1 Introduction

Nature exhibits a captivating interaction amongst various populations, characterized by mutual reliance and restrictions that facilitate survival and propagation. The pioneering work of Lotka and Volterra in the 1920s not only introduced the first predator–prey differential equation model but also made strides towards deciphering the ecological laws that govern nature [1]. It’s notable how predators and prey engage in an intricate pursuit–evasion relationship. Predators naturally move towards regions teeming with prey, whereas prey tactically shift to areas with fewer predators, effectively avoiding their pursuers. Their movements are thus not a matter of chance but are directional, constituting what we term as a chemotaxis phenomenon. Today, as we delve deeper into the complexity of biological phenomena, numerous modified models have been devised and studied by scholars worldwide. Among these models, special attention is placed on the description of cell motion, typically represented by a single operator like \(\Delta (\psi (\cdot ) u).\) Upon expanding the Laplace operator, it reveals that the nonlinear diffusion combines both diffusive and advective flux. This indicates that the density-dependent diffusion function \(\psi (\cdot )\) plays a dual role- adding to the cross-diffusion structure while opening up potential for diffusion degeneracy. Consequently, many conventional methods prove inadequate for analyzing such diffusion mechanisms, requiring a delicate approach to their investigation.

The current paper is devoted to the study of the boundedness of the following fully parabolic indirect pursuit–evasion predator–prey system with density-dependent diffusion

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}=\Delta (\psi _1(w)u)+u(\lambda -u+\alpha v), &{}\quad (x,t)\in \Omega \times (0,\infty ), \\ v_{t}=\Delta (\psi _2(z) v)+v(\mu -v-\beta u), &{}\quad (x,t)\in \Omega \times (0,\infty ),\\ w_{t}=\Delta w -w+v,&{}\quad (x,t)\in \Omega \times (0,\infty ),\\ z_{t}=\Delta z-z+u,&{} \quad (x,t)\in \Omega \times (0,\infty ),\\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }= \frac{\partial w}{\partial \nu }=\frac{\partial z}{\partial \nu }=0, &{}\quad (x,t)\in \partial \Omega \times (0,\infty ),\\ (u,v,w,z)(x,0)=(u_{0}(x),v_{0}(x),w_0(x),z_0(x)),&{}\quad x\in \Omega , \end{array}\right. \end{aligned}$$
(1)

where \(\Omega \subset {\mathbb{R}}^{2}\) denotes a bounded domain with smooth boundary \(\partial \Omega ,\) the parameter \(\lambda ,\mu , \alpha , \beta >0.\) The initial data \((u_{0},v_{0},w_0,z_0)\) satisfy

$$\begin{aligned} (u_{0},v_{0},w_0,z_0)\in C^0(\bar{\Omega })\times C^0(\bar{\Omega })\times W^{1,\infty }(\Omega )\times W^{1,\infty }(\Omega )\quad \text{with} \ u_{0},v_{0},w_0,z_0 \ge 0. \end{aligned}$$
(2)

Here the variables u(x, t) and v(x, t) represent the density of predators and prey, respectively. w(x, t) and z(x, t) denote concentrations of chemical attractants produced by prey and predators. The density-dependent diffusion functions \(\psi _1(w)\) and \(\psi _2(z)\) reflect the influence of the chemical attractant on the predator and prey, which satisfies

$$\begin{aligned} \psi _1(w)\in C^3([0,\infty )), \psi _1(w)>0, \psi ^{\prime }_1(w)<0,\quad \forall \ w\ge 0. \end{aligned}$$
(3)

and

$$\begin{aligned} \psi _2(z)\in C^3([0,\infty )), \psi _2(z)>0, \psi ^{\prime }_2(z)>0 \ \text{and}\ \psi ^{\prime \prime }_2(z)<0, \ \forall \ z\ge 0. \end{aligned}$$
(4)

The aforementioned conditions for the diffusion functions \(\psi _{1}(w)\) and \(\psi _{2}(z)\) imply a monotone property in the resource dependent dispersal rate of the involved species. Specifically, for predators, \(\psi _{1}^{\prime }(w)<0,\) indicating that their diffusion velocity decreases in the presence of chemoattractants secreted by prey. Conversely, prey exhibits increased motility as a survival strategy when faced with chemorepellents secreted by predators, which is reflected by the condition \(\psi _{2}^{\prime }(z)>0.\)

Now we are prepared to explaining the concept of indirect pursuit–evasion. In absence of deeply considering density-dependent diffusion in chemotaxis system, the resulting predator–prey system involving two species has been extensively studied in the literature [2,3,4]. A prototypical version of the pursuit–escape system [5] is given by

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}=D_1\Delta u-\chi \nabla \cdot (u\nabla v)+g(u,v),&{}\quad (x,t)\in \Omega \times (0,\infty ),\\ v_{t}=D_2\Delta v+\xi \nabla \cdot (v\nabla u)+h(u,v),&{}\quad (x,t)\in \Omega \times (0,\infty ),\end{array}\right. \end{aligned}$$

where u and v represent the population densities of predators and preys, respectively. The functions g(u, v) and h(u, v) capture the population interactions between the predator and the prey. The parameters \(D_{1},D_{2},\chi\) and \(\xi\) are assumed to be positive constants. To gain a comprehensive understanding of the pursuit–escape process, Tyutyunov et al. [6] proposed the following indirect pursuit–evasion interaction system

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}=\Delta u-\chi \nabla \cdot (u\nabla w)+g(u,v), &{}\quad (x,t)\in \Omega \times (0,\infty ), \\ v_{t}=\Delta v-\xi \nabla \cdot (v\nabla z)+h(u,v), &{}\quad (x,t)\in \Omega \times (0,\infty ),\\ \tau w_{t}=\Delta w-w+v, &{}\quad (x,t)\in \Omega \times (0,\infty ),\\ \tau z_{t}=\Delta z-z+u, &{}\quad (x,t)\in \Omega \times (0,\infty ), \end{array}\right. \end{aligned}$$
(5)

where \(\Omega \subset {\mathbb{R}}^n\) is a bounded domain with smooth boundary, \(\tau \in \{0,1\}\) and the parameters \(\chi\) and \(\xi\) are positive. In this ecosystem, predators and prey emit distinct signature substances, such as pheromones or odors, as a means of indicating their presence. Instead of directly gravitating towards areas with higher or lower concentrations of species, both predators and prey tactically maneuver along gradients of these specific chemicals secreted by each other. This chemical communication system guides their movements, enabling them to navigate their environment effectively. Considerable research exists surrounding the functions \(g(u,v) = u(\lambda -u+\alpha v),\) \(h(u,v) = v(\mu -v-\beta u),\) where \(\lambda ,\mu ,\alpha ,\beta >0.\) When \(\tau =0,\) Li et al. [7] utilize some prior estimators to obtain a unique globally bounded classical solution of the system (5) under the suitably regular initial data. They demonstrate that if \(\beta \lambda <\mu\) and \(\chi ,\xi\) are explicitly small, any nontrivial bounded classical solution will converge to a spatially homogeneous coexistence state. Moreover, if \(\beta \lambda >\mu ,\) any bounded classical solution (u, v) of (5) will tend to stability \((\lambda , 0)\) as \(t \rightarrow \infty .\) Additionally, when \(\tau =1\) and \(n\le 1,\) under the conditions that \(\chi , \xi\) are small enough and \(\alpha >2,\) asymptotic stabilization of globally bounded solution to the system (5) is well-established [8]. Other variants of predator–prey models for (5) have also been thoroughly analyzed as well [9, 10].

Based on the above analysis, we are in the position to recall some research related to the system (1) involving density-dependent motilities. The system (1) aims to analyze full parabolic cross-diffusion system in which the diffusion rate and the chemotactic sensitivity of predator and prey are nonlinearly dependent on the density-dependent diffusion functions of the respective produced signals. If the third and fourth equation of the system (1) are replaced by elliptic equations \(0=\Delta w-w+v\) and \(0=\Delta z-z+u,\) it has been established that the global solutions of this parabolic-elliptic density-dependent diffusion system are uniformly bounded with respect to time. Interested readers can refer to [11, 12] and the relevant literature therein. However, the existence and boundedness of global solutions for full parabolic predator–prey system (1) remain unclear. Throughout what follows, we will show that the above conclusion is still true under conditions (2) (3) and (4). More precisely, we have:

Theorem 1.1

Assuming \(\Omega \subset {\mathbb{R}}^2\) is a bounded domain with smooth boundary,  and the parameters \(\lambda ,\alpha ,\mu ,\beta >0,\) as well as the initial data \((u_{0},v_{0},w_{0},z_{0})\) satisfying (2), while the diffusion functions \(\psi _1(w)\) and \(\psi _2(z)\) satisfy (3) and (4) respectively, it can be concluded that the system (1) possesses a unique global classical solution (u, v, w, z). This solution is globally bounded, meaning there exists a constant \(C>0\) such that

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

for all \(t>0.\)

Remark 1.1

There exist density-dependent diffusion functions that fulfill the conditions given in (2) and (3). For instance, \(\psi _1(w)=e^{-2 w},\psi _1(w)=\frac{1}{(1+w)^2}\) and \(\psi _{2}(z)=\frac{3z+2}{z+4}, \psi _{2}(z)=\arctan z+1.\)

Remark 1.2

Theorem 1.1 indicates that the hypothesis of diffusion functions is sufficient to guarantee that the system (1) possesses a unique globally bounded classical solution in \({\mathbb{R}}^2.\) However, we encountered challenges in establishing the boundedness of the \(L^p\) norm for v if \(n>2.\) Specifically, dealing with terms such as \(\int _{\Omega } v^{p}\Delta z\) has proven to be a difficult task. Additionally, while using the Neumann heat semigroup on \(\Omega\) to get the desired result in Lemma 3.4, it doesn’t seem feasible to select appropriate parameters if \(n>2.\) As a result, demonstrating the global boundedness of solutions to the system (1) in higher dimensions remains an open problem.

2 Preliminaries

Firstly, the proof of local existence of solutions to the system (1) is similar to that in [11, 17], which is achieved by combining the parabolic regularity theory and the standard contraction mapping argument. Therefore, we give the following lemma directly.

Lemma 2.1

Let \(\Omega \subset {\mathbb{R}}^2\) be bounded domain with smooth boundary and assume that the density-dependent diffusion function \(\psi _1(w)\) and \(\psi _2(z)\) satisfying (3) and (4), respectively. \(\lambda ,\alpha ,\mu ,\beta >0.\) If the initial data \((u_{0},v_{0},w_{0},z_{0})\) satisfy condition (2), then there exist \(T_{\max }\in (0,\infty ]\) and a uniquely determined triple (u, v, w, z) of nonnegative functions \((u, v, w, z)\in \left( C^{0}\left( \bar{\Omega } \times \left[ 0, T_{\max }\right) \right) \cap C^{2,1}\left( \bar{\Omega } \times \left( 0, T_{\max }\right) \right) \right) ^4,\) which solve (1) in the classical sense in \(\Omega \times (0, T_{\max }).\) Moreover, if \(T_{\max }<\infty ,\) then

$$\begin{aligned} \Vert u(\cdot , t)\Vert _{L^{\infty }(\Omega )}+\Vert v(\cdot , t)\Vert _{L^{\infty }(\Omega )}+\Vert w(\cdot , t)\Vert _{W^{1,\infty }(\Omega )}+\Vert z(\cdot , t)\Vert _{W^{1,\infty }(\Omega )}\rightarrow \infty \quad \text{as } t\nearrow \infty . \end{aligned}$$
(6)

The next Gagliardo–Nirenberg inequality will enable us to derive the boundedness of \(L^{p}\) for u.

Lemma 2.2

[13] Let \(p\in (1,\infty ),\) \(r\in (0,p),\) and suppose \(\Omega \subset {\mathbb{R}}^{n}(n\ge 1)\) is a bounded domain with smooth boundary. Then, there exists a positive constant C such that

$$\begin{aligned} \Vert f\Vert _{L^{p}(\Omega )}\le C(\Vert \nabla f\Vert _{L^{2}(\Omega )}^{\lambda }\Vert f\Vert _{L^{r}(\Omega )}^{1-\lambda }+\Vert f\Vert _{L^{r}(\Omega )}) \end{aligned}$$

for any \(f\in W^{1,2}(\Omega )\cap L^{r}(\Omega ),\) where \(\lambda \in (0,1)\) and \(\lambda =\frac{\frac{n}{r}-\frac{n}{p}}{1-\frac{n}{2}+\frac{n}{r}}.\)

Next, we present a set of lemmas that serve as the foundation for the ODE comparison argument, which will be utilized in subsequent discussion.

Lemma 2.3

[14] Let \(\eta\) be a positive absolutely continuous function on \((0,\infty )\) that satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} \eta ^{\prime }(t)+\varepsilon \eta ^p\le \xi \\ \eta (0)= \eta _0 \end{array}\right. \end{aligned}$$

with some constants \(\varepsilon >0,\) \(\xi >0\) and \(p>0.\) Then for \(t>0,\) we have

$$\begin{aligned} \eta (t)\le \max \left\{ \eta _0, \left( \frac{\xi }{\varepsilon }\right) ^\frac{1}{p}\right\} . \end{aligned}$$

Lemma 2.4

[15] Let \(T>0,\) \(\tau \in (0,T)\) and assume that \(\eta\) is a nonnegative absolutely continuous function defined on the interval [0, T),  which satisfies the inequality

$$\begin{aligned} \eta ^{\prime }(t)+\delta \eta (t)\le h(t)\quad \text{for}\ a.e. \ t\in (0,T) \end{aligned}$$

with some \(\delta >0\) and a nonnegative function \(g\in C^{0}([0,T)).\) Additionally, suppose there exists \(\sigma >0\) such that

$$\begin{aligned} \int _{t}^{t+\tau }h(s)ds\le \sigma ,\quad \forall \ t\in [0,T-\tau ). \end{aligned}$$

Then

$$\begin{aligned} \eta (t)\le \max \left\{ \eta (0)+\sigma ,\frac{\sigma }{\delta }+2\sigma \right\} ,\quad \forall \ t\in (0,T). \end{aligned}$$

Lemma 2.5

[16] Let \(T>0,\) \(\tau \in (0,T)\) and assume that \(\eta\) is a nonnegative absolutely continuous function defined on the interval [0, T),  which satisfies the inequality

$$\begin{aligned} \eta ^{\prime }(t)+\delta (t)\eta (t)\le \sigma (t)\eta (t)+\gamma (t)\quad \text{for}\ a.e. \ t\in (0,T) \end{aligned}$$

where \(\delta (t)>0, \sigma (t)\ge 0, \gamma (t)\ge 0\) are functions in \(L_{loc}^{1}([0,T)).\) Furthermore, assume that there exist positive constants \(\sigma _1,\gamma _1,\zeta\) such that

$$\begin{aligned}{} & {} \sup \limits _{t\in (0,T)}\int _{t}^{t+\tau }\sigma (s)ds\le \sigma _1,\quad \forall \ t\in [0,T-\tau ),\\{} & {} \sup \limits _{t\in (0,T)}\int _{t}^{t+\tau }\gamma (s)ds\le \gamma _1,\quad \forall \ t\in [0,T-\tau ) \end{aligned}$$

and

$$\begin{aligned} \int _{t}^{t+\tau }\delta (s)ds-\int _{t}^{t+\tau }\sigma (s)ds>\zeta ,\quad \forall \ t\in [0,T-\tau ). \end{aligned}$$

Then

$$\begin{aligned} \eta (t)\le \eta (0)e^{\sigma _1}+\frac{\gamma _1e^{2\sigma _1}}{1-e^{-\zeta }}+\gamma _1e^{\sigma _1},\quad \forall \ t\in (0,T). \end{aligned}$$

3 Global Boundedness of Solutions

To establish the global boundedness of the solution to the system (1), we begin by establishing some prior estimates regarding \(L^1\) boundedness of the solution.

Lemma 3.1

For \(n\ge 1,\) there exists a constant \(K_1>0\) such that the solution of the system (1) exhibits the following properties

$$\begin{aligned}{} & {} \int _{\Omega }u(\cdot ,t)\le \frac{\alpha }{\beta }m_1,\quad \forall \ t\in (0,T_{max}), \end{aligned}$$
(7)
$$\begin{aligned}{} & {} \int _{\Omega }v(\cdot ,t)\le m_1,\quad \forall \ t\in (0,T_{max}), \end{aligned}$$
(8)
$$\begin{aligned}{} & {} \int _{\Omega }w(\cdot ,t)\le m_2,\quad \forall \ t\in (0,T_{max}) \end{aligned}$$
(9)

and

$$\begin{aligned} \int _{\Omega }z(\cdot ,t)\le m_3, \quad \forall \ t\in (0,T_{max}), \end{aligned}$$
(10)

as well as

$$\begin{aligned} \int _{t}^{t+\tau } \int _{\Omega }u^2+\int _{t}^{t+\tau } \int _{\Omega }v^2\le K_1,\quad \forall \ t\in (0,T_{max}-\tau ), \end{aligned}$$
(11)

where \(m_1:=\max \{\frac{\beta }{\alpha }\int _{\Omega }u_{0}+\int _{\Omega }v_{0}, \left( \frac{\beta (\lambda +1)^2}{2\alpha }+\frac{(\mu +1)^2}{2}\right) \vert \Omega \vert \},\) \(m_2:=\max \{\int _{\Omega }w_0, m_1\},\) \(m_3:=\max \{\int _{\Omega }z_0, \frac{\alpha }{\beta }m_1\},\) \(\tau :=\min \{1,\frac{T_{max}}{2}\}.\)

Proof

Integrating the first and second equation of (1), it follows that

$$\begin{aligned} \frac{d}{dt}\left\{ \frac{\beta }{\alpha }\int _{\Omega }u+\int _{\Omega }v\right\} =\frac{\beta }{\alpha }\lambda \int _{\Omega }u+\mu \int _{\Omega }v-\frac{\beta }{\alpha }\int _{\Omega }u^2-\int _{\Omega }v^2 \end{aligned}$$

for all \(t\in (0,T_{\max }).\) Employing Young’s inequality, we obtain

$$\begin{aligned} \frac{d}{dt}\int _{\Omega }\left( \frac{\beta }{\alpha }u+v\right) +\int _{\Omega }\left( \frac{\beta }{\alpha }u+v\right) \le -\frac{1}{2}\int _{\Omega }\left( \frac{\beta }{\alpha }u^2+v^2\right) +\left\{ \frac{\beta (\lambda +1)^2}{2\alpha }+\frac{(\mu +1)^2}{2}\right\} \vert \Omega \vert \end{aligned}$$
(12)

for all \(t\in (0,T_{\max }).\) By Lemma 2.3, it can be readily derived that Eqs. (7) and (8) hold. Next, by integrating (12) with respect to time from t to \(t+\tau ,\) it also readily derive (11) by taking advantage of the nonnegativity of u and v. Similarly, upon integrating the third and fourth equations of the system (1), we get

$$\begin{aligned} \frac{d}{dt}\int _{\Omega }w+\int _{\Omega }w=\int _{\Omega }v\le m_1 \end{aligned}$$

and

$$\begin{aligned} \frac{d}{dt}\int _{\Omega }z+\int _{\Omega }z=\int _{\Omega }u\le \frac{\alpha }{\beta }m_1, \end{aligned}$$

for all \(t\in (0,T_{\max }).\) Then, by applying Lemma 2.3 once again, we can easily deduce that Eqs. (9) and (10) hold. \(\square\)

We proceed to establish the estimates for \(\Vert \nabla w\Vert _{L^2(\Omega )}\) and \(\Vert \nabla z\Vert _{L^2(\Omega )}.\)

Lemma 3.2

For \(n\ge 1,\) there exists constants \(K_2, K_3>0\) such that the solution of the system (1) has the following estimates

$$\begin{aligned}{} & {} \int _{\Omega }\vert \nabla z\vert ^2\le K_2,\quad \forall \ t\in (0,T_{max}), \end{aligned}$$
(13)
$$\begin{aligned}{} & {} \int _{t}^{t+\tau }\int _{\Omega }\vert \Delta z\vert ^2\le K_2,\quad \forall \ t\in (0,T_{max}-\tau ) \end{aligned}$$
(14)

and

$$\begin{aligned} \int _{\Omega }\vert \nabla w\vert ^2\le K_3,\quad \forall \ t\in (0,T_{max}), \end{aligned}$$
(15)

where \(\tau :=\min \{1,\frac{T_{max}}{2}\}.\)

Proof

By multiplying the fourth equality of (1) by \(-\Delta z,\) applying the integration by parts, and utilizing Young’s inequality, we can obtain

$$\begin{aligned} \frac{1}{2}\frac{d}{d t}\int _{\Omega }\vert \nabla z\vert ^2&=-\int _{\Omega }\vert \Delta z\vert ^2-\int _{\Omega }\vert \nabla z\vert ^2-\int _{\Omega }u\Delta z\\ &\le -\int _{\Omega }\vert \Delta z\vert ^2-\int _{\Omega }\vert \nabla z\vert ^2+\frac{1}{2}\int _{\Omega }u^2+\frac{1}{2}\int _{\Omega }\vert \Delta z\vert ^2\\ &=-\int _{\Omega }\vert \nabla z\vert ^2-\frac{1}{2}\int _{\Omega }\vert \Delta z\vert ^2+\frac{1}{2}\int _{\Omega }u^2 \end{aligned}$$

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

$$\begin{aligned} \frac{d}{d t}\int _{\Omega }\vert \nabla z\vert ^2+2\int _{\Omega }\vert \nabla z\vert ^2\le -\int _{\Omega }\vert \Delta z\vert ^2+\int _{\Omega }u^2 \end{aligned}$$

for all \(t\in (0,T_{\max }).\) By utilizing Lemma 2.4 and (11), it can easily derive that (13) holds true. Then, by integrating the aforementioned inequality with respect to time from t to \(t+\tau ,\) we obtain (14). By employing similar methods, we can establish the validity of (15). \(\square\)

Subsequently, we present the following lemma concerning \(L^p\) boundedness of v,  which will play a crucial role in proving Lemma 3.4.

Lemma 3.3

For \(p\ge 2,\) \(n=2,\) assuming that the conditions in Theorem 1.1are satisfied, there exists a positive constant \(K_4\) such that

$$\begin{aligned} \Vert v(\cdot , t)\Vert _{L^p(\Omega )}\le K_4,\quad \forall \ t\in (0,T_{max}) \end{aligned}$$
(16)

and

$$\begin{aligned} \int _{t}^{t+\tau }\int _{\Omega }v^{p+1}\le K_4,\quad \forall \ t\in (0,T_{max}-\tau ), \end{aligned}$$
(17)

where \(\tau :=\min \{1,\frac{T_{max}}{2}\}.\)

Proof

By testing the second equation of (1) with \(pv^{p-1}\) and applying integration by parts, we derive

$$\begin{aligned} \frac{d}{dt}\int _{\Omega }v^{p}&=-p(p-1)\int _{\Omega }\psi _2(z)v^{p-2}\vert \nabla v\vert ^2-p(p-1)\int _{\Omega }\psi _2^{\prime }(z)v^{p-1}\nabla v\cdot \nabla z\\ &\quad +\mu p\int _{\Omega }v^{p}-p\int _{\Omega }v^{p+1}-\beta p\int _{\Omega }uv^{p}\\ &=-\frac{4(p-1)}{p}\int _{\Omega }\psi _2(z)\vert \nabla v^{\frac{p}{2}}\vert ^2-(p-1)\int _{\Omega }\psi _2^{\prime }(z)\nabla v^p\cdot \nabla z+\mu p\int _{\Omega }v^{p}\\ &\quad -p\int _{\Omega }v^{p+1}-\beta p\int _{\Omega }uv^{p}\\ &=-\frac{4(p-1)}{p}\int _{\Omega }\psi _2(z)\vert \nabla v^{\frac{p}{2}}\vert ^2+(p-1)\int _{\Omega }\psi _2^{\prime \prime }(z)v^p\vert \nabla z\vert ^2+(p-1)\int _{\Omega }\psi _2^{\prime }(z)v^p\Delta z\\ &\quad +\mu p\int _{\Omega }v^{p}-p\int _{\Omega }v^{p+1}-\beta p\int _{\Omega }uv^{p} \end{aligned}$$
(18)

for all \(t\in (0,T_{\max }).\) Based on hypothesis (4), considering that \(\psi _2^{\prime \prime }(z)<0\) and \(\psi _2^{\prime }(z)<\psi _2^{\prime }(0)\) for all \(z\ge 0,\) and taking into account the non-negativity of u and v,  (18) can be rewritten as

$$\begin{aligned} \frac{d}{dt}\int _{\Omega }v^{p}\le -\frac{4(p-1)}{p}\psi _2(0)\int _{\Omega }\vert \nabla v^{\frac{p}{2}}\vert ^2+(p-1)\psi _2^{\prime }(0)\int _{\Omega }v^p\Delta z +\mu p\int _{\Omega }v^{p}-p\int _{\Omega }v^{p+1} \end{aligned}$$
(19)

for all \(t\in (0,T_{\max }).\) Let us proceed to the second term on the right-hand side of (19). By virtue of Gagliardo–Nirenberg inequality from Lemma 2.2, combined with Hölder’s inequality and Young’s inequality, we assert the existence of a positive constant \(c_1\) such that

$$\begin{aligned}&(p-1)\psi _2^{\prime }(0)\int _{\Omega }v^p\Delta z\\ &\quad \le (p-1)\psi _2^{\prime }(0)\Vert v^{\frac{p}{2}}\Vert _{L^4(\Omega )}^{2}\Vert \Delta z\Vert _{L^2(\Omega )}\\ &\quad \le (p-1)\psi _2^{\prime }(0)c_1\left( \Vert \nabla v^{\frac{p}{2}}\Vert _{L^2(\Omega )}\Vert v^{\frac{p}{2}}\Vert _{L^2(\Omega )}+\Vert v^{\frac{p}{2}}\Vert _{L^{\frac{p}{2}}(\Omega )}^{2}\right) \Vert \Delta z\Vert _{L^2(\Omega )}\\ &\quad \le \frac{4(p-1)}{p}\psi _2(0)\Vert \nabla v^{\frac{p}{2}}\Vert _{L^2(\Omega )}^2+\frac{p(p-1)(\psi _2^{\prime }(0))^2c_1^{2}}{16d_2(0)}\Vert v\Vert _{L^p(\Omega )}^{2}\Vert \Delta z\Vert _{L^2(\Omega )}^2\\ &\qquad +\frac{(p-1)\psi _2^{\prime }(0)c_1m_1^p}{2}\left( \Vert \Delta z\Vert _{L^2(\Omega )}^2+\vert \Omega \vert \right) \end{aligned}$$
(20)

for all \(t\in (0,T_{\max }).\) By substituting (20) into (19) and applying Young’s inequality along with (14), we obtain

$$\begin{aligned}&\frac{d}{dt}\int _{\Omega }v^p+\frac{p(p-1)(\psi _2^{\prime }(0))^2c_1^{2}K_2}{8\psi _2(0)}\int _{\Omega }v^p\\ &\quad \le \frac{(p-1)\psi _2^{\prime }(0)c_1m_1^p}{2}\int _{\Omega }\vert \Delta z\vert ^2+\frac{p(p-1)(\psi _2^{\prime }(0))^2c_1^{2}}{16\psi _2(0)}\int _{\Omega }\vert \Delta z\vert ^2\int _{\Omega }v^p\\ &\qquad +\left( \mu p+\frac{p(p-1)(\psi _2^{\prime }(0))^2c_1^{2}K_2}{8\psi _2(0)}\right) \int _{\Omega }v^p-p\int _{\Omega }v^{p+1}\\ &\quad \le \frac{(p-1)\psi _2^{\prime }(0)c_1m_1^p}{2}\int _{\Omega }\vert \Delta z\vert ^2+\frac{p(p-1)(\psi _2^{\prime }(0))^2c_1^{2}}{16\psi _2(0)}\int _{\Omega }\vert \Delta z\vert ^2\int _{\Omega }v^p-\int _{\Omega }v^{p+1}+c_2 \end{aligned}$$
(21)

for all \(t\in (0,T_{\max }),\) where \(c_2>0\) is a constant.

Let \(\eta (t):=\int _{\Omega }v^p,\) \(\delta (t):=\frac{p(p-1)(\psi _2^{\prime }(0))^2c_1^{2}K_2}{8\psi _2(0)},\) \(\sigma (t):=\frac{p(p-1)(\psi _2^{\prime }(0))^2c_1^{2}}{16\psi _2(0)}\int _{\Omega }\vert \Delta z\vert ^2\) and \(\gamma (t):=\frac{(p-1)\psi _2^{\prime }(0)c_1m_1^p}{2}\int _{\Omega }\vert \Delta z\vert ^2+c_2,\) it yields \(\eta '(t)+\delta (t)\eta (t)\le \sigma (t)\eta (t)+\gamma (t)\) for all \(t\in (0,T_{\max }).\) Therefore, by applying Lemma 2.5 and (14), there exists a positive constant \(c_3\) such that \(\eta (t)\le c_3\) for all \(t\in (0,T_{\max }),\) thanks to \(\int _{t}^{t+\tau }\delta (s)ds-\int _{t}^{t+\tau }\sigma (s)ds>\zeta\) for all \(t\in (0,T_{\max }-\tau ),\) where \(\zeta =\frac{p(p-1)(\psi _2^{\prime }(0))^2c_1^{2}K_2}{16\psi _2(0)}>0.\) Subsequently, by integrating (21) with respect to time, we can easily derive (17). \(\square\)

Then we focus on estimating the boundedness of \(\Vert w\Vert _{L^\infty (\Omega )}\) and \(\Vert \nabla w\Vert _{L^\infty (\Omega )}.\)

Lemma 3.4

For \(n=2\) and under the assumption that the conditions in Theorem 1.1are satisfied, there exist positive constants \(K_5,K_6\) such that the followings hold

$$\begin{aligned} \Vert w\Vert _{L^\infty (\Omega )}\le K_5,\quad \forall \ t\in [0,T_{max}) \end{aligned}$$
(22)

and

$$\begin{aligned} \Vert \nabla w \Vert _{L^\infty (\Omega )}\le K_6,\quad \forall \ t\in [0,T_{max}). \end{aligned}$$
(23)

Proof

By recalling the well-known smoothing property of Neumann heat semigroup [17] on \(\Omega ,\) we can assert the existence of a positive constant \(c_1\) such that

$$\begin{aligned} \Vert e^{t\Delta }\phi \Vert _{L^p(\Omega )}\le c_1(1+t^{-\frac{n}{2}(\frac{1}{q}-\frac{1}{p})})e^{-\lambda _1t}\Vert \phi \Vert _{L^q(\Omega )} \end{aligned}$$
(24)

for all \(t>0,\) \(\phi \in L^q(\Omega )\) satisfying \(\int _{\Omega }\phi =0,\) and there exists a constant \(c_2>0\) such that

$$\begin{aligned} \Vert \nabla e^{t\Delta }f\Vert _{L^p(\Omega )}\le c_2(1+t^{-\frac{1}{2}-\frac{n}{2}(\frac{1}{q}-\frac{1}{p})})e^{-\lambda _2t}\Vert f\Vert _{L^q(\Omega )} \end{aligned}$$
(25)

for all \(t>0,\) \(f\in L^q(\Omega ).\) By utilizing a variation-of-constants formula for v,  Lemma 3.3 and (24), we obtain

$$\begin{aligned} \Vert w\Vert _{L^\infty (\Omega )}&\le \Vert e^{t(\Delta -1)}w_{0}\Vert _{L^\infty (\Omega )}+ \int _{0}^{t}\Vert e^{(t-s)(\Delta -1)}\bar{v}\Vert _{L^\infty (\Omega )}ds\\ &\quad +\int _{0}^{t}\Vert e^{(t-s)(\Delta -1)}(v-\bar{v})\Vert _{L^\infty (\Omega )}ds \\ &\le c_3 \Vert w_{0}\Vert _{L^1(\Omega )}+\frac{m_1}{\vert \Omega \vert }+c_4\int _{0}^{t}\bigg(1+(t-s)^{-\frac{1}{p}}\bigg)e^{-(\lambda _1+1)(t-s)}\Vert v\Vert _{L^p(\Omega )}ds\\ &\le c_5 \end{aligned}$$

for all \(t\in (0,T_{\max }),\) where \(c_3, c_4, c_5>0\) are constants and \(m_1\) is taken from Lemma 3.1. By (25) and Lemma 3.3, there exist positive constants \(c_6, c_7, c_8\) such that

$$\begin{aligned} \Vert \nabla w\Vert _{L^\infty (\Omega )}&\le \Vert \nabla e^{t(\Delta -1)}w_{0}\Vert _{L^\infty (\Omega )}+\int _{0}^{t}\Vert \nabla e^{(t-s)(\Delta -1)}v\Vert _{L^\infty (\Omega )}ds \\ &\le c_6 \Vert w_{0}\Vert _{L^\infty (\Omega )}+c_7\int _{0}^{t}\bigg(1+(t-s)^{-\frac{1}{2}-\frac{1}{p}}\bigg)e^{-(\lambda _2+1)(t-s)}\Vert v\Vert _{L^p(\Omega )}ds\\ &\le c_8 \end{aligned}$$

for all \(t\in (0,T_{\max }),\) which implies (22) and (23). \(\square\)

Now, we will use some established results to establish the boundedness of \(\int _{\Omega }u^{p}.\)

Lemma 3.5

For \(p\ge 2,\) assuming that the conditions in Theorem 1.1are satisfied, there exists a positive constant \(K_7\) such that

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^p(\Omega )}\le K_7,\quad \forall \ t\in (0,T_{\max }). \end{aligned}$$
(26)

Proof

Testing the first equation of the system (1) by \(u^{p-1}\) and integrating in parts, we get

$$\begin{aligned} \frac{1}{p}\frac{d}{dt}\int _{\Omega } u^{p}&=-(p-1)\int _{\Omega }\psi _1(w)u^{p-2}\vert \nabla u\vert ^2-(p-1)\int _{\Omega }\psi _1^{\prime }(w)u^{p-1}\nabla u\cdot \nabla w\\ &\quad +\lambda \int _{\Omega } u^{p}-\int _{\Omega } u^{p+1}+\alpha \int _{\Omega } u^{p}v \end{aligned}$$
(27)

for all \(t\in (0, T_{\max }).\) We invoke the Young’s inequality to estimate

$$\begin{aligned} -(p-1)\int _{\Omega }\psi _1^{\prime }(w)u^{p-1}\nabla u\cdot \nabla w\le (p-1)\int _{\Omega }\psi _1(w)u^{p-2}\vert \nabla u\vert ^2+\frac{p-1}{4}\int _{\Omega }\frac{\vert \psi _1^{\prime }(w)\vert ^2}{\psi _1(w)}u^{p}\vert \nabla w\vert ^2 \end{aligned}$$
(28)

for all \(t\in (0, T_{\max }).\) According to hypothesis (3) and (22), there exists a constant \(c_1>0\) such that

$$\begin{aligned} \frac{\vert \psi _1^{\prime }(v)\vert ^2}{\psi _1(v)}\le c_1. \end{aligned}$$
(29)

By virtue of Young’s inequality along with the combination of (27), (28) and (29), we derive

$$\begin{aligned}&\frac{1}{p}\frac{d}{dt}\int _{\Omega } u^{p}+\int _{\Omega } u^{p}\\ &\quad \le \frac{(p-1)c_1}{4}\int _{\Omega }u^{p}\vert \nabla w\vert ^2+(\lambda +1)\int _{\Omega } u^{p}-\int _{\Omega } u^{p+1}+\alpha \int _{\Omega } u^{p}v\\ &\quad \le \frac{1}{3}\int _{\Omega } u^{p+1}+\frac{(3p)^{p}[(p-1)c_1]^{p+1}}{[4(p+1)]^{p+1}}\int _{\Omega }\vert \nabla w\vert ^{2(p+1)}+\frac{1}{3}\int _{\Omega } u^{p+1}+\frac{(3p)^p(\lambda +1)^{p+1}}{(p+1)^{p+1}}\vert \Omega \vert \\ &\qquad -\int _{\Omega } u^{p+1}+\frac{1}{3}\int _{\Omega } u^{p+1}+\frac{(3p)^p\alpha ^{p+1}}{(p+1)^{p+1}}\int _{\Omega }v^{p+1}\\ &\quad \le \frac{(3p)^{p}[(p-1)c_1]^{p+1}}{[4(p+1)]^{p+1}}\int _{\Omega }\vert \nabla w\vert ^{2(p+1)}+\frac{(3p)^p\alpha ^{p+1}}{(p+1)^{p+1}}\int _{\Omega } v^{p+1}+\frac{(3p)^p(\lambda +1)^{p+1}}{(p+1)^{p+1}}\vert \Omega \vert \end{aligned}$$
(30)

for all \(t\in (0, T_{\max }).\) To address the first term on the right side of the aforementioned inequality, we can make use of (15) and (23). By means of the interpolation inequality, it follows that

$$\begin{aligned} \int _{\Omega }\vert \nabla w\vert ^{2(p+1)}=\Vert \nabla w\Vert _{L^{2(p+1)}(\Omega )}^{2(p+1)}\le \Vert \nabla w\Vert _{L^{2}(\Omega )}^{2}\Vert \nabla w\Vert _{L^{\infty }(\Omega )}^{2p}\le c_2 \end{aligned}$$

for all \(t\in (0, T_{\max }),\) where \(c_2>0\) is a constant. By combining (30), we obtain

$$\begin{aligned} \frac{d}{dt}\int _{\Omega } u^{p}+p\int _{\Omega } u^{p}\le c_3\int _{\Omega } v^{p+1}+c_4 \end{aligned}$$

for all \(t\in (0, T_{\max }),\) where \(c_3:=\frac{3^p(\alpha p)^{p+1}}{(p+1)^{p+1}}>0,\) \(c_4:=\left\{ \frac{3^{p}[p(p-1)c_1]^{p+1}c_2}{[4(p+1)]^{p+1}}+\frac{3^p[p(\lambda +1)]^{p+1}}{(p+1)^{p+1}}\right\} \vert \Omega \vert >0.\) In view of (17), this readily yields

$$\begin{aligned} \int _{t}^{t+1}\left\{ c_3\int _{\Omega } v^{p+1}+c_4\right\} \le c_3K_4+c_4 \end{aligned}$$

for all \(t\in (0, T_{\max }-\tau ).\) By invoking Lemma 2.4, we get

$$\begin{aligned} \int _{\Omega } u^{p}\le \max \left\{ \int _{\Omega } u_{0}^{p}+c_5, \frac{c_5}{p}+2c_5\right\} ,\quad \forall t\in (0, T_{\max }), \end{aligned}$$

where \(c_5:=c_3K_4+c_4>0.\) This completes the proof of (26). \(\square\)

Lemma 3.6

Assuming that the conditions in Theorem 1.1are satisfied, there exists a positive constant \(K_8\) such that

$$\begin{aligned} \Vert z(\cdot ,t)\Vert _{W^{1,\infty }(\Omega )}\le K_8,\quad \forall t\in (0,T_{\max }). \end{aligned}$$
(31)

Proof

By considering the variation-of-constants formula for z and utilizing the smoothing property of the Neumann heat semigroup on \(\Omega ,\) we can establish the boundedness of \(\Vert z(\cdot ,t)\Vert _{L^{\infty }(\Omega )}\) for all \(t\in (0,T_{\max }).\) Furthermore, we can verify the validity of (31). The details involved in this process are similar to those outlined in Lemma 3.4. \(\square\)

Proof of Theorem 1.1

By the standard Alikakos–Moser iterative method [18] and using Lemmas 3.3 and 3.5, we infer that there exists a constant \(C>0\) such that

$$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^{\infty }(\Omega )}+ \Vert v(\cdot ,t)\Vert _{L^{\infty }(\Omega )}\le C \end{aligned}$$

for all \(t\in (0,T_{\max }).\) Combining Lemmas 3.4 and 3.6, we have completed the proof of the Theorem 1.1. \(\square\)