CED-Net: A more effective DenseNet model with channel enhancement

1.   Introduction and main results

A taxis is the movement of an organism in response to a stimulus such as chemical signal or the presence of food. Taxes can be classified based on the types of stimulus, such as chemotaxis, prey-taxis, galvanotaxis, phototaxis and so on. According to the direction of movements, the taxis is said to be attractive (resp. repulsive) if the organism moves towards (resp. away from) the stimulus. In the ecosystem, a widespread phenomenon is the prey-taxis, where predators move up the prey density gradient, which is often referred to as the direct prey-taxis. However some predators may approach the prey by tracking the chemical signals released by the prey, such as the smell of blood or specific odo, and such movement is called indirect prey-taxis (cf. ^[1]). Since the pioneering modeling work by Kareiva and Odell ^[2], prey-taxis models have been widely studied in recent years (cf. ^{[3,4,5,6,7,8,9,10,11,12]}), followed by numerous extensions, such as three-species prey-taxis models (cf. ^[13,14,15]) and predator-taxis models (cf. ^[16,17]). The indirect prey-taxis models have also been well studied (cf. ^[18,19,20]).

Recently, a predator-prey model with attraction-repulsion taxis mechanisms was proposed by Bell and Haskell in ^[21] to describe the interaction between direct prey-taxis and indirect chemotaxis, where the direct prey-taxis describes the predator's directional movement towards the prey density gradient, while the indirect chemotaxis models a defense mechanism in which the prey repels the predator by releasing odour chemicals (like a fox breaking wind in order to escape from hunting dogs). The model reads as

where the unknown functions $u(x, t)$, $v(x, t)$ and $w(x, t)$ denote the densities of the prey, predator and prey-derived chemical repellent, respectively, at position $x \in \Omega$ and time $t > 0$. Here, $\Omega\subset \mathbb{R}^n$ is a bounded domain (habitat of species) with smooth boundary $\partial\Omega$, and $\nu$ is the unit outer normal vector of $\partial\Omega$. The parameters $d$, $\eta$, $\chi$, $\xi$, $a_1$, $a_2$, $a_3$, $e$, $\rho$, $r$, $\gamma$ are all positive, where $\chi > 0$ and $\xi > 0$ denote the (attractive) prey-taxis and (repulsive) chemotaxis coefficients, respectively. The predator $v$ is assumed to be a generalist, so that it has a logistic growth term $\rho v(1-v)$ with intrinsic growth rate $\rho > 0$. More modeling details with biological interpretations are referred to in ^[21]. We remark that the predator-prey model with attraction-repulsion taxes has some similar structures to the so-called attraction-repulsion chemotaxis model proposed originally in ^[22], where the species elicit both attractive and repulsive chemicals (see ^{[23,24,25,26]} and references therein for some mathematical studies).

The initial data satisfy the following conditions:

In ^[21], the global existence of strong solutions to (1.1) was established in one dimension ($n = 1$), and the existence of nontrivial steady state solutions alongside pattern formation was studied by the bifurcation theory. The main purpose of this paper is to study the global dynamics of (1.1) in higher dimensional spaces, which are usually more physical in the real world. Specifically, we shall show the existence of global classical solutions in all dimensions and explore the global stability of constant steady states, by which we may see how parameter values play roles in determining these dynamical properties of solutions.

The first main result is concerned with the global existence and boundedness of solutions to (1.1). For the convenience of presentation, we let

and

Then, the result on the global boundedness of solutions to (1.1) is stated as follows.

Theorem 1.1 (Global existence). Let $\Omega\subset \mathbb{R}^n(n\geqslant1)$ be a bounded domain with smooth boundary and parameters $d$, $\eta$, $\chi$, $\xi$, $a_1$, $a_2$, $a_3$, $e$, $\rho$, $r$, $\gamma$ be positive. If

where $K_3(p)$ is defined in $(1.4)$, then for any initial data $(u_0, v_0, w_0)$ satisfying $(1.2)$, the system $(1.1)$ admits a unique classicalsolution $(u, v, w)$ satisfying

and $u, v, w > 0$ in $\Omega\times(0, +\infty)$. Moreover, there exists a constant $C > 0$ independent of $t$ such that

Our next goal is to explore the large-time behavior of solutions to (1.1). Simple calculations show the system (1.1) has four possible homogeneous equilibria as classified below:

with

where the trivial equilibrium $(0, 0, 0)$ is called the extinction steady state, $(0, 1, 0)$ is the predator-only steady state, and $(u_*, v_*, w_*)$ is the coexistence steady state. We shall show that if $a_1 > a_3$, then the coexistence steady state is globally asymptotically stable with exponential convergence rate, provided that $\xi$ and $\chi$ are suitably small, while if $a_1\leqslant a_3$, the predator-only steady state is globally asymptotically stable with exponential or algebraic convergence rate when $\xi$ and $\chi$ are suitably small. To state our results, we denote

and

where $K_1$ is defined in (1.3). Then, the global stability result is stated in the following theorem.

Theorem 1.2 (Global stability). Let the assumptions in Theorem 1.1 hold. Then, the following results hold.

(1) Let $a_1 > a_3$. If $\xi$ and $\chi$ satisfy

where $a = \max\big\{\frac{\xi^2}{\Gamma}, \Phi-\sqrt{\Phi^2-e^2}\big\}, b = \Phi+\sqrt{\Phi^2-e^2}$, then there exist some constants $T_*$, $C$, $\alpha > 0$ such that the solution $(u, v, w)$ obtained in Theorem 1.1 satisfies for all $t\geqslant T_*$

(2) Let $a_1\leqslant a_3$, If $\xi$ and $\chi$ satisfy

then there exist some constants $T^*$, $C$, $\beta > 0$ such that the solution $(u, v, w)$ obtained in Theorem 1.1 satisfies, for all $t\geqslant T^*$,

Remark 1.1. In the biological view, the relative sizes of $a_1$ and $a_2$ determine the coexistence of the system. The results indicated that a large $\frac{a_1}{a_2}$ facilitates the coexistence of the species.

The rest of this paper is organized as follows. In Section 2, we state the local existence of solutions to (1.1) with extensibility conditions. Then, we deduce some a priori estimates and prove Theorem 1.1 in Section 3. Finally, we show the global convergence to the constant steady states and prove Theorem 1.2 in Section 4.

2.   Preliminary

For convenience, in what follows we shall use $C_i(i = 1, 2, \cdots)$ to denote a generic positive constant which may vary from line to line. For simplicity, we abbreviate $\int_{0}^{t}\int_\Omega f(\cdot, s)dxds$ and $\int_\Omega f(\cdot, t)dx$ as $\int_{0}^{t}\int_\Omega f$ and $\int_\Omega f$, respectively. The local existence and extensibility result of problem (1.1) can be directly established by the well-known Amman's theory for triangular parabolic systems (cf. ^[27,28]). Below, we shall present the local existence theorem without proof for brevity, and we refer to ^[21] for the proof in one dimension as a reference.

Lemma 2.1 (Local existence and extensibility). Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with smooth boundary. The parameters $d$, $\eta$, $\chi$, $\xi$, $a_1$, $a_2$, $a_3$, e, $\rho$, r, $\gamma$ are positive. Then, for the initial data $(u_0, v_0, w_0)$ satisfying $(1.2)$, there exists $T_{max}\in(0, \infty]$ such that the system $(1.1)$ admits a unique classicalsolution $(u, v, w)$ satisfying

and $u, v, w > 0$ in $\Omega\times(0, T_{max})$. Moreover, we have

We recall some well-known results which will be used later frequently. The first one is an uniform Grönwall inequality ^[29].

Lemma 2.2. Let $T_{max} > 0$, $\tau\in(0, T_{max})$. Suppose that $c_1$, $c_2$, $y$ are three positive locally integrable functions on $(0, T_{max})$ such that $y'$ is locally integrable on $(0, T_{max})$ and satisfies

If

where $C_{i}(i = 1, 2, 3)$ are positive constants, then

Next, we recall a basic inequality ^[30].

Lemma 2.3. Let $p\in[1, \infty)$. Then, the following inequality holds:

for any $u\in C^{2}(\bar{\Omega})$ satisfying $\frac{\partial u}{\partial \nu} = 0$ on $\partial \Omega$, where $D^{2}u$ denotes the Hessian of $u$.

The last one is a Gagliardo-Nirenberg type inequality shown in [31,Lemma 2.5].

Lemma 2.4. Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary. Then, for any $\varphi\in W^{2, 2}(\Omega)$ satisfying $\frac{\partial \varphi}{\partial \nu}|_{\partial\Omega} = 0$, there exists a positive constant $C$ depending only on $\Omega$ such that

3.   Global existence

In this section, we establish the global boundedness of solutions to the system (1.1). To this end, we will proceed with several steps to derive a priori estimates for the solution of the system (1.1). The first one is the uniform-in-time $L^\infty(\Omega)$ boundedness of $u$.

Lemma 3.1. Let $(u, v, w)$ be the solution of $(1.1)$ and $K_1$ be as defined in $(1.3)$. Then, we have

Furthermore, there is a constant $C > 0$ such that for any $0 < \tau < \min\{T_{max}, 1\}$, it follows that

Proof. The result is a direct consequence of the maximum principle applied to the first equation in (1.1). Indeed, if we let $\bar{u} = \max\left\{\frac{a_{1}}{a_{2}}, \left\|u_{0}\right\|_{L^\infty(\Omega)}\right\}$, then $\bar{u}$ satisfies

Apparently, the comparison principle of parabolic equations gives $u\leqslant \bar{u}$ on $\Omega \times\left(0, T_{\max }\right)$.

Next, we multiply the first equation of (1.1) by $u$ and integrate the result to get

Then, the integration of the above inequality with respect to $t$ over $(t, t+\tau)$ completes the proof by noting that $\int_\Omega u_0^2$ is bounded.

Having at hand the uniform-in-time $L^\infty(\Omega)$ boundedness of $u$, the a priori estimate of $w$ follows immediately.

Lemma 3.2. Let $(u, v, w)$ be the solution of $(1.1)$. We can find a constant $C > 0$ satisfying

Proof. Noting the boundedness of $\|u\|_{L^\infty(\Omega)}$ from Lemma 3.1, we get the desired result from the third equation of (1.1) and the regularity theorem [32,Lemma 1].

Now, the a priori estimate of $v$ can be obtained as below.

Lemma 3.3. Let $(u, v, w)$ be the solution of $(1.1)$. Then, there exists a constant $C > 0$ such that

and

where $\tau$ is a constant such that $0 < \tau < \min\{T_{max}, 1\}$.

Proof. Integrating the second equation of (1.1) over $\Omega$ by parts, using Young's inequality and Lemma 3.1, we find some constant $C_1 > 0$ such that

Hence, (3.1) is obtained by the Grönwall inequality. Integrating (3.3) over $(t, t+\tau)$, we get (3.2) immediately.

Due to the estimates of $u$ and $v$ obtained in Lemmas 3.1 and 3.3 respectively, we have the following improved uniform-in-time $L^2(\Omega)$ boundedness of $\nabla u$ and the space-time $L^2$ boundedness of $\Delta u$ when $n = 2$.

Lemma 3.4. Let $(u, v, w)$ be the solution of $(1.1)$. If $n = 2$, then we can find a constant $C > 0$ such that

and

where $\tau$ is defined in Lemma 3.3.

Proof. Integrating the first equation of (1.1) by parts and using Lemma 3.1, we find a constant $C_1 > 0$ such that

The Gagliardo-Nirenberg inequality in Lemma 2.4, Young's inequality and Lemma 3.1 yield some constants $C_2, C_3 > 0$ satisfying

and

which along with (3.6) imply

Then, applications of Lemma 2.2, 3.1 and 3.3 give (3.4). Finally, (3.5) can be obtained by integrating (3.7) over $(t, t+\tau)$.

Now, the uniform-in-time boundedness of $v$ in $L^2(\Omega)$ can be established when $n = 2$.

Lemma 3.5. Let $(u, v, w)$ be the solution of $(1.1)$. If $n = 2$, then there exists a constant $C > 0$ such that

Proof. Multiplying the second equation of (1.1) by $v$, integrating the result by parts and using Young's inequality, we have

which along with Lemma 3.1 and Lemma 3.2 gives some constant $C_1 > 0$ such that

Using Lemmas 3.1 and 3.3, Hölder's inequality, Lemma 2.4 and Young's inequality, we find some constants $C_2, C_3, C_4 > 0$ such that

Furthermore, Young's inequality yields some constant $C_5 > 0$ such that

Substituting (3.9) and (3.10) into (3.8), we get

which alongside Lemma 2.2, Lemma 3.3 and Lemma 3.4 completes the proof.

To get the global existence of solutions in any dimensions, we derive the following functional inequality which gives an a priori estimate on $\nabla u$.

Lemma 3.6. Let $(u, v, w)$ be the solution of $(1.1)$ and $q\geqslant2$. If $n\geqslant1$, then there exists a constant $C > 0$ such that

where $K_2$ is defined in (1.3).

Proof. From the first equation of (1.1) and the fact $2\nabla u\cdot\nabla\Delta u = \Delta|\nabla u|^2-2|D^2u|^2$, it follows that

which implies

Now, we estimate the right hand side of (3.11). Choosing $s\in(0, \frac{1}{2})$ and

we get

which, along with the Gagliardo-Nirenberg inequality, Young's inequality and the embedding

gives some constants $C_1$, $C_2$, $C_3$, $C_4 > 0$ such that

Therefore, it holds that

Owning to the fact $|\Delta u|\leqslant \sqrt{n}|D^2u|$, Young's inequality and Lemma 3.1, we have

where $K_2$ is defined in (1.3). Hence, substituting the estimates $I_1$ and $I_2$ into (3.11), we finish the proof of the lemma.

Now, we show the following functional inequality to derive the a priori estimate on $v$ in any dimensions.

Lemma 3.7. Let $(u, v, w)$ be the solution of $(1.1)$ and $q\geqslant2$. If $n\geqslant1$, we can find a constant $C > 0$ such that

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

Proof. Utilizing the second equation of (1.1) and integration by parts, we get

Now, we estimate the right hand side of (3.12). An application of Young's inequality and Lemma 3.2 yields some constant $C_1 > 0$ such that

and

which along with (3.12), Lemma 3.1 and the fact

gives a constant $C_2 > 0$ such that

Hence, we finish the proof of the lemma.

Combining Lemmas 3.6 and 3.7, we have the following inequality which can help us to achieve the global existence of solutions in any dimensions.

Lemma 3.8. Let $(u, v, w)$ be the solution of $(1.1)$ and $p\geqslant2$. If $n\geqslant1$, we can find a constant $C > 0$ such that

where $K_3(p)$ is defined in (1.4).

Proof. Combining Lemmas 3.6 and 3.7, we see for any $p = q\geqslant2$ there exists a constant $C_1 > 0$ such that for all $t \in\left(0, T_{max}\right)$

Now, we estimate the right hand side of the above inequality. Indeed, owing to Lemma 2.3 and Young's inequality, for all $t \in\left(0, T_{max}\right)$, we have

and

where $K_1$ and $K_2$ are defined in (1.3). Similarly, we can find a constant $C_2 > 0$ such that

Substituting the above estimates into (3.13), we get

where $K_3(p)$ is given in (1.4). Furthermore, we can use Young's inequality and Lemma 2.3 to get a constant $C_3 > 0$ such that

and

which together with (3.14) finishes the proof.

Next, we shall deduce a criterion of global boundedness of solutions for the system (1.1) inspired by an idea of ^[33].

Lemma 3.9. Let $n\geqslant1$. If there exist $M > 0$ and $p_0 > \frac{n}{2}$ such that

then $T_{max} = +\infty$. Moreover, there exists $C > 0$ such that

Proof. We divide the proof into two steps.

Step 1: We claim that there exists a constant $C_1 > 0$ such that

Indeed, due to Lemma 3.8, for any $p = 2p_0$, there exists a constant $C_2 > 0$ such that

Let

Then, $\frac{2p_0+1}{2p_0}\theta < 1$ due to $p_0 > \frac{n}{2}$. By the Gagliardo-Nirenberg inequality, Young's inequality and (3.15), we can find some constants $C_3, C_4 > 0$ such that

which along with (3.16) implies

Therefore, the claim follows from the Grönwall inequality applied to the above inequality.

Step 2: Thanks to the regularity theorem [32,Lemma 1], we can find a constant $C_5 > 0$ such that $\|\nabla u\|_{L^\infty(\Omega)}\leqslant C_5$ due to $2p_0 > n$. With (3.12) and Lemmas 3.1 and 3.2, we get a constant $C_6 > 0$ such that for any $p\geqslant2$

Thanks to Young's inequality, we find a constant $C_7 > 0$ such that

which together with (3.17) implies

with $C_8 = C_7+\rho+ea_3K_1+1$. Applying $1+p^n\leqslant(1+p)^n$ and the following inequality ^[34]

with $f = v^{\frac{p}{2}}$ and $\varepsilon = \frac{2}{p^2C_8}$, we find a constant $C_{10} > 0$ such that

Substituting (3.19) into (3.18), we have

Then, employing the standard Moser iteration in ^[35] or a similar argument as in ^[34], we can prove that there exists a constant $C_{11} > 0$ such that

Thus, with the help of Lemma 3.2, we finish the proof.

Now, utilizing the criterion in Lemma 3.9, we prove the global existence and boundedness of solutions for the system (1.1).

Proof of Theorem 1.1. If $n\leqslant2$, then the conclusion of the theorem can be obtained by Lemmas 3.3, 3.5 and 3.9. If $n\geqslant3$ and

then according to Lemma 3.8, by fixing $p = [\frac{n}{2}]+1$ we can find a constant $C_1 > 0$ such that

which along with the Grönwall inequality gives a constant $C_2 > 0$,

Together with Lemma 3.9, we finish the proof by Lemma 2.1.

4.   Stabilization

In this section, we will employ suitable Lyapunov functionals to study the large-time behavior of $u$, $v$ and $w$. We first improve the regularity of the solution.

Lemma 4.1. There exist constants $\theta_1, \theta_2, \theta_3\in(0, 1)$ and $C > 0$ such that

In particular, one can find $C > 0$ such that

Proof. The conclusion is a consequence of the regularity of parabolic equations in ^[36].

We split our analysis into two cases: $a_1 > a_3$ and $a_1\leqslant a_3$.

4.1.   Coexistence: $a_1 > a_3$

We know that there are four homogeneous equilibria $(0, 0, 0)$, $(0, 1, 0)$, $\left(\frac{a_1}{a_2}, 0, \frac{ra_1}{\gamma a_2}\right)$ and $(u_*, v_*, w_*)$ when $a_1 > a_3$, where $u_*, v_*$ and $w_*$ are defined in (1.5). In this case, we shall prove the coexistence steady state $(u_*, v_*, w_*)$ is globally exponentially stable under certain conditions. Define an energy functional for (1.1) as follows:

where $\varepsilon_1$ and $\varepsilon_2$ are to be determined below.

Proof of Theorem 1.2–(1). We complete the proof in four steps.

Step 1: The parameters $\varepsilon_1$ and $\varepsilon_2$ can be chosen in the following way. First, we recall from (1.5) and (1.6) that

Let

It is clear that $f\in C^0((0, +\infty))$. Then, if

the following holds:

Under (4.2), we let $a = \max\left\{\frac{\xi^2}{\Gamma}, \Phi-\sqrt{\Phi^2-e^2}\right\}$ and $b = \Phi+\sqrt{\Phi^2-e^2}$ with $a < b$. Then, $f(y)$ is continuous on $[a, b]$ with $f(a) = f(b) = 0$, and consequently $f(y)$ must attain the maximum at some point, say $\varepsilon_1$, in $(a, b)$, namely $f(\varepsilon_1) = \max \limits_{y\in [a, b]} f(y)$. Then, $a < \varepsilon_1 < b$, or equivalently (see (4.1))

Next, we assume $\chi > 0$ is suitably small such that

which implies

Hence, there exists a constant $\varepsilon_2 > 0$ such that

which along with Lemma 3.1 yields

Step 2: We claim

Indeed, using the equations in system (1.1) along with integration by parts, we have

Similarly, we obtain

and

Then, it follows that

where $X = (\nabla u, \nabla v, \nabla w)$, $Y = (u-u_*, v-v_*, w-w_*)$, and

Note that (4.3) yields

and (4.4) gives

The above results indicate that matrix $S$ is positive definite. Using (4.3) and (4.4) again, we observe that

and

which imply that matrix $T$ is positive definite. Therefore, one can choose a constant $C_1 > 0$ such that

Integrating the above inequality with respect to time, we get a constant $C_2 > 0$ satisfying

which together with the uniform continuity of $u, v$ and $w$ due to Lemma 4.1 yields

By the Gagliardo-Nirenberg inequality, we can find a constant $C_3 > 0$ such that

and

which along with (4.6) and Lemma 4.1 prove the claim.

Step 3: From the L'Hôpital rule, it holds that for any $s_0 > 0$

which gives a constant $\eta > 0$ such that for all $|s-s_0|\leqslant\eta$

By (4.6), there exists $T_1 > 1$ such that

Therefore, by (4.10), we get

and

Step 4: From (4.11) and (4.12), it follows that

which alongside (4.5) yields a constant $C_4 > 0$ such that

This immediately gives a constant $C_5 > 0$ such that

Hence, utilizing (4.11) and (4.12) again, one obtains a constant $C_6 > 0$ such that

Finally, by (4.7)–(4.9) and Lemma 4.1, we get the decay rates of $\|u-u_*\|_{L^\infty(\Omega)}$, $\|v-v_*\|_{L^\infty(\Omega)}$ and $\|w-w_*\|_{L^\infty(\Omega)}$, as claimed in Theorem 1.2–(1).

4.2.   Predator-only: $a_1\leqslant a_3$

In this case, there are three homogeneous equilibria $(0, 0, 0)$, $(0, 1, 0)$ and $\left(\frac{a_1}{a_2}, 0, \frac{ra_1}{\gamma a_2}\right)$, and we shall show that the steady state $(0, 1, 0)$ is global asymptotically stable, where the convergence rate is exponential if $a_1 < a_3$ and algebraic if $a_1 = a_3$. Define an energy functional for (1.1) as follows:

where $\zeta_1$ and $\zeta_2$ will be determined below.

Proof of Theorem 1.2–(2). We divide the proof into five steps.

Step 1: We shall choose the appropriate parameters $\zeta_1$ and $\zeta_2$. By the definitions of $A$ and $B$ in (1.7), since $A < B$, we have

Let

Then, $g\in C^1\left(\left(\frac{\xi^2}{4d}, \frac{ea_2}{a_1}\right)\right)$, and $g(y) > 0$ in $\left(\frac{\xi^2}{4d}, \frac{ea_2}{a_1}\right)$. We further observe that

which along with $\chi^2 < D\left(A+B-2\sqrt{AB}\right)$ implies

By the definition of $g$, one has

which alongside (4.13) gives $y_0 = \frac{\xi}{2}\sqrt{\frac{ea_2}{da_1}}\in\left(\frac{\xi^2}{4d}, \frac{ea_2}{a_1}\right)$. Thus, $g(y)$ is increasing in $\left(\frac{\xi^2}{4d}, \frac{\xi}{2}\sqrt{\frac{ea_2}{da_1}}\right)$ and decreasing in $\left(\frac{\xi}{2}\sqrt{\frac{ea_2}{da_1}}, \frac{ea_2}{a_1}\right)$. We can find a constant $\zeta_1 > 0$ such that

and

With the definition of $g$, we get

which implies that there exists $\zeta_2 > 0$ such that

One can verify that

and

Thanks to (4.13) and (4.14), one obtains

Step 2: We claim

Indeed, if $(u, v, w)$ is the solution of system (1.1), then we get

and

Then, combining (4.20), (4.21), (4.22) and (4.23), we have from the definition of $G(t)$ that

where $X = (\nabla u, \nabla v, \nabla w)$, $Y = (u, v-1, w)$,

It can be checked that (4.16) and (4.18) ensure that the matrix $P$ is positive definite while (4.17) and (4.18) guarantee that the matrix $Q$ is positive definite. Thus, there is a constant $C_1 > 0$ such that if $a_1 < a_3$, then

and if $a_1 = a_3$, then

Integrating the above inequalities with respect to time, we find a constant $C_2 > 0$ satisfying

which together with the uniform continuity of $u, v$ and $w$ due to Lemma 4.1 yields

Thus, (4.19) is obtained by the Gagliardo-Nirenberg inequality and Lemma 4.1.

Step 3: By the L'Hôpital rule, we get

which gives a constant $\varepsilon > 0$ such that

By (4.19), there exists $T_1 > 0$ such that

Therefore, it follows from (4.28) that

Step 4: If $a_1 < a_3$, from the definition of $G(t)$ and (4.30), one has

which along with (4.25) yields a constant $C_3 > 0$ such that

This gives a constant $C_4 > 0$ such that

Hence, utilizing (4.30) again, we find a constant $C_5 > 0$ such that

Then, by the Gagliardo-Nirenberg inequality and Lemma 4.1, we get the exponential convergence for $\|u\|_{L^{\infty}(\Omega)}+\|v-1\|_{L^{\infty}(\Omega)}+\|w\|_{L^{\infty}(\Omega)}$.

Step 5: If $a_1 = a_3$, we use (4.29), (4.30) and Young's inequality to find a constant $C_6 > 0$:

which alongside (4.26) implies some constant $C_7 > 0$

Solving the above inequality directly yields a constant $C_8 > 0$ such that

Similar to the case $a_1 < a_3$, we can use (4.30), the Gagliardo-Nirenberg inequality and Lemma 4.1 to get the convergence rate of $\|u\|_{L^{\infty}(\Omega)}+\|v-1\|_{L^{\infty}(\Omega)}+\|w\|_{L^{\infty}(\Omega)}$.

Acknowledgments

The author warmly thanks the reviewers for several inspiring comments and helpful suggestions. The research of the author was supported by the National Nature Science Foundation of China (Grant No. 12101377) and the Nature Science Foundation of Shanxi Province (Grant No. 20210302124080).

Conflict of interest

The author declares there is no conflict of interest.