Global dynamics of a delayed diffusive virus infection model with cell-mediated immunity and cell-to-cell transmission

1.   Introduction

In 1987, Karevia and Odell ^[1] first proposed the following one-predator and one-prey model with prey-taxis in order to explain that an area-restricted search creates the following predator aggregation phenomenon

where $D > 0$ is the diffusivity coefficient of preys, $d(w)$ denotes the motility function of predators, $\chi(w)$ represents the prey-taxis sensitivity coefficient, and the term $-\nabla\cdot(u\chi(w)\nabla w)$ stands for the tendency of the predator moving towards the increasing direction of the prey gradient, and it is viewed as the prey-taxis term. The functions $G_{1}(u, w)$ and $G_{2}(u, w)$ describe the predator-prey interactions, which include both intra-specific and inter-specific interactions. Generally the predator-prey interaction functions $G_{1}(u, w)$ and $G_{2}(u, w)$ possess the following prototypical forms.

where $\gamma > 0$ denotes the intrinsic predation rate, $uF(w)$ represents the inter-specific interaction and $uh(u)$ and $f(w)$ stand for the intra-specific interaction. Specifically, $F(w)$ is the functional response function accounting for the intake rate of predators as a function of prey density; it is often used in the following form in the literature ^[2,3,4]

with constants $\lambda > 0$ and $m > 1$; other types of functional response functions (e.g., Beddington-DeAngelis type in ^[5], Crowley-Martin type in ^[6]) and more predator-prey interactions can be found in ^[7,8,9,10]. The predator mortality rate function $h(u)$ is typically of the form

where $\theta > 0$ accounts for the natural death rate and $\alpha\geq0$ denotes the rate of death resulting from the intra-specific competition, which is also called the density-dependent death ^[11]. The prey growth function $f(w)$ is usually assumed to be negative for large $w$ due to the limitation of resources (or crowding effect), and its typical forms are

where $\mu > 0$ is the intrinsic growth rate of prey, $K > 0$ is called the carrying capacity and $0 < k < K$. Now there exist many interesting results about global existence, uniform boundedness, asymptotic behavior, traveling waves and pattern formation of solutions to System (1.1) or its variants in ^{[11,12,13,14,15]}. When $d(w) = d > 0$ and $\chi(w) = \chi > 0$, $G_{1}(u, v)$ and $G_{2}(u, v)$ have the forms of (1.2), Wu et al. ^[15] obtained the global existence and uniform persistence of solutions to (1.1) in any dimension provided that $\chi$ is suitably small. Then, Jin and Wang^[13] derived the global boundedness and asymptotic stability of solutions for System (1.1) without the smallness assumption on $\chi$ in a two-dimensional bounded domain. Moreover, Wang et al. ^[16] studied the nonconstant positive steady states and pattern formation of (1.1) in a one-dimensional bounded domain. Under the conditions that $d(w)$ and $\chi(w)$ are not constants and $h(u)$ is given by (1.4), Jin and Wang^[14] established the global boundedness, asymptotic behavior and spatio-temporal patterns of solutions for (1.1) under some conditions on the parameters in a two-dimensional smooth bounded domain. For more related results in predator-prey models, we refer the readers to ^{[17,18,19,20,21,22,23,24,25,26,27,28,29]} and the references therein.

However, all the aforementioned works are devoted to studying prey-taxis models with one-predator and one prey. Now let us mention some predator-prey models with two-predator and one-prey. Recently, the following general two-predator and one-prey model with prey-taxis has attracted a lot of attention.

as applied in a smooth bounded domain $\Omega\subset \mathbb{R}^{n}$, $n\geq1$. Given $d_{1}(w) = d_{2}(w) = 1$ and $\beta_{1} = \beta_{2} = 0$, Wang et al. ^[30] derived the global boundedness, nonconstant positive steady states and time-periodic patterns of solutions for System (1.6). Wang and Wang ^[31] studied the uniform boundedness and asymptotic stability of nonnegative spatially homogeneous equilibria for (1.6) in any dimension. Given $d_{1}(w) = d_{2}(w) = 1$, $\chi_{i}(w) = \chi_{i} > 0 \; (i = 1, 2)$ and $\beta_{1} = \beta_{2} = \beta > 0$, Mi et al. ^[32] obtained the global boundedness and stability of classical solutions in any dimension under suitable conditions of parameters. Under the conditions that $d_{1}(w)$ and $d_{2}(w)$ are non-constants and $\beta_{1} = \beta_{2} = 0$, Qiu et al.^[33] rigorously proved the global existence, uniform boundedness and stabilization of classical solutions in any dimension with suitable conditions on motility functions and the coefficients of the logistic source. However, when $\beta_{1}, \beta_{2}\neq0$, the global existence and stabilization of solutions for (1.6) are still open. Given $\chi_{i}(w) = -d'_{i}(w)\geq0$ if $d'_{i}(w)\leq0 \; (i = 1, 2)$, the diffusion-advection terms in (1.6) can respectively become the forms $\Delta(d_{1}(w)u)$ and $\Delta(d_{2}(w)v)$, which could be interpreted as "density-suppressed motility" in ^[34,35]. This means that the predator will reduce its motility when encountering the prey, which is a rather reasonable assumption that has very sound applications in the predator-prey systems. Since the possible degeneracy caused by the density-suppressed motility brings considerable challenges for mathematical analysis, many works have showed various interesting results, which can be found in ^{[36,37,38,39,40,41,42,43,44]}. Given $\chi_{i}(w) = -d'_{i}(w)\geq0 \; (i = 1, 2)$, $F_{1}(w) = F_{2}(w) = w$, $h_{1}(u) = u, h_{2}(v) = v$ and $f(w) = \mu w(m(x)-w)$, System (1.6) can be simplified as

where the parameters $D, \mu, \gamma_{i}, \beta_{i} \; (i = 1, 2)$ are positive, the dispersal rate functions $d_{i}(w) \; (i = 1, 2)$ satisfy following the hypothesis: $d_{i}(w)\in C^{2}([0, \infty)), d'_{i}(w)\leq 0$ on $[0, \infty)$ and $d_{i}(w) > 0$. Wang and Xu ^[45] have found some interesting results for System (1.7) in a two-dimensional smooth bounded domain. More specifically, when $D = 1$ and $m(x) = 1$, System (1.7) has a unique globally bounded classical solution. By constructing appropriate Lyapunov functionals and using LaSalle's invariant principle, the authors proved that the global bounded solution of (1.7) converges to the co-existence steady state exponentially or competitive exclusion steady state algebraically as time tends to infinity in different parameter regimes. For a prey's resource that is spatially heterogeneous (i.e., $m(x)$ is non-constant), the authors used numerical simulations to demonstrate that the striking phenomenon "slower diffuser always prevails" given in ^[46,47] fails to appear if the non-random dispersal strategy is employed by competing species (i.e., either $d_{1}(w)$ or $d_{2}(w)$ is non-constant) while it still holds if both $d_{1}(w)$ and $d_{2}(w)$ are constants. However, there are few results about global boundedness and large time behavior of solutions for (1.7) in the general form.

Inspired by the above works, this paper is concerned with the following two-species competitive predator-prey system with the following density-dependent diffusion

where $\Omega\subset \mathbb{R}^{2}$ is a bounded domain with a smooth boundary $\partial\Omega$, $\frac{\partial}{\partial \nu}$ denotes the derivative with respect to the outward normal vector $\nu$ of $\partial\Omega$, and the parameters $D$ and $\gamma_{i}$, $\beta_{i}$ ($i = 1, 2$) are positive. The unknown functions $u = u(x, t)$ and $v = v(x, t)$ denote the densities of two-competing species (e.g., predators), and $w = w(x, t)$ represents the density of predators' resources (e.g., the prey) at a position $x$ and time $t > 0$. When $d_{1}(w) = d_{1} > 0$ and $d_{2}(w) = d_{2} > 0$, System (1.8) becomes the well-known diffusive predator-prey system, which has been widely studied in ^{[48,49,50,51]}. However, to the best of our knowledge, the results of the two-predator and one-prey system given by (1.8) with density-suppressed motility (i.e., $d_{1}(w)$ and $d_{2}(w)$ are non-constants) indicate that the competition and general predator mortality rate $h_{i}(u)$ are almost vacant. The main aim of this paper is to explore the influence of the predation interaction, competition and general predator mortality on the dynamical behavior of System (1.8). Throughout this paper, we assume that the functions $d_{i}(s), F_{i}(s), h_{i}(s), (i = 1, 2), f(s)$ and initial data $(u_{0}, v_{0}, w_{0})$ mentioned in (1.8) satisfy the following hypotheses:

($H_{1}$) $d_{i}(s)\in C^{2}([0, \infty))$ with $d_{i}(s) > 0$ and $d'_{i}(s)\leq0$, $i = 1, 2$ on $[0, \infty)$. The typical example is $d_{i}(s) = \frac{1}{(1+s)^{\kappa_{i}}}$ or $d_{i}(s) = \exp(-\kappa_{i}s)$ with $\kappa_{i} > 0$, $i = 1, 2$.

($H_{2}$) $F_{i}(s)\in C^{1}([0, \infty))$, $F_{i}(0) = 0$, $F_{i}(s) > 0$ and $F'_{i}(s) > 0$, $i = 1, 2$ in $(0, \infty)$.

($H_{3}$) $h_{i}(s)\in C^{2}([0, \infty))$ and there exist constants $\theta_{i} > 0$ and $\alpha_{i}\geq0$ such that $h_{i}(s)\geq\theta_{i}$ and $h'_{i}(s)\geq\alpha_{i}$, $i = 1, 2$ for all $s > 0$.

($H_{4}$) $f(s)\in C^{1}([0, \infty))$ with $f(0) = 0$, and there exist positive constants $\mu$ and $K$ such that $f(s)\leq\mu s$ for all $s\geq0$, $f(K) = 0$ and $f(s) < 0$ for $s > K$.

($H_{5}$) $(u_{0}, v_{0}, w_{0})\in (W^{1, p}(\Omega))^{3}$ with $p > 2$ and $u_{0}, v_{0}, w_{0}\geq0$.

Here, we note that there are many candidates for the above functions $F_{i}(s)$, $h_{i}(s)$ and $f(s)$ as in (1.3)–(1.5). Due to the presence of the prey's density dependent diffusion coefficient, Model (1.8) is a cross-diffusion system and the parabolic comparison principle is no longer applicable. Moreover, when $\alpha_{1} = \alpha_{2} = 0$, the key $L^{2}$-spatiotemporal estimates of $u$ and $v$ cannot be directly derived; thus, the uniform boundedness of solutions is not an obvious result and needs to be justified. Based on the above hypotheses, the first main result of this paper asserts the global existence and boundedness of solutions for System (1.8) as follows.

Theorem 1.1. Let $D, \gamma_{i}, \beta_{i} > 0 \; (i = 1, 2)$, $\Omega\subset \mathbb{R}^{2}$ be a smooth bounded domain and the hypotheses ($H_{1}$)–($H_{5}$) hold. Then System (1.8) has a unique global nonnegative classical solution $(u, v, w)\in (C(\overline{\Omega}\times[0, \infty))\cap C^{2, 1}(\overline{\Omega}\times (0, \infty)))^{3}$, which is uniformly bounded in time, i.e., there exists a constant $C > 0$ independent of $t$ such that

In particular, one has $0\leq w(x, t)\leq K_{0}$ for all $(x, t)\in \Omega\times(0, \infty)$, where

Remark 1.1. For the special case $F_{1}(w) = F_{2}(w) = w$, $h_{1}(u) = u, h_{2}(v) = v$ and $f(w) = \mu w(1-w)$, the results of Theorem 1.1 have been obtained in ^[45]. However, since $\alpha_{1}$ and $\alpha_{2}$ may be equal to zero in the hypotheses of this paper, the $L^{2}$-spatiotemporal estimates of $u$ and $v$ cannot be directly obtained by using the method in ^[45]. By means of the mechanism "density-suppressed motility", we invoke some ideas used in ^[14] and apply the self-adjoint realization of $\Delta+\delta$ with some $\delta > 0$ in $L^{2}(\Omega)$ to establish the key $L^{2}$-spatiotemporal estimates of $u$ and $v$.

The second main aim of this paper is to study the role of non-random dispersal and competition between two predators in the asymptotic properties of the nonnegative spatial homogeneous equilibria of System (1.8). For simplicity, we let $F_{1}(w) = F_{2}(w) = w$, $h_{1}(u) = \theta_{1}+\alpha_{1} u$, $h_{2}(v) = \theta_{2}+\alpha_{2} v$ and $f(w) = \mu w(1-w)$, where $\theta_{1} = \theta_{2} = \theta > 0$ and $\alpha_{1}, \alpha_{2}, \mu > 0$; then, System system (1.8) can be simplified as

Theorem 1.1 ensures that System (1.11) possesses a unique global bounded nonnegative classical solution $(u, v, w)$ such that $0\leq w(x, t)\leq K_{0}: = \max\{||w_{0}||_{L^{\infty}(\Omega)}, 1\}$ for all $(x, t)\in\Omega\times(0, \infty)$. Now we find some sufficient conditions of parameters so that (1.11) admits a positive constant solution $(u^{*}, v^{*}, w^{*})$ satisfying

i.e.,

where

If the determinant $\Phi$ of the coefficient matrix $A$ in (1.13) does not equal zero, it follows from Cramer's rule that

where

When $\beta_{1} < \alpha_{1}$ and $\beta_{2} < \alpha_{2}$, it follows that $\Phi > 0$ and $\Phi_{w} > 0$, and thus we know $w^{*} > 0$. Next, we shall discuss the sign of $\Phi_{u}$ and $\Phi_{v}$. For convenience, we let

and

where

and

It is not difficult to see that $u^{*} = \frac{\Phi_{u}}{\Phi}$, $v^{*} = \frac{\Phi_{v}}{\Phi}$ and $w^{*} = \frac{\Phi_{w}}{\Phi}$ are positive, if $\gamma_{i} > \theta$, $i = 1, 2$ and one of the following conditions holds:

($H_{6}$) $\gamma_{1} < \gamma_{2}$, $l_{1} > 0$, $\beta_{1} < \min\{\alpha_{1}, \beta_{1}^{*}\}$ and $\beta_{2} < \min\{\alpha_{2}, \beta_{2}^{*}\}$;

($H_{7}$) $\gamma_{1} > \gamma_{2}$, $l_{2} > 0$, $\beta_{1} < \min\{\alpha_{1}, \beta_{1}^{*}\}$ and $\beta_{2} < \min\{\alpha_{2}, \beta_{2}^{*}\}$;

($H_{8}$) $\gamma_{1} = \gamma_{2}$ and $\max\{\beta_{1}, \beta_{2}\} < \min\{\alpha_{1}, \alpha_{2}\}$.

Now, we give our main results on the asymptotic stability properties of the nonnegative spatial homogeneous equilibria of System (1.11) as follows.

Theorem 1.2. Let $\Omega\subset \mathbb{R}^{2}$ be a smooth bounded domain and the parameters $\gamma_{i}, \alpha_{i}, \beta_{i}$, $(i = 1, 2)$, $\theta$, $\mu$ and $D$ be positive. Assume that $d_{1}(w)$ and $d_{2}(w)$ satisfy $(H_{1})$, and that $(u, v, w)$ is a global bounded classical solution of System (1.11). Suppose that $\gamma_{i} > \theta, i = 1, 2,$

and

as well as one of the conditions ($H_{6}$)–($H_{8}$) holds. Then for all initial data $(u_{0}, v_{0}, w_{0})$ satisfying ($H_{5}$), there exist positive constants $C$ and $\lambda$ such that

for all $t > 0$, where $(u^{*}, v^{*}, w^{*})$ is given by (1.14).

Remark 1.2. From a biological point of view, it is well known that the change of the predators comes from predation, competition and mortality in System (1.11). The parameters $\gamma_{i}$, $\beta_{i}$, $\alpha_{i} \; (i = 1, 2)$ and $\theta$ respectively stand for the predation rate, competition strength, density-dependent death and natural death rate of the predators, which play a collective role in studying the dynamical behavior of (1.11). More specifically, when $\gamma_{i} > \theta$, it is called strong predation; otherwise, it is weak predation. Hence the results of Theorem 1.2 can tell us that if the predations of two predators are strong and the prey diffusion coefficient $D$ is suitably large, all species can reach a coexistence state.

Theorem 1.3. Let $\Omega\subset \mathbb{R}^{2}$ be a smooth bounded domain and the parameters $\gamma_{i}, \alpha_{i}, \beta_{i}$, $(i = 1, 2)$, $\theta$, $\mu$ and $D$ be positive. Assume that $d_{1}(w)$ and $d_{2}(w)$ satisfy $(H_{1})$, and the $(u, v, w)$ is a global bounded classical solution of System (1.11). Suppose that we have (1.20) and

as well as one of the following conditions holds:

(i) $\gamma_{1} > \gamma_{2} > \theta$, $l_{2}\leq 0$, $\beta_{1} < \min\{\alpha_{1}, \beta_{1}^{*}\}$ and $\beta_{2} < \alpha_{2}$;

(ii) $\gamma_{1} > \gamma_{2} > \theta$, $l_{2} > 0$, $\beta_{1} < \min\{\alpha_{1}, \beta_{1}^{*}\}$ and $\beta_{2}\in[\beta_{2}^{*}, \alpha_{2})$;

(iii) $\gamma_{1} > \theta\geq \gamma_{2}$,

where

Then for all initial data $(u_{0}, v_{0}, w_{0})$ satisfying ($H_{5}$), one has

exponentially if $\theta > \gamma_{2}\overline{w}-\beta_{2}\overline{u}$ or algebraically if $\theta = \gamma_{2}\overline{w}-\beta_{2}\overline{u}$.

Theorem 1.4. Let $\Omega\subset \mathbb{R}^{2}$ be a smooth bounded domain and the parameters $\gamma_{i}, \alpha_{i}, \beta_{i}$, $(i = 1, 2)$, $\theta$, $\mu$ and $D$ be positive. Assume that $d_{1}(w)$ and $d_{2}(w)$ satisfy $(H_{1})$, and the $(u, v, w)$ is a global bounded solution of System (1.11). Suppose that we have (1.20) and

as well as one of the following conditions holds:

(i) $\gamma_{2} > \gamma_{1} > \theta$, $l_{1}\leq 0$, $\beta_{2} < \min\{\alpha_{2}, \beta_{2}^{*}\}$ and $\beta_{1} < \alpha_{1}$;

(ii) $\gamma_{2} > \gamma_{1} > \theta$, $l_{1} > 0$, $\beta_{2} < \min\{\alpha_{2}, \beta_{2}^{*}\}$ and $\beta_{1}\in[\beta_{1}^{*}, \alpha_{1})$;

(iii) $\gamma_{2} > \theta\geq\gamma_{1}$,

where

Then for all initial data $(u_{0}, v_{0}, w_{0})$ satisfying ($H_{5}$), one has

exponentially if $\theta > \gamma_{1}\widetilde{w}-\beta_{1}\widetilde{v}$ or algebraically if $\theta = \gamma_{1}\widetilde{w}-\beta_{1}\widetilde{v}$.

Remark 1.3. It follows from Theorem 1.3 that if the predator $u$ is superior over $v$ in the competition and the prey diffusion coefficient $D$ is suitably large, the semi-trivial equilibrium $(\overline{u}, 0, \overline{w})$ is globally asymptotically stable. Similarly, we can obtain Theorem 1.4. Hence, we only give the conclusion of Theorem 1.4, without showing the details of the proof for brevity.

Theorem 1.5. Let $\Omega\subset \mathbb{R}^{2}$ be a smooth bounded domain and the parameters $\gamma_{i}, \alpha_{i}, \beta_{i}$, $(i = 1, 2)$, $\theta$, $\mu$ and $D$ be positive. Assume that $d_{1}(w)$ and $d_{2}(w)$ satisfy $(H_{1})$, and the $(u, v, w)$ is a global bounded solution of System (1.11). Suppose that

Then for all initial data $(u_{0}, v_{0}, w_{0})$ satisfying ($H_{5}$), one has

exponentially if $\gamma_{i} < \theta$, $i = 1, 2$ or algebraically if $\gamma_{i} = \theta$, $i = 1, 2$.

Remark 1.4. It follows from Theorem 1.5 that if the capture rates of the two predators are low (i.e. $\gamma_{i}\leq\theta$, $i = 1, 2$), the prey-only steady state $(0, 0, 1)$ is globally asymptotically stable regardless of the size of the prey diffusion coefficient $D$.

Remark 1.5. Compared with the previous results of ^[33] without competitive terms, the results of Theorems 1.2–1.5 indicate that the competition terms play a crucial role in the global stability of the constant steady states in (1.11). Moreover, under the condition of density-suppressed motility, our global stability results of Theorems 1.2–1.5 can also generalize the ranges of parameters $\alpha_{i}$ and $\beta_{i} \; (i = 1, 2)$ for two dimensional cases in ^[32]. However, since the heat-semigroup estimates of $u$ and $v$ are no longer applicable due to the appearance of density-suppressed motility, the global stability in $L^{\infty}$-norm is still open in the higher-dimensional problem.

The rest of this paper is organized as follows. In Section 2, we first state the local existence of the classical solution to (1.8) and collect preliminary lemmas. In Section 3, we derive the global existence and boundedness of classical solutions for (1.8) and prove Theorem 1.1. Finally, we shall study the asymptotic stability of global bounded solutions for (1.11) and prove Theorems 1.2–1.5.

2.   Local existence and preliminaries

In this section, we shall give the local existence and some preliminary lemmas. Firstly, we state the local existence of the classical solution to (1.8), as obtained by means of the abstract theory of quasilinear parabolic systems in ^[52].

Lemma 2.1. Let $D, \gamma_{i}, \beta_{i} > 0 \; (i = 1, 2)$, $\Omega\subset \mathbb{R}^{2}$ be a smooth bounded domain and the hypotheses ($H_{1}$)–($H_{5}$) hold. Then, there exists a $T_{\max } \in(0, \infty]$ such that System (1.8) possesses a unique classical solution $(u, v, w)\in(C(\overline{\Omega}\times[0, T_{\max}))\cap C^{2, 1}(\overline{\Omega}\times(0, T_{\max})))^{3}$ satisfying

In addition, the following extensibility criterion holds, i.e. if $T_{\max } < \infty$, then

Proof. Let $\textbf{z} = (u, v, w)^{T}$; then, System (1.8) can be rewritten as

where

According to the conditions that $D > 0$ and $d_{i}(w) > 0 \; (i = 1, 2)$, the matrix $P(\textbf{z})$ is positively definite for the given initial data $\textbf{z}_{0}$, which asserts that System (1.8) is normally parabolic. Thus it follows from Theorem 7.3 of ^[53] that there exists a $T_{\max } \in(0, \infty]$ such that System (1.8) admits a unique classical solution $(u, v, w)\in(C^{0}(\overline{\Omega}\times[0, T_{\max}))\cap C^{2, 1}(\overline{\Omega}\times(0, T_{\max})))^{3}$. The nonnegativity of $(u, v, w)$ directly comes from the maximum principle ^[14,45]. It similarly follows from Lemma 2.2 of ^[13] that $w\leq K_{0}: = \max\{||w_{0}||_{L^{\infty}(\Omega)}, K\}$. Since $P(\textbf{z})$ is an upper triangular matrix, we can deduce from Theorem 5.2 of ^[54] that the extensibility criterion given by (2.2) holds. The proof of Lemma 2.1 is complete.

Lemma 2.2. Let $\Omega\subset \mathbb{R}^{n}(n\geq 1)$ be a smooth bounded domain, $D > 0$ and $T\in(0, \infty]$. Assume that $\varphi(x, t)\in C(\overline{\Omega}\times[0, T))\cap C^{2, 1}(\overline{\Omega}\times(0, T))$ satisfies

where $\psi\in L^{\infty}((0, T);L^{p}(\Omega))$ with $p\geq1$. Then there exists a positive constant $C$ such that

where

Proof. This lemma directly comes from Lemma 1 of ^[55].

Now, we give the following lemma (^[56], Lemma 3.4) to derive some a priori estimates for $w$.

Lemma 2.3. Let $T > 0, \tau\in(0, T)$ and $a, d > 0$, and assume that $y:[0, T)\rightarrow [0, \infty)$ is absolutely continuous. If there exists a nonnegative function $h\in L_{loc}^{1}([0, T))$ satisfying

and

one has

Next, we give a basic property of the solution components $u$ and $v$ for System (1.8).

Lemma 2.4. Let the assumptions of Lemma 2.1 hold. Then there exists a constant $C > 0$ such that

and

where $\tau = \min\{1, \frac{1}{2}T_{\max}\}$.

Proof. It follows from a direct computation for System (1.8) that

for all $t\in (0, T_{\max})$, where we have applied ($H_{3}$), ($H_{4}$), $\beta_{1}, \beta_{2} > 0$ and (2.1).

Let $\theta: = \min\{\theta_{1}, \theta_{2}\}$, it follows from (2.1) that

which leads to (2.11) by Gronwall's inequality. If $\alpha_{i} > 0$, $i = 1, 2$, then integrating (2.14) over $(t, t+\tau)$, we have (2.12) directly. If $\alpha_{i} = 0$, $i = 1, 2$, we can also prove (2.12) by means of the idea used in ^[14]. For the readers' convenience, we give the sketch of the proof.

Let $\mathcal{A}$ denote the self-adjoint realization of $-\Delta+\delta$ under homogeneous Neumann boundary conditions in $L^{2}(\Omega)$, where $\delta\in\left(0, \min\{\frac{\theta_{1}}{d_{1}(0)}, \frac{\theta_{2}}{d_{2}(0)}\}\right)$. It follows from $\delta > 0$ that $\mathcal{A}$ has an order-preserving bounded inverse $\mathcal{A}^{-1}$ on $L^{2}(\Omega)$. Thus this allows us to obtain a positive constant $c_{1}$ such that

and

By a simple calculation in (1.8), we have

which can be written as

where we have applied ($H_{1}$), ($H_{3}$), ($H_{4}$), (2.1) and $\delta\in\left(0, \min\{\frac{\theta_{1}}{d_{1}(0)}, \frac{\theta_{2}}{d_{2}(0)}\}\right)$. Hence, by multiplying (2.18) by $\mathcal{A}^{-1}\left(\frac{1}{\gamma_{1}}u+\frac{1}{\gamma_{2}}v+w\right)\geq0$ and integrating it over $\Omega$, we derive

According to the fact that $0 < d_{i}(K_{0})\leq d_{i}(w)$, $i = 1, 2$, due to ($H_{1}$) and (2.1), and by letting $c_{3}: = \min\{d_{1}(K_{0}), d_{2}(K_{0}), D\} > 0$, we deduce

It follows from Hölder's and Young's inequality as well as (2.15) that

According to (2.16), we have

By combining (2.20)–(2.22), we derive

By the ordinary differential equations (ODE) argument, there exists a $c_{4} > 0$ such that

which implies that

The proof of Lemma 2.4 is complete.

Finally, we shall give the following key estimate of $w$, which plays a crucial role in the proof of Theorem 1.1.

Lemma 2.5. Let the assumptions of Lemma 2.1 hold. Then there exists a constant $C > 0$ such that

and

where $\tau = \min\{1, \frac{1}{2}T_{\max}\}$.

Proof. Multiplying the third equation of System (1.8) with $-2\Delta w$ and integrating it by parts, we deduce from Young's inequality and (2.1) that

where we have used the hypotheses ($H_{2}$) and ($H_{4}$).

It follows from $\frac{\partial w}{\partial\nu} = 0$, Young's inequality and (2.1) that

Let $c_{5}: = \max\left\{\frac{4F_{1}^{2}(K_{0})}{D}, \frac{4F_{2}^{2}(K_{0})}{D}\right\}$; we infer from (2.28) and (2.29) that

where $c_{6}: = \frac{2(\mu K_{0})^{2}}{D}|\Omega|+\frac{K_{0}^{2}}{2D}|\Omega|$. It follows from Lemma 2.3 and Lemma 2.4 that (2.26) holds. Then integrating (2.30) over $(t, t+\tau)$, we can deduce from (2.12) and (2.26) that (2.27) holds.

3.   Global boundedness of solutions

In this section, we shall study the global existence and uniform boundedness of solutions for system (1.8) when $n = 2$. To do this, we need the following lemmas.

Lemma 3.1. Let the conditions of Theorem 1.1 hold. Then the solution $(u, v, w)$ of system (1.8) satisfies

and

for all $q\in[1, \infty)$ and $t\in(0, T_{\max})$, where $C > 0$ is a constant independent of $t$.

Proof. Multiplying the first equation of System (1.8) by $2u$ and integrating by parts, we deduce from Young's and Hölder's inequalities that

where $\mathcal{K}_{1}: = \max_{w\in[0, K_{0}]}\frac{|d'_{1}(w)|^{2}}{d_{1}(w)}$ and we have applied ($H_{1}$)–($H_{3}$) and (2.1).

By using the Gagliardo-Nirenberg inequality in two dimensions, there exists a $C_{1} > 0$ such that

According to Lemma 2.5 of ^[19] when $n = 2$, it follows from Lemma 2.5 that

for all $t\in(0, T_{\max})$, where $C_{2}, C_{3} > 0$. Thus, by combining (3.4) with (3.5), we infer from Young's inequality that

where $C_{4}, C_{5}$ are positive constants. Thus it follows from (3.3) and (3.6) that

where $C_{6}: = \max\{C_{4}, C_{5}+2\gamma_{1}F_{1}(K_{0})\}$.

By Lemma 2.4, we can find $t_{0} = t_{0}(t)\in((t-\tau)_{+}, t)$ for any $t\in(0, T_{\max})$ such that there exists a $C_{7} > 0$ satisfying

where $\tau$ is defined in Lemma 2.4. It follows from Lemma 2.5 that there exists a $C_{8} > 0$ such that

Therefore, integrating (3.7) over $(t_{0}, t)$, we deduce from $t_{0} < t < t_{0}+\tau\leq t_{0}+1$, (3.8) and (3.9) that

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

Similarly, we obtain

It follows from the third equation of System (1.8), we know that $w$ solves the following problem

where $g(u, v, w) = w-uF_{1}(w)-vF_{2}(w)+f(w)$. According to ($H_{2}$), ($H_{3}$) and (2.1), we infer from (3.10) and (3.11) that

for all $t\in(0, T_{\max})$. Hence, it follows from Lemma 2.2 in two dimensions that (3.2) holds. The proof of Lemma 3.1 is complete.

Next, we shall prove the boundedness of $w$ in $W^{1\infty}(\Omega)$.

Lemma 3.2. Let the conditions of Theorem 1.1 hold. Then the solution component $w$ of system (1.8) satisfies

for all $t\in(0, T_{\max})$, where $C > 0$ is a constant independent of $t$.

Proof. Multiplying the first equation of System (1.8) by $u^{2}$ and integrating by parts, we deduce from Young's and Hölder's inequalities that

for all $t\in(0, T_{\max})$, where $\mathcal{K}_{1}$ is defined in the proof of Lemma 3.1 and we have applied ($H_{1}$)–($H_{3}$) and (2.1).

It follows from Lemma 3.1 that there exist positive constants $C_{1}$ and $C_{2}$ such that $||\nabla w||_{L^{4}(\Omega)}\leq C_{1}$ and $||u||_{L^{2}(\Omega)}\leq C_{2}$ for all $t\in(0, T_{\max})$. Then by using the Gagliardo-Nirenberg inequality and Young's inequality, we can find positive constants $C_{i}$, $i = 3, \cdot\cdot\cdot, 6$ such that

and

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

Combining (3.15)–(3.17), we have

for all $t\in(0, T_{\max})$. By the ODE argument, we can derive

Similarly, we also derive the boundedness of $||v||_{L^{3}(\Omega)}$. Then it follows from Lemma 2.2 in two dimensions that (3.14) holds.

By means of the boundedness of $||w(\cdot, t)||_{W^{1, \infty}(\Omega)}$, it follows from the Moser iteration of ^[45] that we can obtain the boundedness of $||u(\cdot, t)||_{L^{\infty}(\Omega)}$ and $||v(\cdot, t)||_{L^{\infty}(\Omega)}$ for all $t\in(0, T_{\max})$.

Lemma 3.3. Let the conditions of Theorem 1.1 hold. Then the solution component $(u, v)$ of system (1.8) satisfies

for all $t\in(0, T_{\max})$, where $C > 0$ is a constant independent of $t$.

Proof. Multiplying the first equation of System (1.8) by $u^{p-1}$ with $p\geq2$ and integrating by parts, we deduce from Young's inequality that

for all $t\in(0, T_{\max})$, where $\mathcal{K}_{1}$ is defined in the proof of Lemma 3.1 and we have applied ($H_{1}$)–($H_{3}$) and (2.1).

It follows from Lemma 3.2 that there exists a $C_{1} > 0$ such that $||\nabla w(\cdot, t)||_{L^{\infty}(\Omega)}\leq C_{1}$ for all $t\in (0, T_{\max})$. Hence, we deduce from (3.21) that

for all $t\in(0, T_{\max})$, where $C_{2}: = \frac{\mathcal{K}_{1}C_{1}^{2}}{2}+\gamma_{1}F_{1}(K_{0})+1$ is independent of $p$. The rest can be handled exactly as the Moser iteration in Lemma 2.7 of ^[45] to derive the boundedness of $||u(\cdot, t)||_{L^{\infty}(\Omega)}$ for all $t\in(0, T_{\max})$. Similarly, we can obtain the boundedness of $||v(\cdot, t)||_{L^{\infty}(\Omega)}$ for all $t\in(0, T_{\max})$. The proof of Lemma 3.3 is complete.

Proof of Theorem 1.1. Theorem 1.1 is a direct consequence of Lemma 2.1, Lemma 3.2 and Lemma 3.3.

4.   Large time behavior

In this section, we shall study the asymptotic stability of global bounded solutions for System (1.11) by constructing energy functionals used in ^[13,57]. To do this, we first give some regularity results of the solution $(u, v, w)$ for System (1.11).

Lemma 4.1. Let $(u, v, w)$ be a global bounded classical solution for (1.11) ensured in Theorem 1.1. Then there exist $\sigma\in(0, 1)$ and $C > 0$ such that

Proof. This lemma can be verified by a similar argument in Lemma 4.1 of ^[14], so we omit the details here for brevity.

Lemma 4.2. Let $(u, v, w)$ be a global bounded classical solution for (1.11) ensured in Theorem 1.1. Then there exists a $C > 0$ such that

Proof. This lemma can be verified by a similar argument in Lemma 3.6 of ^[45], so we omit the details here for brevity.

In order to prove the asymptotic stabilization of global bounded solutions for system (1.11), we provide the following lemma, which is proved in ^[57].

Lemma 4.3. Let $\phi:(1, \infty)\rightarrow [0, \infty)$ be uniformly continuous such that $\int_{1}^{\infty}\phi(t)dt < \infty$. Then

4.1.   Proof of Theorem 1.2

In this subsection, we are devoted to studying the stabilization of the coexistence steady state $(u^{*}, v^{*}, w^{*})$ for some parameters cases. Let us introduce the following functionals

and

where $(u^{*}, v^{*}, w^{*})$ is given by (1.14).

Lemma 4.4. Let the conditions of Theorem 1.2 hold. Then there exists a positive constant $\varepsilon_{1}$ independent of $t$ such that

and

Proof. Let

then $\mathcal{E}_{1}(t)$ can be rewritten as

Step 1: We shall prove the nonnegativity of $\mathcal{E}_{1}(t)$ for all $t > 0$. Let $H(\xi): = \xi-u^{*}\ln \xi$ for $\xi > 0$; it follows from Taylor's formula for all $x \in \Omega$ and each $t > 0$ that there exists a $\tau = \tau(x, t) \in(0, 1)$ such that

Hence, we immediately derive that $I_{1}(t) = \int_{\Omega}\left(H(u)-H\left(u^{*}\right)\right)dx \geq 0$. Similarly, we know that $I_{2}(t) \geq 0$ and $I_{3}(t) \geq 0$ for all $t > 0$. Thus, we know that (4.4) holds.

Step 2: Now, we further prove (4.5). By a series of simple calculations, we get

where we have used the fact that $\theta = \gamma_{1}w^{*}-\alpha_{1}u^{*}-\beta_{1}v^{*}$.

Similarly, it follows from the identities $\theta = \gamma_{2}w^{*}-\beta_{2}u^{*}-\alpha_{2}v^{*}$ and $\mu = u^{*}+v^{*}+\mu w^{*}$ that

and

Hence, by combining (4.6)–(4.8), we derive

where $X_{1} = (u-u^{*}, v-v^{*}, w-w^{*})$ and $Y_{1} = \left(\frac{\nabla u}{u}, \frac{\nabla v}{v}, \nabla w\right)$, as well as

It follows from (1.20) that

as well as

which implies that the matrix $A_{1}$ is positive definite as according to Sylvester's criterion. Similarly, we deduce from (1.21) that

as well as

which implies that the matrix $B_{1}$ is positive definite. Thus there exist positive constants $\kappa_{1}$ and $\kappa_{2}$ such that

for all $x\in \Omega$ and $t > 0$. Let $\varepsilon_{1}: = \min\{\kappa_{1}, \kappa_{2}\}$, we have

which implies that (4.5) holds. The proof of Lemma 4.4 is complete.

With the aid of Lemma 4.4, we shall give the following large time behavior of global solutions for system (1.11).

Lemma 4.5. Let the assumptions of Theorem 1.2 hold. Then the global bounded solution of (1.11) converges to the coexistence steady state $(u^{*}, v^{*}, w^{*})$ given by (1.14), i.e.,

as $t\rightarrow \infty$.

Proof. It follows from Lemma 4.4 and integration over $(1, \infty)$ that

According to Theorem 1.1 and Lemma 4.1, the bounded solution $u, v$ and $w$ are Hölder continuous in $\bar{\Omega} \times[t, t+1]$ with respect to $t > 1$. Thus we conclude that $\mathcal{F}_{1}(t)$ is uniformly continuous in $(1, \infty)$. Thus we infer from Lemma 4.3 that

as $t \rightarrow \infty$. By the Gagliardo-Nirenberg inequality in two dimensions, there exists a $C_{1} > 0$ such that

Moreover, it follows from Lemma 4.2 that $u(\cdot, t)-u^{*}$ is bounded in $W^{1, 4}(\Omega)$; thus, we conclude from (4.18) that $u(\cdot, t) \rightarrow u^{*}$ in $L^{\infty}(\Omega)$ as $t \rightarrow \infty$. By the similar arguments for $v$ and $w$, we derive (4.17). The proof of Lemma 4.5 is complete.

Now, we give the convergence rate of the coexistence state $(u^{*}, v^{*}, w^{*})$ for System (1.11).

Lemma 4.6. Let the assumptions of Theorem 1.2 hold; the global bounded solution $(u, v, w)$ of (1.11) exponentially converges to the coexistence state $(u^{*}, v^{*}, w^{*})$, i.e. there exist $C > 0$ and $\lambda > 0$ such that

for all $t > T_{1}$, where $T_{1} > 0$ is some fixed time.

Proof. It follows from Lemma 4.5 that $||u-u^{*}||_{L^{\infty}(\Omega)}\rightarrow 0$ as $t\rightarrow \infty$. Therefore, we apply L'Hôpital's rule to obtain

which implies that there exists a $t_{1} > 0$ such that

for all $t > t_{1}$. Similarly, we can find $t_{2} > 0$ satisfying

and

for all $t > t_{2}$. Let $T_{1}: = \max\{t_{1}, t_{2}\}$; by means of the definitions of $\mathcal{E}_{1}(t)$ and $\mathcal{F}_{1}(t)$, it follows from the second inequalities in (4.21)–(4.23) that there exists a $C_{1} > 0$ such that

By Lemma 4.4, we derive

which implies that there exist $C_{2} > 0$ and $C_{3} > 0$ such that

Thus we deduce from the first inequalities in (4.21)–(4.23) that there exists a $C_{4} > 0$ such that

It follows from the Gagliardo-Nirenberg inequality in two dimensions, Lemma 4.2 and Lemma 3.2 that there exist positive constants $C_{5}$ and $C_{6}$ such that

for all $t > T_{1}$. The proof of Lemma 4.6 is complete.

Proof of Theorem 1.2. The statement of Theorem 1.2 is a straightforward consequence of Lemma 4.6.

4.2.   Proof of Theorems 1.3 and 1.4

In this subsection, we shall study the stabilization of the semi-trivial steady state $(\overline{u}, 0, \overline{w})$ or $(0, \widetilde{v}, \widetilde{w})$ for some parameters cases. Since the methods of the proofs of Theorem 1.3 and Theorem 1.4 are very similar, we only give the proof of Theorem 1.3 for brevity. To do this, let us introduce the following functionals

and

where $\overline{u} = \frac{\mu(\gamma_{1}-\theta)}{\alpha_{1}\mu+\gamma_{1}}\; \; \text{and}\; \; \overline{w} = \frac{\alpha_{1}\mu+\theta}{\alpha_{1}\mu+\gamma_{1}}$.

Lemma 4.7. Let the conditions of Theorem 1.3 hold. Then there exists a positive constant $\varepsilon_{2}$ independent of $t$ such that

and

Proof. Let

then $\mathcal{E}_{2}(t)$ can be represented as

Firstly, we can prove the nonnegativity of $\mathcal{E}_{2}(t)$ for all $t > 0$ by the similar arguments used in Step 1 in Lemma 4.4. For brevity, we omit the details here. Now, we shall prove (4.30). By a series of simple calculations, we get

where we have used the fact that $\theta = \gamma_{1}\overline{w}-\alpha_{1}\overline{u}$.

Similarly, we can derive

and

where we have used the fact that $\mu = \overline{u}+\mu \overline{w}$. Thus it follows from (4.31)–(4.33) that

where $X_{2} = (u-\overline{u}, v, w-\overline{w})$ and $Y_{2} = \left(\frac{\nabla u}{u}, \nabla w\right)$, as well as

It follows from (1.20) that

as well as

which implies that the matrix $A_{2}$ is positive definite as according to Sylvester's criterion. Similarly, we deduce from (1.23) that

which implies that the matrix $B_{2}$ is positive definite. Thus there exist positive constants $\iota_{1}$ and $\iota_{2}$ such that

for all $x\in \Omega$ and $t > 0$. Let $\varepsilon_{2}\in (0, \min\{\iota_{1}, \iota_{2}\})$; we obtain

which implies that (4.30) holds. The proof of Lemma 4.7 is complete.

With the help of Lemma 4.7, we shall give the following stabilization of the semi-trivial steady state $(\overline{u}, 0, \overline{w})$ for System (1.11).

Lemma 4.8. Let the assumptions of Theorem 1.3 hold. Then the global bounded solution $(u, v, w)$ of (1.11) converges to the semi-trivial steady state $(\overline{u}, 0, \overline{w})$ given by (1.24), i.e.,

as $t\rightarrow \infty$.

Proof. The proof of this lemma is similar to that of Lemma 4.5; here we omit the details.

Now, we give the convergence rate of the semi-trivial steady state $(\overline{u}, 0, \overline{w})$ for System (1.11).

Lemma 4.9. Let the assumptions of Theorem 1.3 hold; then, there exist positive constants $C$ and $\lambda$ such that:

(a) when $\theta = \gamma_{2}\overline{w}-\beta_{2}\overline{u}$, then

(b) when $\theta > \gamma_{2}\overline{w}-\beta_{2}\overline{u}$, then

where $T_{2} > 0$ is some fixed time.

Proof. Let

then it follows from Lemma 4.7 that there exists a $\varepsilon_{2} > 0$ such that

We deduce from Lemma 4.8 that $||u(\cdot, t)-\overline{u}||_{L^{\infty}(\Omega)}+||v(\cdot, t)||_{L^{\infty}(\Omega)}+||w(\cdot, t)-\overline{w}||_{L^{\infty}(\Omega)}\rightarrow 0$ as $t\rightarrow \infty$. Hence, we apply L'Hôpital's rule to obtain

which implies that there exists a $t'_{1} > 0$ such that

for all $t > t'_{1}$. Similarly, we can find $t'_{2} > 0$ satisfying

for all $t > t'_{2}$.

By using the fact that $\lim_{s\rightarrow 0}\frac{s}{s^{2}+s} = 1$, it follows from $||v(\cdot, t)||_{L^{\infty}(\Omega)}\rightarrow 0$ as $t\rightarrow \infty$ that there exists a $t'_{3} > 0$ such that

for all $t > t'_{3}$.

$(a)$ When $\theta = \gamma_{2}\overline{w}-\beta_{2}\overline{u}$, (4.45) can be turned into

Let $T_{2}: = \max\{t'_{1}, t'_{2}, t'_{3}\}$; by means of the definitions of $\mathcal{E}_{2}(t)$ and $\mathcal{F}_{2}^{*}(t)$, it follows from the second inequalities in (4.47) and (4.48) that there exist positive constants $C_{1}$ and $C_{2}$ such that

for all $t > T_{2}$, where we have used Hölder's inequality and the boundedness of $(u, v, w)$ asserted by Theorem 1.1. Thus, we deduce from (4.50) that

which implies

with some positive constant $C_{3}$. Hence we infer from the first inequalities in (4.47)–(4.49) that there exists a $C_{4} > 0$ such that

It follows from the Gagliardo-Nirenberg inequality in two dimensions, Lemma 4.2 and Lemma 3.2 that there exist positive constants $C_{5}$ and $C_{6}$ such that

for all $t > T_{2}$.

$(b)$ When $\theta > \gamma_{2}\overline{w}-\beta_{2}\overline{u}$, let $T_{2}: = \max\{t'_{1}, t'_{2}, t'_{3}\}$; by means of the definitions of $\mathcal{E}_{2}(t)$ and $\mathcal{F}_{2}^{*}(t)$, it follows from the second inequalities in (4.47) and (4.48) that there exists a positive constant $C_{7}$ such that

for all $t > T_{2}$.

By combining (4.45) with (4.56), we have

Since $\theta > \gamma_{2}\overline{w}-\beta_{2}\overline{u}$, then we can select $\varepsilon_{2}\leq \frac{1}{\gamma_{2}}(\theta-\gamma_{2}\overline{w}+\beta_{2}\overline{u})$ such that

which means that there exist $C_{8} > 0$ and $C_{9} > 0$ satisfying

Thus we deduce from the first inequalities in (4.47)–(4.49) that there exists a $C_{10} > 0$ such that

It follows from the Gagliardo-Nirenberg inequality in two dimensions, Lemma 4.2 and Lemma 3.2 that there exist positive constants $C_{11}$ and $C_{12}$ such that

for all $t > T_{2}$. The proof of Lemma 4.9 is complete.

Proof of Theorem 1.3. The statement of Theorem 1.3 is a direct consequence of Lemma 4.9.

4.3.   Proof of Theorem 1.5

In this subsection, we are devoted to discussing the asymptotic stability of the prey-only steady state $(0, 0, 1)$ under some suitable parameters conditions. To do this, let us denote the following functionals

and

we can derive the following estimates of $\mathcal{E}_{3}(t)$ and $\mathcal{F}_{3}(t)$.

Lemma 4.10. Let the conditions of Theorem 1.5 hold. Then there exists a $\varepsilon_{3} > 0$ independent of $t$ such that

and

Proof. By the similar arguments as in the proofs of Lemma 4.4 and Lemma 4.7, we can derive $(4.62)$ and

where we have used the fact that $w\leq K_{0} = \max\{||w_{0}||_{L^{\infty}(\Omega)}, 1\}$. Let $\varepsilon_{3}\in \left(0, \min\{\frac{\alpha_{1}}{\gamma_{1}}, \frac{\alpha_{2}}{\gamma_{2}}, \mu, \frac{D}{K_{0}^{2}}\}\right)$; we obtain

for all $t > 0$. The proof of Lemma 4.10 is complete.

With the help of Lemma 4.10, we shall give the following stabilization of the prey-only steady state for System (1.11).

Lemma 4.11. Let the assumptions of Theorem 1.5 hold. Then the global bounded solution of (1.11) converges to the prey-only steady state $(0, 0, 1)$, i.e.,

as $t\rightarrow \infty$.

Proof. The proof of this lemma is similar to that of Lemma 4.5; here we omit the details.

Now, we give the convergence rate of the prey-only steady state $(0, 0, 1)$ for System (1.11).

Lemma 4.12. Let the assumptions of Theorem 1.5 hold; then there exist positive constants $C$ and $\lambda$ such that:

(a) when $\gamma_{i} = \theta$, $i = 1, 2$, then

(b) when $\gamma_{i} < \theta$, $i = 1, 2$, then

where $T_{3} > 0$ is some fixed time.

Proof. Let

then it follows from Lemma 4.10 that there exists a $\varepsilon_{3} > 0$ such that

By using the facts that $\lim_{s\rightarrow 0}\frac{s}{s^{2}+s} = 1$ and $\lim_{s\rightarrow 1}\frac{s-1-\ln s}{(s-1)^{2}} = \frac{1}{2}$, it follows from $||u(\cdot, t)||_{L^{\infty}(\Omega)}+||v(\cdot, t)||_{L^{\infty}(\Omega)}+||w(\cdot, t)-1||_{L^{\infty}(\Omega)}\rightarrow 0$ as $t\rightarrow \infty$, as asserted in Lemma 4.11 that there exists a $T_{3} > 0$ such that

and

as well as

for all $t > T_{3}$.

$(a)$ When $\gamma_{i} = \theta$, $i = 1, 2$, (4.70) can be simplified as

By means of the definitions of $\mathcal{E}_{3}(t)$ and $\mathcal{F}_{3}^{*}(t)$, it follows from the second inequality in (4.73) that there exist positive constants $C_{1}$ and $C_{2}$ such that

for all $t > T_{3}$, where we have used Hölder's inequality and the boundedness of $(u, v, w)$ asserted by Theorem 1.1. Thus we deduce from (4.74) that

which implies

with some positive constant $C_{3}$. Hence we infer from the first inequalities in (4.71)–(4.73) that there exists a $C_{4} > 0$ such that

It follows from the Gagliardo-Nirenberg inequality in two dimensions, Lemma 4.2 and Lemma 3.2 that there exist positive constants $C_{5}$ and $C_{6}$ such that

for all $t > T_{3}$.

$(b)$ When $\gamma_{i} < \theta$, $i = 1, 2$, by means of the definitions of $\mathcal{E}_{3}(t)$ and $\mathcal{F}_{3}^{*}(t)$, it follows from the second inequalities in (4.71)–(4.73) that there exists a positive constant $C_{7}$ such that

for all $t > T_{3}$.

By combining (4.70) with (4.80), we derive

for all $t > 0$. Since $\gamma_{i} < \theta$, $i = 1, 2$, we can select $\varepsilon_{3}\leq \min\left\{\frac{1}{\gamma_{1}}(\theta-\gamma_{1}), \frac{1}{\gamma_{2}}(\theta-\gamma_{2})\right\}$ such that

which means that there exist $C_{8} > 0$ and $C_{9} > 0$ satisfying

Thus we deduce from the first inequalities in (4.71)–(4.73) that there exists a $C_{10} > 0$ such that

It follows from the Gagliardo-Nirenberg inequality in two dimensions, Lemma 4.2 and Lemma 3.2 that there exist positive constants $C_{11}$ and $C_{12}$ such that

for all $t > T_{3}$. The proof of Lemma 4.12 is complete.

Proof of Theorem 1.5. The statement of Theorem 1.5 is a direct consequence of Lemma 4.12.

Acknowledgments

The author is grateful to the editor and three anonymous referees for their valuable comments, which greatly improved the exposition of our paper. The author is also deeply grateful to Professor Renjun Duan for the support and friendly hospitality in CUHK. The work was partially supported by the National Natural Science Foundation of China (Grant Nos: 11601053, 11526042), Natural Science Foundation of Chongqing (Grant No. cstc2019jcyj-msxmX0082), Chongqing Municipal Education Commission Science and Technology Research Project (Grant No. KJZD-K202200602), China-South Africa Young Scientist Exchange Project in 2020, The Hong Kong Scholars Program (Grant Nos: XJ2021042, 2021-005) and Young Hundred Talents Program of CQUPT from 2022–2024.

Conflict of interest

The author declares that there is no conflict of interest.