Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop

1.   Introduction

In this paper, we consider the two-species chemotaxis- competition system with two chemicals

where $\Omega\subset \mathbb{R}^n$ ($n\geq1$) is a bounded domain with smooth boundary $\partial\Omega$ and $\partial/\partial\nu$ denotes the derivative with respect to the outer normal of $\partial\Omega$; $a_1,a_2$, $\chi_1, \chi_2, \chi_3$, $\mu_1$ and $\mu_2$ are positive constants.

Chemotaxis describes oriented movement of cells along the concentration gradient of a chemical signal produced by the cells, which is important in a large variety of fields within the life cycle of most multicellular organisms. Based on a well-known chemotaxis model for the chemotactic movement of one specie [11], a generalization of chemotaxis model for multi-species or multi-chemical-signal chemotaxis system was proposed [9,24,42]. System (1.1) describes the communication between breast tumor cells and macrophages in close proximity via a short-ranged chemical signaling loop according to the classical Lotka-Volterra dynamics [22], which was proposed by Knútsdóttir et al. [12]. The key physical variables in (1.1) are assumed to be the density of the macrophages (denoted by $u$) and the tumor cells (denoted by $w$), the concentration of the colony-stimulating factor 1 (CSF-1, denoted by $v$) and the epidermal growth factor (EGF, denoted by $z$). In their experiment, it was shown that in a short-term chemical signalling loop, the process is promoted by the macrophages. More precisely, the tumor cells $w$ secrete CSF-1 $v$, which can bind to CSF-1 receptors on the macrophages $w$. This activates the macrophages to increasing concentrations of CSF-1 gradient and to secrete EGF $z$. Then the EGF can bind to receptors on tumor cells, continuing the chain of activation of them in return. Furthermore, activated tumor cells release more CSF-1 and partially direct their movement toward both concentration gradients of the EGF and the CSF-1, respectively.

From a mathematical point of view, the system (1.1) contains mainly two subsystems. When $u = z\equiv0$ and $a_2\equiv0$, (1.1) is reduced to the chemotaxis-only system [11]:

which describe the aggregation of the paradigm species Dictyostelium discoideum. Model (1.2) has been extensively studied during past four decades. Boundedness of global solutions and unbounded solutions for (1.2) have been extensively investigated (see the surveys [2,6,8] and the references therein). When $\chi_3\equiv0$, (1.1) becomes the two-species chemotaxis system with two chemicals:

which describes the spatio-temporal evolution of two populations which on the one hand proliferate and mutually compete with Lotka-Volterra kinetics and on the other hand the individuals shall move according to random diffusion and migrate toward a chemical signal produced by the opposing species. Without kinetic terms, that is, when $\mu_1 = \mu_2 = 0$, if $\chi_1,\chi_2\in\{-1,1\}$, global boundedness and finite time blow-up was constructed by Tao and Winkler [32]. In particular, when $n = 2,$ global bounded solutions are proved if $\max \{m_1, m_2\}<C$ with some $C>0$, where $m_1: = \int_{\Omega} u_{0}$ and $m_2: = \int_{\Omega} w_{0}$; if $\min \{m_1, m_2\}>4\pi$, the solution may be blow up in finite time. These results were improved in [45]. When $\mu_i>0$ $(i = 1,2)$, global bounded solution and stabilization have been addressed by Tu et al. [36]. Recently, this results were improved by Wang et al. [38]. There are some another results on various the system (1.3) ([3,5,10,18,23,26,29,27,43,47,46]).

Compared with the chemotaxis-only system and the two-species chemotaxis system with two chemicals, the coupled a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop the system (1.1) is much less understood. For the simplified version (1.1) in the unit disk $\Omega = B_{R}(0)\subset \mathbb{R}^2$ with $\mu_i = 0$ $(i = 1,2)$ and the second and fourth equation replaced by

respectively, where $m_1: = \int_{\Omega}u_0$ and $m_2: = \int_{\Omega}w_0$, global boundedness and blow-up of solutions were constructed in [16]. In particular, a critical mass phenomenon has been found: solutions remain bounded if $2m_1 + \frac{ \chi_3}{\chi_2}m_2 < \frac{8\pi}{\chi_2}$, whereas blow-up may occur if $\frac{8\pi}{\chi_1} m_1 + \frac{8\pi}{\chi_2} m_2 < 2 m_1 m_2 + \frac{m_2^2\chi_3}{\chi_2}$ and $\int_{\Omega}u_0|x|^2$ and $\int_{\Omega}w_0|x|^2$ are sufficiently small. Recently, (1.1) with $\mu_i = 0$ $(i = 1,2)$ and $\Omega\subset \mathbb{R}^2$ has been considered in [17]: global boundedness was constructed when $\chi_1 \chi_2 m_1 m_2 < \left( \pi^{\star} - \chi_3 m_2 \right) \pi^{\star}$, where $\pi^{\star} = 8\pi$ when $\Omega = B_{R}(0)$ is a disk, otherwise, $\pi^{\star} = 4\pi$; for the fully parabolic system (1.1) with $\mu_i = 0$ $(i = 1,2)$ and $\Omega\subset \mathbb{R}^2$, global bounded solution was considered when $\chi_1 \chi_2 m_1 m_2 < \frac{ \sqrt{1-4\chi_3 C_{GN}^{4} m_2} }{4C_{GN}^{8}}$, where $C_{GN}>0$ is a constant. Moreover, gradient estimates, blow-up in finite time and asymptotic behavior have also been established. When $n = 3$ and $\mu_i>0$ $(i = 1,2)$ are sufficiently large, for the fully parabolic system (1.1), global bounded solutions were proved by Pan et al.[25]. Very recently, a symmetric model of (1.1) has been investigated [34,35], i.e., replacing the first equation in (1.1) by $u_t = \Delta u-\chi_1\nabla\cdot(u\nabla v)-\chi_4\nabla\cdot(u\nabla z)+\mu_1 u(1-u-a_1w)$.

In order to better understand (1.1), we should mention two biological species which compete for the resources and migrate towards a higher concentration of a chemical produced by themselves was proposed by Tello and Winkler [33]

this model has been extensively studied. When $a_1,a_2\in(0,1)$, global existence and the large time behavior were established under some conditions by Tello and Winkler [33], which was partially improved in [4], and by Stinner et al. [30] for the case of $a_1\geq1>a_2$. For all $a_1,a_2>0$, global boundedness solutions were derived by Mizukami [20], which covers the case that $a_1,a_2\geq1$. Moreover, the convergence rates has been also obtained in the case that $a_1,a_2\in(0,1)$ and $a_1\geq1>a_2$. For the fully parabolic version of model (1.4), which is obtained by replacing the equation with $w_t = \Delta w-w+u+v$, global existence and boundedness has been established for the space dimension does not exceed two by Bai and Winkler [1] and the $n$-dimensional setting by [14,15,44]. Moreover, the convergence rates has been established [1], this conditions were improved by Mizukami [19]. Recently, the conditions for asymptotic behavior in the case of $a_1,a_2\in(0,1)$ were once more improved by Mizukami [21].

The focus of this paper is to establish the global existence, large time behavior, and the rates of convergence of solution to (1.1). The first of our result asserts global existence of a bounded solution based on the comparison methods in [38,37,4,30].

Theorem 1.1. Suppose that $\Omega\subset \mathbb{R}^n$ ($n\geq1$) is a bounded domain with smooth boundary, and $a_1,a_2$, $\chi_1, \chi_2, \chi_3$, $\mu_1, \mu_2$ are positive constants. If one of the following conditions holds:

(i) $\frac{\chi_1}{\mu_1}\leq a_1$ and $\frac{\chi_3}{\mu_2}<1$;

(ii) $\frac{\chi_2}{\mu_2}\leq a_2$ and $\frac{\chi_3}{\mu_2}<1$;

(iii) $\frac{\chi_1}{\mu_1}>a_1$, $\frac{\chi_2}{\mu_2}> a_2$ and $\frac{\chi_3}{\mu_2}<1$ as well as $\frac{\chi_1\chi_2}{\mu_1\mu_2}+a_1a_2-a_1\frac{\chi_2}{\mu_2}-a_2\frac{\chi_1}{\mu_1}+\frac{\chi_3}{\mu_2}<1$.

Then for the nonnegative initial data $(u_{0}, w_{0})\in C^{0}(\overline{\Omega}) \times C^{1}(\overline{\Omega})$, the model (1.1) possesses a unique global classical solution $(u,v,w,z)$ which is uniformly bounded in $\Omega\times(0,\infty)$. Moreover, the solutions $u, v, w,z$ are the Höder continuous functions, i.e., there exists some $\theta\in(0,1)$ and $M>0$ such that

The large time behavior and convergence rates of solutions to (1.1) is mathematically and biologically interesting. We first give the result of competitive coexistence case $a_1,a_2\in(0,1)$.

Theorem 1.2. Let $a_1,a_2\in(0,1)$ and $\Omega\subset \mathbb{R}^n$ ($n\geq1$) be a bounded domain with smooth boundary. Assume that the nonnegative initial data $(u_{0}, w_{0})\in C^{0}(\overline{\Omega}) \times C^{1}(\overline{\Omega})$ with $u_0\not\equiv 0\not\equiv w_0$ and $(u,v,w,z)$ is a global bounded classical solution of (1.1). Let $\chi_1, \chi_2, \chi_3$, $\mu_1$ and $\mu_2$ are positive constants and satisfy

and

then one can find $C>0$ and $\mu>0$ such that

for all $t>0$, where

Remark 1. Since $a_1,a_2\in(0,1)$, it is easy to check $f(\mu_1,\mu_2): = \mu_1\mu^2_2a_1a_2(1-a_1a_2)-\frac{a^2_{2} \mu^2_{2} \chi_{1}^{2} u_{*}}{16}-\frac{\mu_1\mu_2 a_1 a_2\chi_3^2 w_*}{8} -\frac{\mu_1\mu_2a^2_1\chi^2_2w_*}{8} +\frac{\mu_{2}a_1a_{2} \chi_{1}^{2} \chi^2_2u_{*}w_*}{128}+\frac{\mu_1a^2_1\chi^2_2\chi_3^2 w_*}{64}$ satisfies

Hence, there exist some constants $\mu_{10},\mu_{20}>0$ such that $f(\mu_1,\mu_2)>0$ for all $[\mu_{10},\infty)\times[\mu_{20},\infty)$. Thanks to a continuity argument implies (1.7) holds for all $[\mu_{10},\infty)\times[\mu_{20},\infty)$.

The following result is on the competitive exclusion case $a_1\geq1>a_2>0$.

Theorem 1.3. Let $a_1\geq1>a_2>0$, $\chi_1, \chi_2, \chi_3>0$, $\mu_1,\mu_2>0$, and let $\Omega\subset \mathbb{R}^n$ ($n\geq1$) be a bounded domain with smooth boundary. Assume that the nonnegative initial data $(u_{0}, w_{0})\in C^{0}(\overline{\Omega}) \times C^{1}(\overline{\Omega})$ with $w_0\not\equiv 0$ and $(u,v,w,z)$ is a global bounded solution of (1.1).

(i) If $a_1>1$ and assume that for some $a'_1\in (1,a_1]$ such that $a'_1a_2<1$. Moreover, let

there exist $C>0$ and $\lambda>0$ such that

(ii) If $a_1 = 1$ and (1.9) holds with $a'_1 = 1$, there exist $C>0$ and $\kappa>0$ such that

This paper is organized as follows. In Section 2, we show local existence of a solution to (1.1) and use comparison methods to prove global existence and boundedness of (1.1) (Theorems 1.1). Section 3 is devoted to the proof of asymptotic stability to (1.1) (Theorems 1.2 and 1.3).

2.   Boundedness by comparison methods

The local existence of solutions to (1.1) which can be achieved similarly by using well-established methods in [30,31,41].

Lemma 2.1. Let $\Omega\subset \mathbb{R}^n$ ($n\geq1$) be a bounded domain with smooth boundary, $a_1,a_2>0$, $\chi_1, \chi_2, \chi_3>0$, $\mu_1,\mu_2>0$. Then for the nonnegative initial data $(u_{0}, w_{0})\in C^{0}(\overline{\Omega}) \times C^{1}(\overline{\Omega})$, there exists $T_{max}\in(0,\infty]$ such that (1.1) has a unique local non-negative classical solution

and which is such that either $T_{max} = \infty$ or $T_{max}<\infty$ and

Moreover, if the nonnegative initial data $(u_{0}, w_{0})\in C^{0}(\overline{\Omega}) \times C^{1}(\overline{\Omega})$ with $u_0\not\equiv0\not\equiv w_0$, then the solution of (1.1) satisfies $u>0$, $v\geq0$ $w>0$ and $z\geq0$ in $\Omega\times(0,T_{max})$.

According to comparison methods in [38,37,4,30], we will prove boundedness of solution to model (1.1). First we let the parabolic operator $\mathcal{L}_i = \mathcal{L}_i(x,t)$ ($i = 1,2$) satisfy

and

Thus from the model (1.1) shows that

and

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

We first consider the conditions $\frac{\chi_1}{\mu_1}\leq a_1$ and $\frac{\chi_3}{\mu_2}<1$.

Lemma 2.2. Let $\tau = 0$ and the nonnegative initial data $(u_{0}, w_{0})\in C^{0}(\overline{\Omega}) \times C^{1}(\overline{\Omega})$. Assume that $\frac{\chi_1}{\mu_1}\leq a_1$ and $\frac{\chi_3}{\mu_2}<1$. Then $T_{max} = \infty$ and $u,v,w,z$ are bounded in $\Omega\times(0,\infty)$.

Proof. Since $\frac{\chi_1}{\mu_1}\leq a_1$ and $u,v,w,z$ are nonnegative, in view of (2.4) and (2.5), we have

and

where $\mathcal{L}_1u$ and $\mathcal{L}_2w$ are given in (2.2) and (2.3), respectively. Next, in order to prove Lemma 2.2, we have two cases to proceed as below:

Case 1. $\chi_2-a_2\mu_2\leq0$. We choose $A>0$ and let $\xi>0$ sufficiently large such that

Hence, we can define constant functions $\overline{u}$ and $\overline{w}$ as follows

which satisfy

Due to the definition of $\mathbf{p}_1$ in (2.6), using (2.8) and (2.9), we have

According to (2.6), (2.10) and (2.11), and from the comparison principle for classical (sub-/super-) solutions [28,Proposition 52.6], we infer that

Using the definition of $\mathbf{p}_2$ in (2.7), we deduce from (2.8), $\frac{\chi_3}{\mu_2}<1$ and $\chi_2-a_2\mu_2\leq0$ that

According to (2.7), (2.10) and (2.13), and from the comparison principle for classical (sub-/super-) solutions [28,Proposition 52.6], we infer that

Case 2. $\chi_2-a_2\mu_2>0$. Due to $\frac{\chi_3}{\mu_2}<1$ and $\chi_2-a_2\mu_2>0$, we can find $A$ fulfilling

Let $\xi$ large enough such that

We again let $\overline{u} = \overline{u}(x,t): = A\xi$ and $\overline{w} = \overline{w}(x,t): = \xi$ for all $(x,t)\in \overline{\Omega}\times[0,T_{max})$, as in (2.9), we deduce from (2.16)

Similar to (2.11) and (2.12), we infer

Hence, using $\chi_2-a_2\mu_2>0$, (2.7), (2.16) and (2.18) yields

thus by (2.7) and (2.17) we have

Using (2.12), (2.14), (2.18) and (2.20) along with extensibility criterion in Lemma 2.1, this entails that $T_{max} = \infty$ and $u,v,w,z$ are globally bounded.

Next, we consider the conditions $\frac{\chi_2}{\mu_2}\leq a_2$ and $\frac{\chi_3}{\mu_2}<1$.

Lemma 2.3. Let $\tau = 0$ and the nonnegative initial data $(u_{0}, w_{0})\in C^{0}(\overline{\Omega}) \times C^{1}(\overline{\Omega})$. Assume that $\frac{\chi_2}{\mu_2}\leq a_2$ and $\frac{\chi_3}{\mu_2}<1$. Then $T_{max} = \infty$ and $u,v,w,z$ are bounded in $\Omega\times(0,\infty)$.

Proof. Since $\frac{\chi_2}{\mu_2}\leq a_2$, $\frac{\chi_3}{\mu_2}<1$ and $u,v,w,z$ are nonnegative, in view of (2.4) and (2.5), we have

and

where $\mathcal{L}_1u$ and $\mathcal{L}_2w$ are given in (2.2) and (2.3), respectively. In the case of $\chi_1-a_1\mu_1\leq0$ and $\frac{\chi_3}{\mu_2}<1$, Lemma 2.2 showed (1.1) has global bounded solution. Hence, here we need only consider the case $\chi_1-a_1\mu_1>0$. We can find $A>0$ and $\xi>0$ fulfilling

and

We again let $\overline{u} = \overline{u}(x,t): = A\xi$ and $\overline{w} = \overline{w}(x,t): = \xi$ for all $(x,t)\in \overline{\Omega}\times[0,T_{max})$, as in (2.9), we deduce

Using (2.22) and (2.24) we conclude

so according to (2.22) and (2.25) imply

Hence, using $\chi_1-a_1\mu_1>0$, (2.21), (2.23) and (2.27) yields

From (2.21), (2.25) and (2.28) and the comparison principle, we infer

Then using (2.27) and (2.29) along with extensibility criterion in Lemma 2.1, this entails that $T_{max} = \infty$ and $u,v,w,z$ are globally bounded.

Finally, we consider the conditions $\frac{\chi_1}{\mu_1}>a_1$, $\frac{\chi_2}{\mu_2}> a_2$ and $\frac{\chi_3}{\mu_2}<1$.

Lemma 2.4. Let $\tau = 0$ and the nonnegative initial data $(u_{0}, w_{0})\in C^{0}(\overline{\Omega}) \times C^{1}(\overline{\Omega})$. Assume that $\frac{\chi_1}{\mu_1}>a_1$, $\frac{\chi_2}{\mu_2}> a_2$ and $\frac{\chi_3}{\mu_2}<1$ as well as $\left(\frac{\chi_1}{\mu_1}-a_1\right)\left(\frac{\chi_2}{\mu_2}-a_2\right)+\frac{\chi_3}{\mu_2}<1$. Then $T_{max} = \infty$ and $u,v,w,z$ are bounded in $\Omega\times(0,\infty)$.

Proof. The conditions $\frac{\chi_1}{\mu_1}>a_1$, $\frac{\chi_2}{\mu_2}> a_2$, $\frac{\chi_3}{\mu_2}<1$ and $\left(\frac{\chi_1}{\mu_1}-a_1\right)\left(\frac{\chi_2}{\mu_2}-a_2\right)+\frac{\chi_3}{\mu_2}<1$ imply that

Hence, we can choose $\xi>0$ large enough such that

and

which enables us to choose $A>0$ satisfying

Hence, (2.32) implies

We again let $\overline{u} = \overline{u}(x,t): = A\xi$ and $\overline{w} = \overline{w}(x,t): = \xi$ for all $(x,t)\in \overline{\Omega}\times[0,T_{max})$, as in (2.9), we deduce

Due to $u,v,w,z$ are nonnegative, from (2.4) and (2.5), and in view of (2.33), (2.34) imply that

in $\Omega\times(0,T_{max})$, by the comparison principle for this cooperative systems [28,Proposition 52.22], it follows that

Therefore, using the extensibility criterion in Lemma 2.1, we have $T_{max} = \infty$ and $u,v,w,z$ are globally bounded.

Proof of Theorem 1.1. We only need to use Lemmas 2.2-2.4 to obtain the global bounded solutions of (1.1). Finally, the Hölder continuity of the solution $(u, v, w, z)$ comes from standard parabolic regularity theory [13].

3.   Stabilization

In this section, we will derive the asymptotic behavior of the solutions to the model (1.1), the ideas mainly come from [1,21,38,39]. To achieve our goals, we first recall the following lemma which is important for the proof of Theorems 1.2 and 1.3 (see [7,Lemma 4.6] or [20,Lemma 3.1]).

Lemma 3.1. Suppose that $\varphi(x, t) \in C^{0}(\overline{\Omega} \times[0, \infty))$ and there exist constant $C>0$ and $\sigma>0$ such that

Moreover, assume that there exists some constant $M>0$ such that

Then

This following lemma is a straightforward result from [40,Lemma 5.1].

Lemma 3.2. Let $a_{11}, a_{22}, a_{33}, a_{44}, a_{13}, a_{14}, a_{23}\in \mathbb{R}$ and satisfy

and

Then there exists $\varepsilon>0$ such that

for all $x_1,x_2,x_3,x_4\in \mathbb{R}$.

In order to prove stabilization of solutions to (1.1), we will divide the proof into two cases.

3.1.   Competitive coexistence: $a_1,a_2\in(0,1)$

In this subsection, we will study the asymptotic behavior of the solution of (1.1) with $a_1,a_2\in(0,1)$ based on the following energy functional

where $(u_*,v_*,w_*,z_*)$ is the constant steady state defined by (1.8).

Lemma 3.3. Under the assumptions of Theorem 1.2, then there exists $\varepsilon>0$ such that

where the function $E_1(t)$ defined by (3.1) and $F_1(t)$ satisfies

Proof. From Taylor's formula we see that $E_1(t)\geq0$ for $t>0$ (for more details, see [1,Lemma 3.2]). From straightforward calculations we infer that

In order to obtain (3.2), we first deal with the last five parts on the right of (3.4), by a simple computation we conclude

and

Since $v_{*} = w_{*}$, by the second equation in (1.1) we have

Similar to the fourth equation in (1.1) yields

Inserting (3.5)-(3.8) into (3.4) we conclude

To see (3.2), we will show that there exists $\varepsilon>0$ such that

To confirm that the assumptions of Lemma 3.2 are satisfied, let

and

Since $\mu_1,\mu_2,a_2,\chi_{1}$ and $u_{*}$ are positive constants, we obtain

Thanks to (1.6) we have

Using (1.7), we obtain

Hence, combining (3.11)-(3.13) and Lemma 3.2 we have (3.10), which concludes the proof of Lemma 3.3.

Next, we will establish convergence rates for the solution to the model (1.1).

Lemma 3.4. Under the assumptions of Theorem 1.2, then there exist $C,\mu>0$ such that

for all $t>0$.

Proof. The proof is similar to the corresponding proofs of [20,Lemmas 3.5 and 3.6] or [38,Lemmas 3.4 and 3.5], for the convenience of the readers, we give a sketch the proof. We divide the proof into two steps.

Step 1. We derive the large time behavior of solution to (1.1).

Integrating the second part of (3.2) and using (3.3) to see that

According to Lemma 3.1 and (1.5), we obtain

Step 2. We derive the convergence rates of solution to (1.1).

Let $H(s): = s-u_*\ln s$ for $s>0$, by L'Hôpital's theorem to see that

Hence, using (3.14) and (3.15) and the definitions ${E}_1(t)$ in (3.1), there exists $t_1>0$ such that

for all $t>t_1$. Using (3.16) and (3.2) and the definitions ${F}_1(t)$ in (3.3), one can find some $c_1>0$ such that

which implies

with some $c_2>0$. Then by the Gagliardo-Nirenberg inequality with some $C_{GN}>0$

we can find some $c_3, \mu>0$ such that

Then by the application of the elliptic maximum principle (see more details (3.21)-(3.23) in [38]) enables us to obtain

with some $c_4>0$. Hence, combining (3.19) and (3.20) we can complete the proof of this lemma.

3.2.   Competitive exclusion: $a_1\geq1>a_2$

When $a_1\geq1>a_2$, we will prove the competitive exclusion will occur based on the following energy functional

where $a'_1\in [1,a_1]$ satisfies $a'_1a_2<1$.

Lemma 3.5. Under the assumptions of Theorem 1.3, then there exists $\varepsilon>0$ such that

where $E_2(t)$ is defined by (3.21) and

Proof. Using Taylor's formula to see that $\int_\Omega(w-1-\ln w)\geq0$ for all $t>0$ (for more details, see [1,Lemma 3.2]), then using $u$ is nonnegative, which implies $E_2(t)\geq0$ for all $t>0$. By the straightforward calculations we infer

Using the third equation in (1.1), we obtain

Combining (3.23) and (3.24), we have

Using Young's inequality we infer that

Multiplying the second equation in (1.1) with $\frac{\mu_1a'_1\chi_3^2}{2}(v-1)$, we have

Testing the fourth equation in (1.1) with $\frac{\mu_1a'_1\chi_2^2}{2}z$, we see that

Substituting (3.26)-(3.28) into (3.25), we have

In order to prove (3.22), we will show that there exists $\varepsilon>0$ such that

Then using the same argument as in the proof of Lemma 3.4, we put

and

Since $\mu_1, \mu_2, a'_1, a_2,\chi_{3}$ are positive constants, we obtain

Thanks to the second part of (1.9) we have

By the first part of (1.9), we infer that

Then collecting (3.31)-(3.33) we have (3.30), which implies the end of the proof.

According ideas come from [20,Lemmas 3.8,3.9 and 3.10] or [38,Lemmas 3.7 and 3.8], we shall give the convergence rates for the case $a_1\geq1>a_2$.

Lemma 3.6. Under the assumptions of Theorem 1.3.

(i) Let $a_1>1$ and $a_2\in(0,1)$, then there exist $C>0$ and $\lambda>0$ such that

(ii) Suppose that $a_1 = 1$ and $a_2\in(0,1)$, then there exist $C>0$ and $\kappa>0$ such that

Proof. (i) This part can be proved by a similar proof in Lemma 3.4.

(ii) Since $a' = 1$, then the second part in (3.22) can be rewritten as

Similar to (3.14), integrating (3.34) and using Lemma 3.1 enable us to obtain

Hence, by the definition of $E_2(t)$ in (3.21), using L'Hôpital's theorem and Hölder's inequality, we can find $t_1>0$ such that

Combining (3.34) and (3.36), using the definition of $F_2(t)$ in Lemma 3.5, one can find some $C_1>0$ such that

which implies there exists $C_2>0$ such that

By L'Hôpital's theorem and (3.37), we can find $C_2>0$ such that

By (1.5) and the Gagliardo-Nirenberg inequality, we can find $C_3,\kappa>0$ such that

Using the application of the elliptic maximum principle (see more details (3.21)-(3.23) in [38]) implies

with some $C_4>0$. Hence, the proof of Lemma 3.6 is completed.

Proof of Theorem 1.2. The proof of Theorem 1.2 follows from Lemma 3.4.

Proof of Theorem 1.3. We only need to use Lemma 3.6 to obtain the proof of Theorem 1.3.

Acknowledgments

The authors are very grateful to the anonymous reviewers for their carefully reading and valuable suggestions which greatly improved this work.