Global existence for a two-species chemotaxis-Navier-Stokes system with $p$-Laplacian

1.   Introduction

In this paper, we are concerned with the following two-species chemotaxis-Navier-Stokes system:

in $Q = \Omega\times[0, \infty)$, where $\Omega\subset{\mathbb R}^3$ is a bounded domain with smooth boundary, $\mu_1, \mu_2, \alpha$, $\beta, \chi_1, \chi_2,$ are positive constants, and $a_1, a_2\geq 0$ are constants. This system describes the evolution of two kinds of aerobic bacteria, that compete according to Iotka-Volterra competitive kinetics in a liquid surrounding environment. Here $n_1$ and $n_2$ represent the population densities of two species respectively, $c$ stands for the concentration of oxygen, $u$ shows the fluid velocity field, and $P$ represents the pressure of the fluid. The given function $\Phi$ represents the gravitational potential.

The problem (1.1) is a generalized system to the chemotaxis-fluid system, which is proposed by Tuval et al. in [21]. The chemotaxis-Navier-Stokes system models have been widely studied by many researchers ([22,23,25]).

What's more, the investigation of the problems involving chemotaxis-Navier-Stokes system models with p-Laplacian has been addressed by several authors. Tao and Li [19] discussed the following chemotaxis-Navier-Stokes system:

They got that if $p>\frac{32}{15}$, under appropriate assumptions on $f$ and $\chi$, for all sufficiently smooth initial data $(n_0, c_0, u_0)$, the system owns at least one global weak solution in three dimensional spaces. Furthermore, Tao and Li [20] proved that global bounded weak solutions of the chemotaxis-Stokes system exist whenever $p > \frac{23}{11}$. Liu [10] investigated the following problem:

where $\Omega\subset\mathbb R^{N}$ is a bounded domain with smooth boundary. It is proved that if either $p>2$ for $\kappa\in\mathbb R$, $N = 2$ or $p>\frac{94}{45}$ for $\kappa = 0$, $N = 3$ is satisfied, then for each properly chosen initial data and associated initial-boundary problem admits a global weak solution which is bounded. The relevant equations have also been studied in [11,12].

On the other hand, two-species competitive chemotaxis systems have been studied by many authors [1,15] recently, mainly about the global existence and asymptotic stability of solution. Cao, Kurima and Mizukami [3] considered the following two-species chemotaxis-Stokes system:

They proved the global existence, boundedness and stabilization of solutions to the above system in the $3$-dimensional case. Hirata et al [6] gave more complete stabilization of solutions for (1.2) in the $2$-dimensional case. Moreover Liu and Li [13] proved that the system (1.2) admits a time periodic solution under some conditions. The relevant equations have also been studied in [8].

As mentioned above, two-species chemotaxis-Stokes system and one species chemotaxis-Stokes system with p-Laplacian were studied by many authors. However the combination of these two kinds of problems has not been studied. Thus, we are inspired to investigate the case that the two species have different diffusion law, namely one according to the p-Laplacian diffusion and the other according to standard Laplacian diffusion. From a physical point of view, in the same liquid surrounding environment, one species is influenced by ions and molecules and thus its mobility is described by a nonlinear function of the cells, but the other species is not affected by ions or molecules thus diffuse by linear Laplacian diffusion.

Obviously, Cao, Kurima and Mizukami solved the problem (1.1) when $p = 2$. If $p>2$, from a mathematical point of view, p-Laplacian diffusion term $\nabla \cdot(|\nabla n_1|^{p-2} \nabla n_1)$ lead to a lot of difficulties in our proof of main result because of its nonlinear character. To be specific, a main difficulty arises in the estimate of $n_1$ in contrast to the case of Laplacian, and the property of the Neumann heat semigroup becomes useless and so on. In order to overcome the difficulties bring by the p-Laplacian diffusion term, we consider a regularized problems of (1.1) and establish an energy-type inequality in Section $3$ as a starting point to discuss the problem.

In this paper, we shall consider (1.1) along with the boundary conditions

where $Q_{\Gamma} = \partial\Omega\times[0, \infty)$, and the initial conditions

Assume that $n_{1, 0}$, $n_{2, 0}$, $c_0$, $u_0$ and $\Phi$ are given functions satisfying

and

where $L\log L(\Omega)$ is the standard Orlicz space associated with the Young function $(0, \infty) \ni z \mapsto z \ln(1 + z)$ and $L_\sigma^2(\Omega): = \{\varphi\in(L^2(\Omega))^3|\nabla\cdot\varphi = 0\}$.

Let us first give the definition of weak solution.

Definition 1.1. We call $(n_1, n_2, c, u)$ a global weak solution of (1.1), (1.3) and (1.4) if

such that $n_1\geq 0, n_2\geq 0$ and $c\geq 0 \;\; a.e$ in $Q$ and that

and that

hold for all $\phi_1, \phi_2, \phi_3 \in C_0^{\infty}(\bar{\Omega}\times [0, \infty))$ and $\phi_4\in(C_0^{\infty}(\Omega\times[0, \infty)))^3$ satisfying $\nabla\cdot\phi_4 = 0$.

The plan of this paper is as follows. In Section 2, we list some lemmas, which will be used throughout this paper. In Section 3, we consider a family of regularized problems and show the global existence of the regularized problems, by establishing an energy-type inequality and using the Moser-Alikakos iteration procedure. Finally, in Section 4, we show that the problem (1.1), (1.3) and (1.4) admits a global-in-time weak solution.

2.   Preliminaries

In this section, we recall some lemmas, which will be used throughout the paper. Before going further, we first list the Gagliardo-Nirenberg interpolation inequality [16] for the convenience of application.

Lemma 2.1. For functions $u: \Omega \rightarrow \mathbb{R}$ defined on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^{N},$ we have

where $1 \leq q, r \leq \infty, \frac{j}{m} \leq b\leq 1,$

and $s>0$ is arbitrary.

Next, we list the following Lemma 2.2 [18].

Lemma 2.2. Let $T>0, \tau \in(0, T), a>0, b>0,$ and suppose that $y:[0, T) \rightarrow$ $[0, \infty)$ is absolutely continuous such that

where $h \geq 0, h(t) \in L_{l o c}^{1}([0, T))$ and

Then

Finally, we also give a generalized lemma of Lemma 2.2 [7].

Lemma 2.3. Let $T>0, \tau \in(0, T), \sigma \geq 0, a>0, b \geq 0,$ and suppose that $f:[0, T) \rightarrow [0, \infty)$ is absolutely continuous and satisfies

where $h \geq 0, \;h(t) \in L_{\mathit{\text{loc}}}^{1}([0, T))$ and

Then

3.   Regularized problem

Inspired by the idea from [22], in order to construct a global weak solution of (1.1), (1.3) and (1.4), we first consider the following appropriately regularized problem:

for $\varepsilon\in(0, 1)$. We take $F_{\varepsilon}(s): = \frac{1}{\varepsilon}\ln(1+\varepsilon s)$, for $s\geq 0$ and utilize the standard Yosida approximation $Y_\varepsilon$ [14,17], which was defined by

By $\textbf{P}$ we mean the Helmholtz projection in $L^2(\Omega)$, then we represent $A$ as the realization of Stokes operator $-\textbf{P}\Delta$ in $L^2_\sigma(\Omega)$, with domain $D(A) = W^{2, 2}(\Omega)\cap W^{1, 2}_{0, \sigma}(\Omega)$, where $W^{1, 2}_{0, \sigma}(\Omega) = W^{1, 2}_{0}(\Omega)\cap L^2_\sigma(\Omega) = \overline{C_{0, \sigma}^{\infty}(\Omega)}^{\|\cdot\|_{W^{1, 2}(\Omega)}},$ with $C_{0, \sigma}^{\infty}(\Omega) = C_{0}^{\infty}(\Omega)\cap L^2_\sigma(\Omega)$. Thereafter, it is obvious to see that our choice of $F_{\varepsilon}$ ensures that

and

The families of approximate initial date $n_{01\varepsilon}(x)> 0$, $n_{02\varepsilon}(x)> 0$, $c_{0\varepsilon}(x)> 0$ and $u_{0\varepsilon}$ satisfy that

and

Firstly, we give the local smooth solutions existence result of the above approximate problem as follows.

Lemma 3.1. Taking $p\geq 2$, then for each $\varepsilon \in(0, 1)$, there exist $T_{max, \varepsilon}\in(0, \infty]$ and uniquely determined functions

such that $(n_{1\varepsilon}, n_{2\varepsilon}, c_{\varepsilon}, u_{\varepsilon})$ is a classical solution of (3.1) in $\Omega \times(0, T_{\max , \varepsilon})$ with some $P_{\varepsilon} \in C^{1, 0}(\Omega \times(0, T_{\max , \varepsilon})).$ Moreover, we have $n_{1\varepsilon}> 0$, $n_{2\varepsilon}> 0$, and $c_{\varepsilon}> 0$ in $\bar{\Omega}\times (0, T_{\max, \varepsilon})$ and if $T_{\max , \varepsilon}<\infty,$ then

as $t \nearrow T_{\max , \varepsilon}$, for all $q>3$ and $\gamma>\frac{3}{4}$.

Proof. Combination of arguments Lemma 2.1 in [23] and Lemma 2.1 in [19], which is based on a standard Schauder fixed point argument and a parabolic regularity theory, entails the existence of classical solution. Since $(n_{1\varepsilon}, n_{2\varepsilon}, c_{\varepsilon}, u_{\varepsilon})$ is a smooth solution of (3.1), the nonnegativity of $n_{1\varepsilon}, n_{2\varepsilon}$ and $c_{\varepsilon}$ follows from the maximum principle [23,5,2].

We are now ready to construct some basic estimates of $(n_{1\varepsilon}, n_{2\varepsilon}, c_{\varepsilon}, u_{\varepsilon})$. In what follows, without special explanation, we take $\tau = \min\left\{1, \frac{T_{\text{max}, \varepsilon}}2\right\}$, what's more, all constants $C$ denote some different constants from line to line, which are only depend on the given coefficients, initial date and $\Omega$. All these estimates of this form $\int_{t-\tau}^{t}\cdots ds$ in this paper can be replaced by $\int_0^{T_{\text{max}, \varepsilon}}\cdots ds$ if $\tau<1$, so these estimates do not dependent on $\tau$. The following estimates of $n_{1\varepsilon}$, $n_{2\varepsilon}$ and $c_\varepsilon$ are obvious but important in the proof of our result.

Lemma 3.2. For each $\varepsilon\in(0, 1)$, the solution of (3.1) satisfies

and

Proof. Integrating the first equation in (3.1) to see

in both cases $a_1 = 0\text{ and }a_1 > 0$, we have

with some $C(\Omega)>0$. We derive from the Hölder inequality that

hence we further have

By Lemma 2.3, we can obtain (3.9). Then completely similar to the proof of (3.10). Finally, an application of the maximum principle to the third equation in (3.1) gives (3.11).

Now we are in the position to derive an energy-type inequality which will be used in the reduction of further estimates. To this end, we first list an inequality which is crucial in the proof of the energy-type inequality. More details of the proof please refer to [4].

Lemma 3.3. Suppose that $h\in C^2(\mathbb{R})$. Then for all $\varphi\in C^2(\bar{\Omega})$ fulfilling $\frac{\partial\varphi}{\partial\nu} = 0$ on $\partial\Omega$ we have

Then we derive the decisive energy-type inequality from the first three equations in (3.1). The main idea of the proof is similar to the strategy introduced in [22] (see also [19]).

Lemma 3.4. Assume that $p\geq 2$. There exist constants $C > 0, \;K > 0$ such that for any $\varepsilon\in(0, 1)$, the solution of (3.1) satisfies

where $\bar{A} = \frac{\alpha}{\chi_1}, \bar{B} = \frac{\beta}{\chi_2}$.

Proof. Testing the first equation of (3.1) by $\ln n_{1\varepsilon}$ and integrating by parts to compute

for all $t \in (0, T_{\text{max}, \varepsilon})$. Here we use that $s\ln s\geq -\frac{1}{e} ,$ for all $s \geq 0$ to see that

for all $t \in (0, T_{\text{max}, \varepsilon}).$ Moreover, due to the fact that $(1-n_{1\varepsilon})\ln n_{1\varepsilon}\leq 0$ for all $(x, t)\in \Omega\times(0, T_{\text{max}, \varepsilon})$, we have

Furthermore, we see that

Proceeding similarly, we obtain

Finally, we have the following inequality

for all $t \in (0, T_{\text{max}, \varepsilon}).$ Indeed, by a straightforward calculation and integration by parts we see

for all $t \in (0, T_{\text{max}, \varepsilon})$. In view of the third equation of (3.1) and integration by parts, we further have

for all $t \in (0, T_{\text{max}, \varepsilon})$. On the other hand, considering the Lemma 3.3 with $\varphi = c_{\varepsilon}$ and $h(\varphi) = -\frac{1}{\varphi}$ to observe that

What's more, we observe that

Combining the above three equations, we further have

Then applying Lemma 3.3 in [23], the inequality

holds for all $t \in (0, T_{\text{max}, \varepsilon})$. Since the convexity of $\Omega$ and $\frac{\partial c_{\varepsilon}}{\partial\nu} = 0$ on $\partial\Omega,$ it follows that $\frac{\partial}{\partial\nu}|\nabla c_{\varepsilon}|^2\leq 0$ on $\partial\Omega$([4]). Moreover because of $\nabla\cdot u_{\varepsilon} = 0$, integration by parts and the Young's inequality we find

Combined the above discussion with (3.2), we arrive at (3.15).

Finally, considering the consequence of Lemma 3.2 and combining (3.13)-(3.15) with $C = \frac{\bar{A}\mu_1a_1}{e}\int_{\Omega}n_{2\varepsilon} + \frac{\bar{B}\mu_2a_2}{e}\int_{\Omega}n_{1\varepsilon},$ and $K = \max \left\{\frac{2+\sqrt{3}}{2}s_0, 2(2+\sqrt{3}), \frac{1}{\bar{A}}, \frac{1}{\bar{B}}\right\}$, we arrive at (3.12).

Lemma 3.5. Assume that $p \geq 2$. $\bar{A}, \bar{ B}$ and $K$ be given in Lemma 3.4. Then for any $\varepsilon \in (0, 1)$ there exists $K_0 > 0$ large enough such that

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

Proof. Testing the fourth equation in (3.1) with $u_\varepsilon$ and considering Hölder's inequality, the sobolev embedding $W^{1, 2}(\Omega)\hookrightarrow L^6(\Omega)$ as well as Minkowski inequality show that

for all $t \in (0, T_{\text{max}, \varepsilon}).$ Here we have used that

since $\nabla\cdot u_{\varepsilon} = 0$ and $\nabla\cdot(1+\varepsilon A)^{-1}u_{\varepsilon} = 0.$

Let $\theta: = \frac{p-1}{4(2 p-3)} \in(0, 1),$ then $\theta$ satisfies $\frac{5(p-1)}{6 p} = \theta(\frac{1}{p}-\frac{1}{3})+(1-\theta) \frac{p-1}{p} .$ An application of the Gagliardo-Nirenberg inequality shows that

for all $t \in(0, T_{\max , \varepsilon})$, with some $C_{1}>0 .$ Due to our assumption $p \geq 2>\frac{7}{4},$ we have $\frac{2 p}{p-1} \theta<p$. Using the Young's inequality together with (3.9) to estimate

for all $t \in(0, T_{\max , \varepsilon})$, with some $C_2 > 0$.

Let $\eta: = \frac{1}{4} \in(0, 1),$ then $\eta$ satisfies $\frac{5}{12} = \eta\left(\frac{1}{2}-\frac{1}{3}\right)+\frac{1}{2}(1-\eta) .$ Once again we use the Gagliardo-Nirenberg inequality to see that

for all $t \in(0, T_{\max , \varepsilon})$, with some $C_{3}>0 .$ Since $4\eta<2$, we use Young's inequality together with (3.10) to estimate

for all $t \in(0, T_{\max , \varepsilon})$, with some $C_4 > 0$.

Noticing that

Then considering the consequence of Lemma 3.4 and combining (3.17)-(3.19) with $\delta = \frac{p^p}{4K\|\nabla\Phi\|_{\infty}^2(p-1)^p}$ and $\vartheta = \frac{1}{K\|\nabla\Phi\|_{\infty}^2},$ we arrive at (3.16).

We can thereby establish the following consequences.

Lemma 3.6. Assume that $p \geq 2$. $\bar{A}, \bar{B}$ and $K$ be given in Lemma 3.4. Then for any $\varepsilon \in (0, 1)$, we have

for all $t \in (0, T_{\mathit{\text{max}}, \varepsilon}),$ and

Proof. Set

for all $t \in (0, T_{\text{max}, \varepsilon}),$ and

for all $t \in (0, T_{\text{max}, \varepsilon}).$ Obviously, Lemma 3.5 shows that

Now, we are going to show that $y_{\varepsilon}(t)$ is dominated by $h_{\varepsilon}(t)$. Firstly, employing the standard Poincaré inequality, there exists $C_1>0$ such that

Obviously, using that $s\ln s\leq 2s^{\frac65}$, for all $s>0$ and combining the Young's inequality together with (3.18), we see that

with some $C_2>0$. Similar to the proof of (3.23), due to the Young's inequality and (3.19), we can also obtain

with some $C_3>0$. Finally, according to the Young's inequality and (3.11), there exists $C_4$ such that

(3.22)-(3.25) ensure that $y_{\varepsilon}\leq C_5 h_{\varepsilon}(t)+C_5$, with some $C_5>0$. And then

By Lemma 2.2, it is easy to see that (3.20) and (3.21) hold. The proof is complete.

Lemma 3.7. Assume that $p\geq 2$, $\xi \in(1, 11-\frac{12}{p}]$, $\zeta\in(1, 5]$. For any fixed $T\in (0, T_{\mathit{\text{max}}, \varepsilon})$, there exists $C>0$ such that the solution of (3.1) satisfies

and

Proof. By Lemma 3.6, we have

When multiplying the first equation of (3.1) by $\xi n_{1\varepsilon}^{\xi-1},$ and integrating by parts, due to Young's inequality and (3.3) we can estimate

with $C_2 = 2^{\frac{1}{p-1}}(\frac{\chi_1}{\varepsilon})^{\frac{p}{p-1}}\xi(\xi-1)$. Using the Young's inequality again, we further have

where $C_3 = (\frac{2C_2}{\xi\mu_1})^{\frac{3p-4}{p}}, \;C_4 = 4^{\xi}\xi\mu_1$. If $\xi \in(1, 11-\frac{12}{p}]$, we have $(\xi-2)\frac{4(p-1)}{3p-4}\leq \xi+1$, this implies that

where $C_5 = \frac{\xi\mu_1}{2}+C_4$. Considering Lemma 2.3 and (3.28), then there exists $C_6 > 0$ such that $\int_{\Omega}n_{1\varepsilon}^{\xi}\leq C_6$, for all $t \in (0, T_{\text{max}, \varepsilon})$ and $\xi \in(1, 11-\frac{12}{p}]$. Since $p \geq 2$, ensuring that $11-\frac{12}{p}\geq 5$, we obtain $\int_{\Omega}n_{1\varepsilon}^{5}\leq C_6$, for all $t \in (0, T_{\text{max}, \varepsilon}).$

Next we test the second equation from (3.1) by $\zeta n_{2\varepsilon}^{\zeta-1}$ with $\zeta\in(1, 5]$, to see by neglecting several non positive contributions and employing (3.3) as well as Young's inequality that

Considering the fact that $2(\zeta-2)\leq\zeta+1,$ since $\zeta\in(1, 5]$, and combining the Young's inequality, we see that

and that

where $C_7 = \frac{8\zeta(\zeta-1)^2\chi_2^4}{\mu_2{\varepsilon}^4}$. Letting $C_8 = \frac{\zeta\mu_2}{2}+4^\zeta\zeta\mu_2$, then we have

Finally, by Lemma 2.3 and (3.28), we also have that there exists $C_{10}>0$ such that $\int_{\Omega}n_{2\varepsilon}^5\leq C_{10}$, for all $t \in (0, T_{\text{max}, \varepsilon}).$

Lemma 3.8. Assume that $p \geq 2$. For all $\varepsilon\in (0, 1)$, the solution of (3.1) is global in time; that is, we have $T_{\mathit{\text{max}}, \varepsilon} = \infty$.

Proof. Assume that $T_{\text{max}, \varepsilon} < \infty$, for some $\varepsilon\in(0, 1).$ By Lemma 3.6, it is easy to see that the following inequality holds

for all $t\in (0, T_{\text{max}, \varepsilon})$, with some $C_1, C_2 > 0$. Combining (3.26), (3.27) with (3.29), we obtain

and

with some $C_3>0, \;C_4>0.$ Because the proof of (3.30) and (3.31) is similar to [22] Lemma 3.9, we refer the reader to it for more details. Noticing the third equation of (3.1) and using the variation-of-constant formula, we can obtain

for all $t \in (0, T_{\text{max}, \varepsilon}),$ where the $\{e^{t\Delta}\}_{t\geq 0}$ is the Neumann heat semigroup in $\Omega$. For more details of Neumann heat semigroup, please refer to [24]. In conjunction with (3.2), (3.7), (3.26), (3.27), (3.30) and (3.31), we further have

for all $t \in (\tau, T_{\text{max}, \varepsilon})$ and any fixed $\tau\in(0, T_{\text{max}, \varepsilon})$, with some $C_5, \;C_6, \;C_7>0$ since $1-\frac12-\frac32\cdot\frac14 > 0$.

In what follows, we are in the position to estimate $\|n_{1\varepsilon}\|_{L^{\infty}(\Omega)}$ and $\|n_{2\varepsilon}\|_{L^{\infty}(\Omega)}.$ We first combine the estimates for Neumann heat semigroup and Young's inequality to obtain that

for all $t \in (2\tau, T_{\text{max}, \varepsilon})$ and any fixed $\tau\in(0, T_{\text{max}, \varepsilon})$, with some $C_8, \;C_9, \;C_{10}>0,$ since (3.10), (3.26), (3.27), (3.30) and (3.32) hold.

Once again, we multiply the first equation in (3.1) by $\xi n_{1\varepsilon}^{\xi-1},$ $\xi>1$ to see upon integrating by parts that again by Young's inequality, (3.3) and (3.32)

for all $t \in (\tau, T_{\text{max}, \varepsilon})$. Then by Lemma 2.3, we have $\int_{\Omega}n_{1\varepsilon}^{\xi}\leq C_{15}$, for all $t \in (\tau, T_{\text{max}, \varepsilon})$, with any $\xi> 1$ and some $C_{15}>0$.

Finally, based on a Moser-type iteration method, we achieve $L^{\infty}$ bounds for $n_{1\varepsilon}$. Taking $r_k = 2r_{k-1}+2-p,$ $k = \{1, 2, 3\cdots\cdots\}$ and $r_0>p$ is a positive constant. It is not hard to see that $\{r_k\}_{k\in N}$ is a nonnegative strictly increasing sequence, $r_k>p$, for all $k$ and $r_k\nearrow \infty$, as $k\rightarrow \infty$. What's more, there exist $c_1, c_2>0$, which are independent of $k$ such that

Letting

for all $x\in\bar{\Omega}$ and $t\text{ }\in(\tau, T_{\text{max}, \varepsilon})$.

We multiply the first equation of (3.1) by $r_k n_{1\varepsilon}^{r_k-1}$ and invoke the Young's inequality along with (3.3) and (3.32) to see that

for all $t \in (\tau, T_{\text{max}, \varepsilon}).$ Taking $\theta_k: = 2\cdot\frac{r_k+p-2}{r_k+p^{\prime}-2}\geq2, \quad \lambda_k: = \frac{2(r_k+p-2)}{r_k}\geq 2$ and considering the Hölder inequality, there must exist $C_{18}>0$ fulfilling

for all $t \in (\tau, T_{\text{max}, \varepsilon})$, since $r_k-1>1,$ for all $k$. Using the Gagliardo-Nirenberg inequality and the Young's inequality, there exists $C_{19}>0$ such that

for all $t\in (\tau, T_{\text{max}, \varepsilon})$, with $a = \frac{9}{2p+6}\in(0, 1)$ satisfying $\frac{1}{2p} = (\frac1p-\frac13)a+\frac2p(1-a)$.

Similarly, we also have

for all $t \in (\tau, T_{\text{max}, \varepsilon})$, with some $C_{27}>0$. Taking $\eta = \frac{1}{4C_{19}}(\frac{p}{r_k+p-2})^p, \delta = \frac{1}{4C_{20}}(\frac{p}{r_k+p-2})^p$ and substituting (3.35) and (3.36) into (3.34), we obtain

where

Letting

then by (3.33) we have

A quite similar computation gives

with some $C_{22}>0$. Noticing that $\frac{4(1-a)}{\theta_k-2a}, \frac{4}{\theta_k}, \frac{4(1-a)}{\lambda_k-2a}$ and $\frac{4}{\lambda_k}<2$, then we have

An integration of this ODI shows that

On the one hand, if

holding for infinitely many $k\geq 1$, we see

for all such k, and hence conclude that $\|n_{1\varepsilon}(t)\|_{L^{\infty}(\Omega)}\leq \|n_{01\varepsilon}\|_{L^{\infty}(\Omega)}$, for all $t > 0.$

On the other hand, in the opposite case, upon enlarging $C_{19}$ and $C_{20}$ if necessary we have that $M_k\leq \left(2 C_{19}C_{21}+2C_{20}C_{22}\right)r_k(r_k-1)\tilde b^k M_{k-1}^{2}$, for all $k\geq 1.$ Let $C_{23} : = \left(2 C_{19}C_{21}+2C_{20}C_{22}\right)c_2^2$, then from the definition of $\tilde{b}$ and (3.33), we see

for all $k\geq 1.$ Furthermore, we have

for all $k\geq 1$. A straight calculation shows that $\sum_{j = 0}^{k-1}2^j = 2^k-1\leq 2^{k}$ and $\sum_{j = 0}^{k-1}(k-j)2^j = 2^{k+1}-2-k\leq 2^{k+1}$, then we have

Finally, combining the definition of $M_k$, we concludes that

with $C_{24} = C_{23}(4\tilde b)^2$, which confirms $\|n_{1\varepsilon}\|_{L^{\infty}(\Omega)}$ is bounded. By Lemma 3.1, $T_{\max, \varepsilon} = \infty.$

Lemma 3.9. There exists $C>0$ such that for any $\varepsilon\in(0, 1)$, the solution of (3.1) satisfies

for all $T>0$. And

for all $T>0$, where $r\in[1, \frac{4p}{3}+3].$ And

for all $T>0$, where $m\in[1, \frac{17}{3}].$

Proof. Fixing $C_1>0$ and $C_2>0$ such that in accordance with Lemma 3.6, we have that

and that

Then we combine the Gagliardo-Nirenberg inequality with Poincaré inequality to fix $C_3, C_4>0$ such that

Recalling Lemma 3.7, particularly, when $\xi = 5$, we have

with some $C_5>0$. Taking $\tilde{a} = \frac{\frac{p+3}{pr}-\frac{(p+3)r}{pr}}{\frac1p-\frac13-\frac{p+3}{p}} = \frac{3(p+3)(r-1)}{(4p+6)r}$, when $r\leq \frac{4p}{3}+3$, we see that $\tilde{a}\in[0, 1)$ and $\frac{pr}{p+3}\cdot\tilde{a}\leq p.$ Thus, invoking the Gagliardo-Nirenberg inequality along with (3.9), we obtain $C_6>0$ and $C_7 > 0$ such that

with $\sigma = \frac{p+3}{p+3-r\tilde{a}}.$

Again by Lemma 3.7 we also have

with some $C_8>0$. Upon another application of the Gagliardo-Nirenberg inequality in precisely the same way we see

where $\hat{a} = \frac{\frac{5}{2m}-\frac52}{\frac12-\frac13-\frac52} = \frac{15}{14}(1-\frac1m).$ Since $m\leq \frac{17}{3}$, we have $\frac{2m}{5}\cdot\hat{a}\leq 2$. Then employing (3.10) and Young's inequality, we have

Lemma 3.10. There exists $C>0$ such that for any $\varepsilon\in(0, 1)$ and any $T>0$, the solution of (3.1) satisfies

Proof. Testing the first equation in (3.1) by any $\varphi\in C^{\infty}(\bar{\Omega})$, and integrating by parts, due to Hölder's inequality, and (3.3) we can estimate

with some $C_1>0$. Then we use Young's inequality (3.26), (3.27), (3.28), (3.37) and the fact that $\frac15+\frac14<\frac15+\frac{3}{10}<\frac{1}{p^{\prime}}$ to see

By Lemma 3.6, we see there exists $C_{4}>0$ such that

Upon the Hölder's inequality, we furthermore see that

with some $C_5>0$. We next test the second equation in (3.1) by any $\varphi\in C^{\infty}(\bar{\Omega})$, and integrate by parts, due to Hölder's inequality and (3.3) we can obtain

We notice that $\frac15+\frac14<\frac15+\frac{3}{10}<\frac35$. By the Young's inequality, we can find $C_7>0$ such that

Recalling (3.26), (3.27), (3.28) with (3.37), we obtain (3.43).

Likewise, given any $\varphi\in C^{\infty}(\bar{\Omega})$, testing the third equation in (3.1) by $\varphi$ to obtain that

Thereafter we use (3.11) to see

Combining (3.26), (3.27) (3.28) and (3.37) entail (3.44).

Finally, noticing that $\|Y_{\varepsilon}v\|_{L^2(\Omega)}\leq \|v\|_{L^2(\Omega)}$ for all $v\in L_{\sigma}^2(\Omega)$ and the Young's inequality as well as that $\nabla\Phi\in L^{\infty}(\Omega),$ there exist constants $C_{10}, C_{11}>0$ such that for any $\varphi\in (C_{0, \sigma}^{\infty}(\Omega); R^3)$

The proof is complete.

4.   The main result and its proof

In this section, we are going to proof the existence of weak solution for the problem (1.1), (1.3) and (1.4). With the above compactness properties at hand, by means of a standard extraction procedure we can now derive the following theorem which actually is our main result.

Theorem 4.1. Let $\Omega \subset R^3$ be a bounded convex domain with smooth boundary. Suppose that the assumptions (1.5) and (1.6) hold. Then for $p\geq 2$, there exists at least one global weak solution (in the sense of Definition 1.1) of (1.1), (1.3) and (1.4).

Proof. If $(n_{1\varepsilon}, n_{2\varepsilon}, c_{\varepsilon}, u_{\varepsilon})$ is the global solution of (3.1), considering Lemma 3.6-Lemma 3.10 to see that

and that

According to the Aubin-Lions lemma ([9]), there exists $(\varepsilon_j)_{j\in N}\in(0, 1)$ such that $\varepsilon_j\searrow 0$ as $j\rightarrow \infty$, and such that as $\varepsilon = \varepsilon_j\searrow 0$, we have

Considering Lemma 3.7-Lemma 3.9, we also have

as well as

as $\varepsilon\searrow 0$. Combining the Lemma 6.1 and Lemma 6.2 in [19], we further see

and

as well as

as $\varepsilon = \varepsilon_j\searrow 0$. According to these convergence properties, by using the standard arguments and letting $\varepsilon = \varepsilon_j\searrow 0$ in each term of the natural weak formulation of (3.1) separately. Then we can verify that $(n_1, n_2, c, u)$ is a weak solution of (1.1), (1.3) and (1.4). The proof is complete.

Acknowledgment

The authors would like to express their deep thanks to the referee's valuable suggestions for the revision and improvement of the manuscript.