+1)^{\frac{p_{2}-\alpha _{2}}{2}}\right| ^{2}\\&\quad \le C_{10}\int \limits _{\Omega }(u+1)^{p_{1}+\beta _{1}+\gamma -1}+C_{11}\int \limits _{\Omega }(v+1)^{p_{2}+\beta _{2}+\gamma -1} \end{aligned} \end{aligned}$$
(4.7)

due to \(\gamma =\max \left\{ \gamma _{1},\gamma _{2}\right\} \), \(p_{1}+\beta _{1}+\gamma _{2}=p_{2}+\beta _{2}+\gamma _{1}\) and \(f_{i}=0(i=1,2)\). Since \(\alpha _i<1+\frac{2}{n}-\gamma -\beta _i\) for \(i=1,2\), using the similar techniques as in (3.12) we obtain

$$\begin{aligned} \begin{aligned}&(C_{10}+1)\int \limits _{\Omega }(u+1)^{p_{1}+\beta _{1}+\gamma _{1}-1}\\&\quad \le \frac{4 \left( p_{1}-1\right) }{\left( p_{1}-\alpha _{1}\right) ^2} \int \limits _{\Omega }\left| \nabla \left( u+1\right) ^{\frac{p_{1}-\alpha _{1}}{2}}\right| ^2+{\tilde{C}}_{10} \\ \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&(C_{11}+1)\int \limits _{\Omega }(v+1)^{p_{2}+\beta _{2}+\gamma _{2}-1}\\&\quad \le \frac{4 \left( p_{2}-1\right) }{\left( p_{2}-\alpha _{2}\right) ^2} \int \limits _{\Omega }\left| \nabla \left( v+1\right) ^{\frac{p_{2}-\alpha _{2}}{2}}\right| ^2+{\tilde{C}}_{11}, \\ \end{aligned} \end{aligned}$$

where \({\tilde{C}}_{10}, {\tilde{C}}_{11}\) are some positive constants. Together them with (4.7) that we have

$$\begin{aligned} \begin{aligned}&\frac{\textrm{d}}{\textrm{d} t} \left\{ \frac{1}{p_{1}}\int \limits _{\Omega }(u+1)^{p_{1}}+\frac{1}{p_{2}} \int \limits _{\Omega }(v+1)^{p_{2}}\right\} \\&\qquad +\int \limits _{\Omega }(u+1)^{p_{1}+\beta _{1}+\gamma -1}+\int \limits _{\Omega }(v+1)^{p_{2}+\beta _{2}+\gamma -1}\\&\quad \le C_{12}, \end{aligned} \end{aligned}$$
(4.8)

where \(C_{12}\) is a positive constant. This along with the ODE comparison argument, we obtain the existence of positive constant \(C_{13}\) such that

$$\begin{aligned} \frac{1}{p_{1}}\int \limits _{\Omega }(u+1)^{p_{1}}(\cdot , t)+\frac{1}{p_{2}}\int \limits _{\Omega }(v+1)^{p_{2}}(\cdot , t) \le C_{13} \end{aligned}$$
(4.9)

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

Part 2 We deal with the hypothesis (ii) in Theorem 1.2 by using the effect of damping terms.

Together (4.1) with (4.2) and adding \(\int \limits _{\Omega }(u+1)^{\frac{p_{1}}{2}}+ \int \limits _{\Omega }(v+1)^{\frac{p_{2}}{2}}\) to the both sides, since \(p_{1}, p_{2}>1\) and \(p_{1}+\beta _{1}+\gamma _{2}=p_{2}+\beta _{2}+\gamma _{1}\), we have

$$\begin{aligned} \begin{aligned}&\frac{\textrm{d}}{\textrm{d} t} \left\{ \frac{1}{p_{1}}\int \limits _{\Omega }(u+1)^{p_{1}} + \frac{1}{p_{2}}\int \limits _{\Omega }(v+1)^{p_{2}}\right\} +\int \limits _{\Omega }(u+1)^{\frac{p_{1}}{2}} + \int \limits _{\Omega }(v+1)^{\frac{p_{2}}{2}} \\&\quad \le \int \limits _{\Omega }\varrho _{1}(u)+\int \limits _{\Omega }\varrho _{2}(v), \end{aligned} \end{aligned}$$
(4.10)

where

$$\begin{aligned} \begin{aligned} \varrho _{1}(u)&=\left( \frac{C_{\chi _{1}}(p_{1}-1)}{p_{1}+\beta _{1}+\gamma _{2}-1} +\frac{C_{\chi _{2}}\gamma _{1}(p_{2}-1)}{(p_{2}+\beta _{2}+\gamma _{1}-1)(p_{2} +\beta _{2}-1)}\right) (u+1)^{p_{1}+\beta _{1}+\gamma _{2}-1}\\&\quad +\frac{C_{\chi _{1}}(p_{1}-1)}{(p_{1}+\beta _{1}-1)}(u+1)^{p_{1}+\beta _{1} +\gamma _{1}-1}+(u+1)^{\frac{p_{1}}{2}}-f_{1}(u)(u+1)^{p_{1}-1} \end{aligned} \end{aligned}$$
(4.11)

and

$$\begin{aligned} \begin{aligned} \varrho _{2}(v)&=\left( \frac{C_{\chi _{2}}(p_{2}-1)}{p_{2}+\beta _{2}+\gamma _{1}-1} +\frac{C_{\chi _{1}}\gamma _{2}(p_{1}-1)}{(p_{1}+\beta _{1}+\gamma _{2}-1)(p_{1} +\beta _{1}-1)}\right) (v+1)^{p_{2}+\beta _{2}+\gamma _{1}-1}\\&\quad +\frac{C_{\chi _{2}}(p_{2}-1)}{(p_{2}+\beta _{2}-1)}(v+1)^{p_{2}+\beta _{2} +\gamma _{2}-1}+ (v+1)^{\frac{p_{2}}{2}}-f_{2}(v)(v+1)^{p_{2}-1}. \end{aligned} \end{aligned}$$
(4.12)

Let \(p_{i}>2(\beta _{i}+\gamma -2)\) \((i=1,2)\) and

$$\begin{aligned} \tilde{\iota _{1}}={\left\{ \begin{array}{ll}\frac{C_{\chi _{1}}(p_{1}-1)}{p_{1}+\beta _{1} +\gamma _{2}-1}+\frac{C_{\chi _{2}}\gamma _{1}(p_{2}-1)}{(p_{2}+\beta _{2} +\gamma _{1}-1)(p_{2}+\beta _{2}-1)}, & \text{ when } \gamma _{1}<\gamma _{2}; \\ \frac{C_{\chi _{1}}(p_{1}-1)}{p_{1}+\beta _{1}+\gamma _{2}-1}+\frac{C_{\chi _{2}} \gamma _{1}(p_{2}-1)}{(p_{2}+\beta _{2}+\gamma _{1}-1)(p_{2}+\beta _{2}-1)}+\frac{C_{\chi _{1}}(p_{1}-1)}{(p_{1}+\beta _{1}-1)}, & \text{ when } \gamma _{1}=\gamma _{2};\\ \frac{C_{\chi _{1}}(p_{1}-1)}{(p_{1}+\beta _{1}-1)}, & \text{ when } \gamma _{1}>\gamma _{2},\end{array}\right. } \end{aligned}$$
(4.13)

and

$$\begin{aligned} \tilde{\iota _{2}}={\left\{ \begin{array}{ll}\frac{C_{\chi _{2}}(p_{2}-1)}{p_{2}+\beta _{2} +\gamma _{1}-1}+\frac{C_{\chi _{1}}\gamma _{2}(p_{1}-1)}{(p_{1}+\beta _{1}+\gamma _{2}-1)(p_{1} +\beta _{1}-1)}, & \text{ when } \gamma _{1}>\gamma _{2}; \\ \frac{C_{\chi _{2}}(p_{2}-1)}{p_{2}+\beta _{2}+\gamma _{1}-1}+\frac{C_{\chi _{1}}\gamma _{2}(p_{1}-1)}{(p_{1} +\beta _{1}+\gamma _{2}-1)(p_{1}+\beta _{1}-1)}+\frac{C_{\chi _{2}}(p_{2}-1)}{(p_{2}+\beta _{2}-1)}, & \text{ when } \gamma _{1}=\gamma _{2}; \\ \frac{C_{\chi _{2}}(p_{2}-1)}{(p_{2}+\beta _{2}-1)}, & \text{ when } \gamma _{1}<\gamma _{2},\end{array}\right. } \end{aligned}$$
(4.14)

then we can divide the three cases (a) \(\gamma _{1}<\gamma _{2}\), (b) \(\gamma _{1}=\gamma _{2}\) and (c) \(\gamma _{1}>\gamma _{2}\) to deal with \(\varrho _{1}\), and \(\varrho _{2}\) is treated in a similar way. Here we only give the proof of case (a).

If \(\gamma _{1}<\gamma _{2}\), then \(\gamma =\gamma _{2}\), it follows from \(\lim _{s \rightarrow \infty } \inf \left\{ -\frac{f_{1}(s)}{s^{\beta _{1}+\gamma }}\right\} =: \iota _{1} \in \left( \tilde{\iota _{1}}, \infty \right] \) for \(\tilde{\iota _{1}}=\frac{C_{\chi _{1}}(p_{1}-1)}{p_{1}+\beta _{1} +\gamma _{2}-1}+\frac{C_{\chi _{2}}\gamma _{1}(p_{2}-1)}{(p_{2}+\beta _{2}+\gamma _{1}-1)(p_{2}+\beta _{2}-1)}\) that

$$\begin{aligned} \begin{aligned} \lim _{s \rightarrow \infty } \inf \frac{1}{(s+1)^{p_{1}+\beta _{1} +\gamma -1}} \varrho _1(s)&=\frac{C_{\chi _{1}}(p_{1}-1)}{p_{1}+\beta _{1} +\gamma _{2}-1}+\frac{C_{\chi _{2}}\gamma _{1}(p_{2}-1)}{(p_{2}+\beta _{2}+\gamma _{1}-1)(p_{2}+\beta _{2}-1)}\\&\;\;+\frac{C_{\chi _{1}}(p_{1}-1)}{(p_{1}+\beta _{1}-1)} \lim _{s \rightarrow \infty } \inf (s+1)^{\gamma _{1}-\gamma _{2}}-\iota _{1} \\&<0, \end{aligned} \end{aligned}$$
(4.15)

so that

$$\begin{aligned} \exists {\bar{s}}_2>0 \text{, } \text{ s.t. } \varrho _1(s)<0 \text{ for } \text{ all } s>{\bar{s}}_2 \text{. } \end{aligned}$$

Therefore, we further have

$$\begin{aligned} \int \limits _{\Omega }\varrho _1(u)(\cdot , \tau )= \int \limits _{u\le {\bar{s}}_2}\varrho _1(u)(\cdot , \tau )+ \int \limits _{u>{\bar{s}}_2}\varrho _1(u)(\cdot , \tau )\le \sup _{0<s\le {\bar{s}}_2}\varrho _1(s)|\Omega |<\infty . \end{aligned}$$
(4.16)

In this similar way, we can also get

$$\begin{aligned} \int \limits _{\Omega }\varrho _1(u)(\cdot , \tau )<\infty \quad \text { when case } (b), (c) \end{aligned}$$
(4.17)

and

$$\begin{aligned} \int \limits _{\Omega }\varrho _2(v)(\cdot , \tau )<\infty \quad \text { when case } (a), (b), (c) \end{aligned}$$
(4.18)

under the condition \(\lim _{s \rightarrow \infty } \inf \left\{ -\frac{f_{i}(s)}{s^{ \beta _{i}+\gamma }}\right\} =: \iota _{i} \in \left( \tilde{\iota _{i}}, \infty \right] \) by the definition of \(\tilde{\iota _{i}}\) in (4.13) and (4.14) with \(i=1,2\). Combining this with (4.10), we obtain the existence of positive constant \(C_{14}\) such that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t} \left\{ \frac{1}{p_{1}}\int \limits _{\Omega }(u+1)^{p_{1}}+ \frac{1}{p_{2}}\int \limits _{\Omega }(v+1)^{p_{2}}\right\} +\int \limits _{\Omega }(u+1)^{\frac{p_{1}}{2}}+ \int \limits _{\Omega }(v+1)^{\frac{p_{2}}{2}} \le C_{14} \end{aligned}$$
(4.19)

for all \(t\in (0, T_{\max })\). This along with the Lemma 5.1 in Chapter III of [30], we obtain the existence of positive constant \(C_{15}\) such that

$$\begin{aligned} \frac{1}{p_{1}}\int \limits _{\Omega }(u+1)^{p_{1}}(\cdot , t)+\frac{1}{p_{2}}\int \limits _{\Omega }(v+1)^{p_{2}}(\cdot , t) \le C_{15} \end{aligned}$$
(4.20)

for all \(t\in (0, T_{\max })\). The proof of Lemma 4.2 is complete. \(\square \)

Remark 4.1

When \(\beta _{i}\ge 0\) with \(i=1,2\), for any \(p_{i}>0\), it is interesting that (4.13) and (4.14) can be simplified to

$$\begin{aligned} \tilde{\iota _{1}}={\left\{ \begin{array}{ll}C_{\chi _{1}}, & \text{ when } \gamma _{1}<\gamma _{2}; \\ 2C_{\chi _{1}}, & \text{ when } \gamma _{1}=\gamma _{2}; \\ C_{\chi _{1}}, & \text{ when } \gamma _{1}>\gamma _{2},\end{array}\right. }\quad \text { and }\quad \tilde{\iota _{2}}={\left\{ \begin{array}{ll}C_{\chi _{2}}, & \text{ when } \gamma _{1}>\gamma _{2}; \\ 2C_{\chi _{2}}, & \text{ when } \gamma _{1}=\gamma _{2}; \\ C_{\chi _{2}}, & \text{ when } \gamma _{1}<\gamma _{2},\end{array}\right. } \end{aligned}$$
(4.21)

due to \(p_{1}+\beta _{1}+\gamma _{2}=p_{2}+\beta _{2}+\gamma _{1}\), and the condition (ii) in Theorem 1.2 can be changed into \(\lim _{s \rightarrow \infty } \inf \left\{ -\frac{f_{i}(s)}{s^{ \beta _{i}+\gamma }}\right\} =: \iota _{i} \in \left[ \tilde{\iota _{i}}, \infty \right] \). In the other hand, when \(\beta _{i}<0\), since the continuity and

$$\begin{aligned} \lim _{p_{1}, p_{2}\rightarrow \infty }\left\{ \frac{C_{\chi _{1}}(p_{1}-1)}{p_{1}+\beta _{1}+\gamma _{2}-1} +\frac{C_{\chi _{2}}\gamma _{1}(p_{2}-1)}{(p_{2}+\beta _{2}+\gamma _{1}-1)(p_{2} +\beta _{2}-1)}\right\} =C_{\chi _{1}}, \\ \lim _{p_{1}\rightarrow \infty }\left\{ \frac{C_{\chi _{1}}(p_{1}-1)}{(p_{1} +\beta _{1}-1)}\right\} =C_{\chi _{1}}, \\ \lim _{p_{1}, p_{2}\rightarrow \infty }\left\{ \frac{C_{\chi _{2}}(p_{2}-1)}{p_{2} +\beta _{2}+\gamma _{1}-1}+\frac{C_{\chi _{1}}\gamma _{2}(p_{1}-1)}{(p_{1}+\beta _{1} +\gamma _{2}-1)(p_{1}+\beta _{1}-1)}\right\} =C_{\chi _{2}}, \end{aligned}$$

and

$$\begin{aligned} \lim _{ p_{2}\rightarrow \infty }\left\{ \frac{C_{\chi _{2}}(p_{2}-1)}{(p_{2}+\beta _{2}-1)}\right\} =C_{\chi _{2}}, \end{aligned}$$

(4.13) and (4.14) can also be simplified to (4.21) such that the condition (ii) in Theorem 1.2 is reasonable for sufficiently large \(p_{1}, p_{2}\).

By Lemma 4.2, we can complete the proof of Theorem 1.2.

Proof of Theorem 1.2

applying the elliptic estimate to the third equation in (1.3) and combining Sobolev embedding theorem, we obtain

$$\begin{aligned} \Vert w(\cdot , t)\Vert _{W^{1, \infty }(\Omega )} \le C_{16}\Vert w(\cdot , t)\Vert _{W^{2, p}(\Omega )} \le C_{17}\Vert u(\cdot , t)+v(\cdot , t)\Vert _{L^p(\Omega )} \le C_{18} \end{aligned}$$
(4.22)

for some positive constants \(C_{16}, C_{17}, C_{18}\). Then, using Moser iteration technique (Appendix A of [27]), there exists a constant \(K_{5}>0\) independent of t such that

$$\begin{aligned} \Vert u(\cdot , t)\Vert _{L^{\infty }}+\Vert v(\cdot , t)\Vert _{L^{\infty }}+\Vert w(\cdot , t)\Vert _{W^{1,\infty }} \le K_{5}, \end{aligned}$$

which together with the extension criterion in Lemma 2.1 proves Theorem 1.2.

5 Asymptotic behavior

In this section, we show the asymptotic behavior of solutions by constructing the proper Lyapunov functionals under some assumptions to prove Theorems 1.3 and 1.4, separately. To achieve our goals and apart from constructing the energy functionals, we first give the following key lemma

Lemma 5.1

(see [2]) Let (u, v, w) be a nonnegative global bounded classical solution of (1.3) with initial data \(\left( u_0, v_0, w_0\right) \) satisfying (1.4). Then there exist \(\theta \in (0,1)\) and \(K_{6}>0\) such that

$$\begin{aligned} \Vert u\Vert _{C^{2+\theta , 1+\frac{\theta }{2}}({\bar{\Omega }} \times [t, t+1])}+\Vert v\Vert _{C^{2+\theta , 1+\frac{\theta }{2}}({\bar{\Omega }} \times [t, t+1])}+\Vert w\Vert _{C^{2+\theta , 1+\frac{\theta }{2}}({\bar{\Omega }} \times [t, t+1])} \le K_{6} \quad \text{ for } \text{ all } t \ge 1. \end{aligned}$$

In addition, Let \(f(t):(1, \infty ) \rightarrow {\mathbb {R}}\) be a nonnegative and uniformly continuous function that satisfies \(\int \limits _1^{\infty } f(t) d t<\infty \). Then, \(f(t) \rightarrow 0\) as \(t \rightarrow \infty \).

5.1 Proof of Theorem 1.3

Lemma 5.2

Let \(\tau =1\) and (u, v, w) is global bounded solution of (1.3). Suppose that \(\rho _1, \rho _2>0\) are arbitrary, then the function

$$\begin{aligned} E_{1}(t):=\int \limits _{\Omega } u+\rho _1 \int \limits _{\Omega }(v-1-\ln v)+\frac{\rho _2}{2} \int \limits _{\Omega }(w-1)^2 \end{aligned}$$

satisfies

$$\begin{aligned} \begin{aligned}&\frac{\textrm{d}}{\textrm{d} t} E_{1}(t)\\&\quad \le -\left( \mu _{1}-\rho _1\mu _{2}\right) \int \limits _{\Omega } u^2-\left( \mu _{1}a_{1}+\rho _1\mu _{2}a_{2}-2\rho _1\mu _{2}\right) \int \limits _{\Omega } u v -\left( 2\rho _{1}\mu _{2}-\mu _{1}-\rho _1\mu _{2}a_{2}\right) \int \limits _{\Omega }u\\&\qquad -\left( 2\rho _1\mu _{2}-\frac{k_{2}\rho _1 C_{\chi _{2}}^2}{4}\right) \int \limits _{\Omega }|\nabla w|^2- \rho _1\mu _{2} \int \limits _{\Omega }\left( u+v-w\right) ^2 -\rho _1\mu _{2}\int \limits _{\Omega }(w-1)^2 \end{aligned} \end{aligned}$$
(5.1)

for all \(t>0\) when \(\rho _2=2\rho _1\mu _{2}\), where \(k_{2}\) is defined in (1.5).

Proof

Let A(t), B(t) and D(t) be defined as

$$\begin{aligned} A_{1}(t):=\int \limits _{\Omega } u, \quad B_{1}(t):=\int \limits _{\Omega }(v-1-\ln v) \quad \text{ and } \quad G_{1}(t):=\frac{1}{2} \int \limits _{\Omega }(w-1)^2. \end{aligned}$$

From the first equation in (1.3) and \((S_{6})\), by a straightforward calculation that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t} A_{1}(t)=-\mu _1 \int \limits _{\Omega } u^2+\mu _1 \int \limits _{\Omega } u\left( 1-a_1 v\right) \end{aligned}$$
(5.2)

and

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t}G_{1}(t)=- \int \limits _{\Omega } \left| \nabla w\right| ^{2}+\int \limits _{\Omega }\left( w-1\right) \left( u+v-w\right) . \end{aligned}$$
(5.3)

Together with \((S_{1})\), \((S_{2})\) and \((S_{6})\), we obtain

$$\begin{aligned} \begin{aligned} \frac{\textrm{d}}{\textrm{d} t}B_{1}(t)&=- \int \limits _{\Omega }D_{2}(v) \frac{|\nabla v|^{2}}{v^{2}}+\int \limits _{\Omega } \frac{S_{2}(v)}{v^{2}} \nabla v \cdot \nabla w+\mu _{2}\int \limits _{\Omega }\left( v-1\right) \left( 1-v-a_{2}u\right) \\&\le \frac{C_{\chi _{2}}^{2}}{4}k_{2} \int \limits _{\Omega } \left| \nabla w\right| ^{2}+\mu _{2}\int \limits _{\Omega }\left( v-1\right) \left( 1-v-a_{2}u\right) \end{aligned} \end{aligned}$$
(5.4)

by using Young’s inequality and the definition of \(k_{2}\) in (1.5). For arbitrary \(\rho _1, \rho _2>0\), a direct linear combination (5.2)\(+\rho _2 \times \) (5.3)\(+\rho _1 \times \) (5.4), the consequence thus obtained then reads

$$\begin{aligned} \begin{aligned} \frac{\textrm{d}}{\textrm{d} t} E_{1}(t) \le&-\left( \mu _{1}-\frac{\rho _2}{2}\right) \int \limits _{\Omega } u^2-\left( \mu _{1}a_{1}+\rho _1\mu _{2}a_{2}-\rho _2\right) \int \limits _{\Omega } u v -\left( \rho _2-\frac{k_{2}\rho _1 C_{\chi _{2}}^2}{4}\right) \int \limits _{\Omega }|\nabla w|^2 \\&\quad - \int \limits _{\Omega }\left( \frac{\rho _{2}}{2}u^{2}+\rho _1\mu _{2}v^{2}+\frac{\rho _{2}}{2}w^{2}-\rho _{2}uw-\rho _{2}vw+\rho _{2}uv\right) -\int \limits _{\Omega }(\frac{\rho _{2}}{2}w^{2}-\rho _{2}w+p_{1}\mu _{2})\\&\quad -\left( 2\rho _1\mu _{2}-\rho _{2}\right) \int \limits _{\Omega }v-\left( \rho _{2}-\mu _{1}-\rho _1\mu _{2}a_{2}\right) \int \limits _{\Omega }u. \end{aligned} \end{aligned}$$

Then, let \(\rho _{2}=2\rho _{1}\mu _{2}\), it immediately complete the proof of this lemma. \(\square \)

Next we will use the energy functional constructed in Lemma 5.2 to obtain the large time behavior of solutions to (1.3) when \(a_1\ge 1>a_2>0\).

Lemma 5.3

Suppose that the conditions in Theorem 1.3 hold and \(\tau =1\), then there exist some \(\rho _{1}, \sigma _1>0\) such that the functions

$$\begin{aligned} E_{1}(t):=\int \limits _{\Omega } u+\rho _1 \int \limits _{\Omega }(v-1-\ln v)+\rho _{1}\mu _{2} \int \limits _{\Omega }(w-1)^2 \end{aligned}$$

and

$$\begin{aligned} F_1(t):=\int \limits _{\Omega } u^2+\int \limits _{\Omega }\left( v+u-w\right) ^2+\int \limits _{\Omega }(w-1)^2 \end{aligned}$$

fulfill

$$\begin{aligned} E_1^{\prime }(t) \le -\sigma _1 F_1(t) \quad \text{ for } \text{ all } t>0. \end{aligned}$$
(5.5)

Proof

Since \(a_{1}\ge 1\) and \(a_{2}<1\), we have

$$\begin{aligned} \frac{\mu _{1}}{\mu _{2}(2-a_{2})}\le \frac{\mu _{1}a_{1}}{\mu _{2}(2-a_{2})} \text { and } \frac{\mu _{1}}{\mu _{2}(2-a_{2})}<\frac{\mu _{1}}{\mu _{2}}, \end{aligned}$$
(5.6)

which ensures the existence of \(\rho _{1}\) satisfying

$$\begin{aligned} \rho _{1}<\frac{\mu _{1}}{\mu _{2}} \text { and } \rho _{1}\in \left[ \frac{\mu _{1}}{\mu _{2}(2-a_{2})},\frac{\mu _{1}a_{1}}{\mu _{2}(2-a_{2})} \right] \end{aligned}$$
(5.7)

such that \(\mu _{1}-\rho _1\mu _{2}>0\) and \(\min \left\{ \mu _{1}a_{1}+\rho _1\mu _{2}a_{2}-2\rho _1\mu _{2}, 2\rho _{1}\mu _{2}-\mu _{1}-\rho _1\mu _{2}a_{2}\right\} \ge 0\). Therefore, by choosing suitable \(\rho _{1}\) and combining (1.6), we have

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t} E_{1}(t) \le -\left( \mu _{1}-\rho _1\mu _{2}\right) \int \limits _{\Omega } u^2- \rho _1\mu _{2} \int \limits _{\Omega }\left( u+v-w\right) ^2 -\rho _1\mu _{2}\int \limits _{\Omega }(w-1)^2. \end{aligned}$$
(5.8)

\(\square \)

Lemma 5.4

Suppose that the conditions in Theorem 1.3 hold and \(\tau =0\), \(\tau =1\), then there exist some \(\rho _{1}, \sigma _2>0\) such that the function

$$\begin{aligned} E_{2}(t):=\int \limits _{\Omega } u+\rho _1 \int \limits _{\Omega }(v-1-\ln v) \end{aligned}$$

fulfills

$$\begin{aligned} E_2^{\prime }(t) \le -\sigma _2 F_1(t) \quad \text{ for } \text{ all } t>0, \end{aligned}$$
(5.9)

where \(F_{1}(t)\) is same in Lemma 5.3.

Proof

Testing the third equation in (1.3) by \(w-1\), we deduce that

$$\begin{aligned} 0=-\int \limits _{\Omega } \left| \nabla w\right| ^{2}+\int \limits _{\Omega }\left( w-1\right) \left( u+v-w\right) , \end{aligned}$$
(5.10)

then by a direct linear combination (5.2)\(+2\rho _{1}\mu _{2} \times \)(5.10) \(+\rho _1 \times \) (5.4) and choosing the \(\rho _{1}\) satisfying (5.6), we can complete the proof of Lemma 5.4 through the same steps as in Lemma 5.3. \(\square \)

Proof of Theorem 1.3

Because of \(s-1-\ln s \ge 0\) for all \(s>0\), \(E_{1}(t), E_{2}(t)\) are nonnegative. Integrating (5.5) or (5.9) over (0, t) and letting \(t \rightarrow \infty \), we obtain

$$\begin{aligned} \int \limits _0^{\infty } \int \limits _{\Omega } u^2+\int \limits _0^{\infty } \int \limits _{\Omega }\left( v+u-w\right) ^2+\int \limits _0^{\infty } \int \limits _{\Omega }(w-1)^2 \le K_{7} \end{aligned}$$

with some \(K_{7}>0\). Therefore, combining this with Lemma 5.1 then yields

$$\begin{aligned} \Vert u(\cdot , t)\Vert _{L^2(\Omega )}+\left\| v(\cdot , t)+u(\cdot , t)-w(\cdot , t)\right\| _{L^2(\Omega )}+\Vert w(\cdot , t)-1\Vert _{L^2(\Omega )} \rightarrow 0 \quad \text{ as } t \rightarrow \infty , \end{aligned}$$

this implies

$$\begin{aligned} \Vert u(\cdot , t)\Vert _{L^2(\Omega )}+\Vert v(\cdot , t)-1\Vert _{L^2(\Omega )}+\Vert w(\cdot , t)-1\Vert _{L^2(\Omega )} \rightarrow 0 \quad \text{ as } t \rightarrow \infty . \end{aligned}$$

Invoking the Gagliardo-Nirenberg inequality to find \(K_{8}>0\) fulfilling

$$\begin{aligned} \Vert \varphi \Vert _{L^{\infty }(\Omega )} \le K_{8}\Vert \varphi \Vert _{W^{1, \infty }(\Omega )}^{n /(n+2)}\Vert \varphi \Vert _{L^2(\Omega )}^{2 /(n+2)} \quad \text{ for } \text{ all } \varphi \in W^{1, \infty }(\Omega ). \end{aligned}$$

Applying the above Gagliardo-Nirenberg inequality to u, v, w for \(t>0\), and using Lemma 5.1, we conclude that

$$\begin{aligned} \Vert u(\cdot , t)\Vert _{L^{\infty }(\Omega )}+\Vert v(\cdot , t)-1\Vert _{L^{\infty }(\Omega )}+\Vert w(\cdot , t)-1\Vert _{L^{\infty }(\Omega )} \rightarrow 0 \quad \text{ as } t \rightarrow \infty . \end{aligned}$$

Hence the proof of Theorem 1.3 is completed.

5.2 Proof of Theorem 1.4

Lemma 5.5

Let \(\tau =1\) and (u, v, w) is global bounded solution of (1.3). Suppose that \(\varsigma _1, \varsigma _2>0\) are arbitrary, then the function

$$\begin{aligned} E_{3}(t):=\int \limits _{\Omega } (u-u_{\star }-u_{\star }\ln \frac{u}{u_{\star }})+\varsigma _1 \int \limits _{\Omega }(v-v_{\star }-v_{\star }\ln \frac{v}{v_{\star }})+\frac{\varsigma _2}{2} \int \limits _{\Omega }(w-w_{\star })^2 \end{aligned}$$

satisfies

$$\begin{aligned} \begin{aligned} \frac{\textrm{d}}{\textrm{d} t} E_{3}(t)&\le -\left( \mu _{1}-\varsigma _1\mu _{2}\right) \int \limits _{\Omega } (u-u_{\star })^2-\left( \mu _{1}a_{1}+\varsigma _1\mu _{2}a_{2}-2\varsigma _1\mu _{2}\right) \int \limits _{\Omega } (u-u_{\star }) (v-v_{\star })\\&\quad -\left( 2\varsigma _1\mu _{2}-\frac{k_{1}u_{\star } C_{\chi _{1}}^2}{4}-\varsigma _1\frac{k_{2}v_{\star } C_{\chi _{2}}^2}{4}\right) \int \limits _{\Omega }|\nabla w|^2- \varsigma _1\mu _{2} \int \limits _{\Omega }\left( u+v-w\right) ^2 \\&\quad -\varsigma _1\mu _{2}\int \limits _{\Omega }(w-w_{\star })^2 \end{aligned} \end{aligned}$$
(5.11)

for all \(t>0\) when \(\varsigma _2=2\varsigma _1\mu _{2}\), where \(k_{1}, k_{2}\) are defined in (1.5).

Proof

Let \(A_{2}(t), B_{2}(t)\) and \(G_{2}(t)\) be defined as

$$\begin{aligned} A_{2}(t):=\int \limits _{\Omega } (u-u_{\star }-u_{\star }\ln \frac{u}{u_{\star }}), \quad B_{2}(t):= \int \limits _{\Omega }(v-v_{\star }-v_{\star }\ln \frac{v}{v_{\star }}) \quad \text{ and } \quad G_{2}(t):=\frac{1}{2} \int \limits _{\Omega }(w-w_{\star })^2. \end{aligned}$$

Together with \((S_{1})\), \((S_{2})\) and \((S_{6})\), by using Young’s inequality and the definitions of \(k_{1}, k_{2}\) in (1.5) that we obtain

$$\begin{aligned} \begin{aligned} \frac{\textrm{d}}{\textrm{d} t}A_{2}(t)&=-u_{\star } \int \limits _{\Omega }D_{1}(u) \frac{|\nabla u|^{2}}{u^{2}}+\int \limits _{\Omega } \frac{u_{\star }S_{1}(u)}{u^{2}} \nabla u \cdot \nabla w+\mu _{1}\int \limits _{\Omega }\left( u-u_{\star }\right) \left( 1-u-a_{1}v\right) \\&\le \frac{C_{\chi _{1}}^{2}}{4}k_{1}u_{\star } \int \limits _{\Omega } \left| \nabla w\right| ^{2}-\mu _{1}\int \limits _{\Omega }\left( u-u_{\star }\right) ^{2} -a_{1}\mu _{1}\int \limits _{\Omega }\left( u-u_{\star }\right) \left( v-v_{\star }\right) \end{aligned} \end{aligned}$$
(5.12)

and

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t}B_{2}(t)\le \frac{C_{\chi _{2}}^{2}}{4}k_{2}v_{\star } \int \limits _{\Omega } \left| \nabla w\right| ^{2}-\mu _{2}\int \limits _{\Omega }\left( v-v_{\star }\right) ^{2} -a_{2}\mu _{2}\int \limits _{\Omega }\left( v-v_{\star }\right) \left( u-u_{\star }\right) \end{aligned}$$
(5.13)

as well as

$$\begin{aligned} \begin{aligned} \frac{\textrm{d}}{\textrm{d} t}G_{2}(t)&=- \int \limits _{\Omega } \left| \nabla w\right| ^{2}+\int \limits _{\Omega }\left( w-w_{\star }\right) \left( u+v-w\right) \\&=- \int \limits _{\Omega } \left| \nabla w\right| ^{2}-\int \limits _{\Omega }\left( w-w_{\star }\right) ^{2} +\int \limits _{\Omega }\left( w-w_{\star }\right) \left( u-u_{\star }\right) +\int \limits _{\Omega }\left( w-w_{\star }\right) \left( v-v_{\star }\right) . \end{aligned} \end{aligned}$$
(5.14)

For some \(\varsigma _{1}, \varsigma _2>0\), a direct linear combination (5.12)\(+\varsigma _1 \times \) (5.13)\(+\varsigma _2 \times \)(5.14) and let \(\varsigma _2=2\varsigma _1\mu _{2}\), it immediately obtain (5.11) due to \(u_{\star }+v_{\star }-w_{\star }=0\). \(\square \)

Next we will use the energy functional constructed in Lemma 5.5 to obtain the large time behavior of solutions to (1.3) for \(a_{1}, a_{2}<1\).

Lemma 5.6

Suppose that the conditions in Theorem 1.4 hold and \(\tau =1\), let \(\varsigma _1:=\frac{a_{1}\mu _{1}}{\mu _{2}(2-a_{2})}\), then there exists \(\sigma _3>0\) such that the functions

$$\begin{aligned} E_{3}(t):=\int \limits _{\Omega } (u-u_{\star }-u_{\star }\ln \frac{u}{u_{\star }})+\varsigma _1 \int \limits _{\Omega }(v-v_{\star }-v_{\star }\ln \frac{v}{v_{\star }})+\varsigma _1\mu _{2} \int \limits _{\Omega }(w-w_{\star })^2 \end{aligned}$$

and

$$\begin{aligned} F_2(t):=\int \limits _{\Omega } (u-u_{\star })^2+\int \limits _{\Omega }\left( v+u-w\right) ^2+\int \limits _{\Omega }(w-w_{\star })^2 \end{aligned}$$

fulfill

$$\begin{aligned} E_3^{\prime }(t) \le -\sigma _3 F_2(t) \quad \text{ for } \text{ all } t>0. \end{aligned}$$
(5.15)

Proof

Since \(a_{1}, a_{2}<1\), we have

$$\begin{aligned} \varsigma _1=\frac{a_{1}\mu _{1}}{\mu _{2}(2-a_{2})}<\frac{\mu _{1}}{\mu _{2}}, \end{aligned}$$
(5.16)

which ensures that \(\mu _{1}-\varsigma _1\mu _{2}>0\) and \(\mu _{1}a_{1}+\varsigma _1\mu _{2}a_{2}-2\varsigma _1\mu _{2}=0\). Therefore, by choosing this \(\varsigma _1\) and combining (1.7), we have

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d} t} E_{3}(t) \le -\left( \mu _{1}-\varsigma _1\mu _{2}\right) \int \limits _{\Omega } (u-u_{\star })^2- \varsigma _1\mu _{2} \int \limits _{\Omega }\left( u+v-w\right) ^2 -\varsigma _1\mu _{2}\int \limits _{\Omega }(w-w_{\star })^2. \end{aligned}$$
(5.17)

\(\square \)

Lemma 5.7

Suppose that the condition in Theorem 1.4 hold and \(\tau =0\), let \(\varsigma _1:=\frac{a_{1}\mu _{1}}{\mu _{2}(2-a_{2})}\), then there exists \(\sigma _4>0\) such that the function

$$\begin{aligned} E_{4}(t):=\int \limits _{\Omega } (u-u_{\star }-u_{\star }\ln \frac{u}{u_{\star }})+\varsigma _1 \int \limits _{\Omega }(v-v_{\star }-v_{\star }\ln \frac{v}{v_{\star }}) \end{aligned}$$

fulfills

$$\begin{aligned} E_4^{\prime }(t) \le -\sigma _4 F_2(t) \quad \text{ for } \text{ all } t>0, \end{aligned}$$
(5.18)

where \(F_{2}(t)\) is same in Lemma 5.6.

Proof

Testing the third equation in (1.3) by \(w-w_{\star }\), we deduce that

$$\begin{aligned} 0=-\int \limits _{\Omega } \left| \nabla w\right| ^{2}+\int \limits _{\Omega }\left( w-w_{\star }\right) \left( u+v-w\right) , \end{aligned}$$
(5.19)

then by a direct linear combination (5.12)\(+\varsigma _{1} \times \) (5.13) \(+2\varsigma _{1}\mu _{2}\times \)(5.19), we can complete the proof of Lemma 5.7 through the same steps as in Lemma 5.6. \(\square \)

We are now in the position to prove our main result.

Proof of Theorem 1.4

Firstly, we show the nonnegativity of \(E_3(t), E_4(t)\). Let \(y(s):=s-u_* \ln s\) for \(s>0\). By applying Taylor’s formula, there exists \(\sigma \in (0,1)\) such that

$$\begin{aligned} \begin{aligned} y(u)-y\left( u_*\right)&=y^{\prime }\left( u_*\right) \cdot \left( u-u_*\right) +\frac{1}{2} y^{\prime \prime }\left[ \sigma u+(1-\sigma ) u_*\right] \cdot \left( u-u_*\right) ^2 \\&=\frac{u_*}{2\left[ \sigma u+(1-\sigma ) u_*\right] ^2}\left( u-u_*\right) ^2 \ge 0 \end{aligned} \end{aligned}$$

for \(x \in \Omega \) and \(t>0\), which implies that \(A_{2}(t)=\int \limits _{\Omega }\left( y(u)-y\left( u_*\right) \right) \ge 0\). Similarly, we can obtain \(B_{2}(t) \ge 0\) for all \(t \ge 0\). Thus, \(E_3(t)\) and \(E_4(t)\) are nonnegative. Integrating (5.15) or (5.18) over (0, t) and letting \(t \rightarrow \infty \), we obtain

$$\begin{aligned} \int \limits _0^{\infty } \int \limits _{\Omega } (u-u_{\star })^2+\int \limits _0^{\infty } \int \limits _{\Omega }\left( v+u-w\right) ^2+\int \limits _0^{\infty } \int \limits _{\Omega }(w-w_{\star })^2 \le K_{8} \end{aligned}$$

with some \(K_{8}>0\). combining this with Lemma 5.1 then yields

$$\begin{aligned} \Vert u(\cdot , t)-u_{\star }\Vert _{L^2(\Omega )}+\left\| v(\cdot , t)+u(\cdot , t)-w(\cdot , t)\right\| _{L^2(\Omega )}+\Vert w(\cdot , t)-w_{\star }\Vert _{L^2(\Omega )} \rightarrow 0 \quad \text{ as } t \rightarrow \infty , \end{aligned}$$

this implies

$$\begin{aligned} \Vert u(\cdot , t)-u_{\star }\Vert _{L^2(\Omega )}+\Vert v(\cdot , t)-v_{\star }\Vert _{L^2(\Omega )}+\Vert w(\cdot , t)-w_{\star }\Vert _{L^2(\Omega )} \rightarrow 0 \quad \text{ as } t \rightarrow \infty . \end{aligned}$$

Applying the above Gagliardo-Nirenberg inequality to u, v, w for \(t>0\), and using Lemma 5.1, we conclude that

$$\begin{aligned} \Vert u(\cdot , t)\Vert _{L^{\infty }(\Omega )}+\Vert v(\cdot , t)-1\Vert _{L^{\infty }(\Omega )}+\Vert w(\cdot , t)-1\Vert _{L^{\infty }(\Omega )} \rightarrow 0 \quad \text{ as } t \rightarrow \infty . \end{aligned}$$

Hence the proof of Theorem 1.4 is completed.