Global existence and boundedness in a quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type

This paper deals with the global existence and boundedness of solutions to the following quasilinear attraction-repulsion chemotaxis system:

under homogeneous Neumann boundary conditions in a bounded domain \(\Omega\subset\mathbb{R}^{n}\) (\(n\geq2\)) with smooth boundary, where \(D(u)\geq c_{D} u^{m-1}\) with \(m\geq1\) and some constant \(c_{D}>0\). It is proved that if \(\xi\gamma-\chi\alpha>0\) or \(m>2-\frac{2}{n}\), then for any sufficiently regular initial data, this system possesses a unique global bounded classical solution for the case of nondegenerate diffusion (i.e., \(D(u)>0\) for all \(u\geq 0\)), whereas for the case of degenerate diffusion (i.e., \(D(u)\geq0\) for all \(u\geq0\)), it is shown that there exists a global bounded weak solution under the same assumptions.

Chemotaxis is widespread in nature. It describes the oriented migration of cells or bacteria toward the concentration gradient of a chemical substance. In 1970, Keller and Segel [1] derived the well-known and widely studied Keller-Segel attractive model. The most obvious feature of this system is that the solution may blow up in finite time (see [2–9] and references therein). Hillen and Painter [5] suggested the chemotaxis model with nonlinear diffusion and aggregation by considering the volume-filling effect. Therefore, there are many papers on the global existence or finite time blow-up of solutions (e.g., see [10–23]).

In many biological processes, the migration of cells or bacteria is generally influenced by a combination of attractive and repulsive chemicals [24, 25]. The scholars in [26, 27] have proposed the corresponding attraction-repulsion chemotaxis model

under no-flux boundary conditions, where \(\Omega\subset\mathbb{R}^{n}\) is a bounded domain with smooth boundary. Here \(\chi\geq0\), \(\xi\geq0\), \(\alpha>0\), \(\beta>0\), \(\gamma>0\), \(\delta>0\), and \(\tau=0, 1\) are parameters. The unknown functions \(u(x,t)\), \(v(x,t)\), and \(w(x,t)\) denote the cell density, the concentration of an attractive signal, and the concentration of a repulsive chemical, respectively. If we take \(\xi =0\), then model (1.1) is the classical attractive Keller-Segel model. The first cross-diffusive term and the second in the first equation of (1.1) mean that the movement of the bacteria is directed toward the increasing concentration of an attractive substance and away from the increasing concentration of a repulsive chemical, respectively. The second and third equations in model (1.1) indicate that chemoattractant and chemorepellent are produced by cells and have attenuation. There are fewer results for (1.1) than the classical attractive Keller-Segel model, mainly since the latter possesses a useful Lyapunov functional whereas the former does not admit such a functional. When \(n=1\) and \(\tau=1\), the global existence and asymptotic dynamics of solutions of (1.1) were studied by [28, 29]. When \(n=2\), \(\tau=1\), and \(\xi\gamma-\chi\alpha>0\), the model (1.1) possesses a unique global bounded classical solution with any sufficient regular initial data (see [30, 31]). When \(n=2 \mbox{ or } 3\), \(\tau=1\), and \(\xi\gamma=\chi\alpha\), Lin et al. [32] proved that (1.1) admits a unique global bounded classical solution, and large time-behavior is considered. When \(\tau=0\), the global solvability, critical mass phenomenon, blow-up, and asymptotic behavior were studied in [33, 34]. Recently, Jin and Wang [35] studied the boundedness, blow-up, and critical mass phenomenon of solutions to a variant of (1.1) for \(n=2\). Liu et al. [36] also studied the pattern formation of model (1.1) with \(\tau=1\) from both analytical and numerical aspects.

To the best of our knowledge, presently, there is no rigorous result on the attraction-repulsion chemotaxis model with nonlinear diffusion. Thus, this paper mainly aims to understand the competition among the repulsion, the attraction, and the nonlinear diffusion. Precisely, we will consider the global existence and boundedness of solutions to the following quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type:

where \(\Omega\subset\mathbb{R}^{n} \) (\(n\geq2\)) is a bounded domain with smooth boundary ∂Ω, and \(\partial/\partial\nu\) represents the derivative with respect to the outer normal of ∂Ω. As usual, we assume that \(\chi, \xi\ge0\) and that α, β, γ, and δ are positive parameters. For the diffusion coefficient D, we assume that

and there exist some constants \(c_{D}>0\) and \(m\ge1\) such that

In addition to (1.3) and (1.4), we will require that \(D(u)\) satisfy

in some places. In particular, when \(D(u)\) does not satisfy (1.5) (i.e., \(D(u)\geq0\) for all \(u\geq0\)), equation (1.2)₁ may be degenerate at \(u=0\).

We will show that we can allow for the case of attraction dominating the repulsion (i.e., \(\xi\gamma-\chi\alpha<0\)) and still obtain global existence results due to the nonlinear diffusion. Thus, our results confirm that the attraction-repulsion system with nonlinear diffusion can prevent blow-up of solutions in higher dimensions as mentioned before.

We now state the main results of this paper.

Theorem 1.1

Let \(\Omega\subset\mathbb{R}^{n}\) (\(n \geq2\)) be a bounded domain with smooth boundary. Assume that \(u_{0}\in W^{1,\infty}(\Omega)\) is a nonnegative function and \(D(u)\) satisfies (1.3), (1.4), and (1.5). Suppose that

Then there exists a unique nonnegative bounded solution \((u,v,w)\) belonging to \(C^{0} (\overline{\Omega} \times[0,\infty) )\cap C^{2,1} (\overline{\Omega} \times(0,\infty) )\) that solves system (1.2) classically.

Remark 1.1

Theorem 1.1 shows that the solution is still global, provided that the diffusion is strong enough even if the attraction prevails over the repulsion, which provides a supplement to the dichotomy boundedness vs. blow-up in attraction-repulsion chemotaxis equations of parabolic-elliptic type with nonlinear diffusion.

Remark 1.2

For \(n=2\), Theorem 1.1 also shows that both the attraction and repulsion cannot result in blow-up when the linear diffusion is replaced by a nonlinear one.

For the case of \(D(u)\) only fulfilling (1.3) and (1.4), since equation (1.2)₁ with \(m >1\) may be degenerate at \(u =0\), system (1.2) does not admit classical solutions in general as the porous medium equation does. However, we can prove that system (1.2) in this case possesses at least one nonnegative global bounded solution \((u, v, w)\) in the following weak sense.

Definition 1.1

Let \(T>0\). Then a triple of nonnegative functions \((u,v,w)\) defined on \(\Omega\times(0,T)\) is called a weak solution to (1.2) if

1. (1)

   \(u\in L^{\infty} ((0,T);L^{\infty}(\Omega) )\) and \(D(u)\nabla u \in L^{2}_{\mathrm{loc}} ((0,T);L^{2} (\Omega) )\),

2. (2)

   \(v\in L^{\infty} ((0,T);W^{1,\infty}(\Omega) )\) and \(w\in L^{\infty} ((0,T);W^{1,\infty}(\Omega) )\),

3. (3)

   \((u,v,w)\) satisfies (1.2) in the distributional sense, that is, for every \(\varphi\in C_{0} ^{\infty} (\Omega\times[0,T) )\),

   $$\begin{aligned}& \begin{aligned}[b] &\int_{0} ^{T} \int_{\Omega} \bigl(D(u)\nabla u\cdot\nabla\varphi-\chi u \nabla v \cdot\nabla\varphi+\xi u \nabla w\cdot\nabla\varphi-u\varphi _{t} \bigr)\,dx\,dt \\ &\quad= \int_{\Omega}u_{0}(x)\varphi(x,0)\,dx, \end{aligned}\\& \int_{0} ^{T} \int_{\Omega} (\nabla v\cdot\nabla\varphi+\beta v \varphi )\,dx\,dt= \int_{0} ^{T} \int_{\Omega} \alpha u \varphi \,dx\,dt, \\& \int_{0} ^{T} \int_{\Omega} (\nabla w\cdot\nabla\varphi+\delta w \varphi )\,dx\,dt= \int_{0} ^{T} \int_{\Omega} \gamma u \varphi \,dx\,dt. \end{aligned}$$

If \((u,v,w)\) is a weak solution to (1.2) on \(\Omega\times(0,T)\) for all \(T\in(0,\infty)\), then \((u,v,w)\) is called a global weak solution to (1.2).

Theorem 1.2

Let \(\Omega\subset\mathbb{R}^{n}\) (\(n \geq2\)) be a bounded domain with smooth boundary. Assume that \(u_{0}\in W^{1,\infty}(\Omega)\) is a nonnegative function and that \(D(u)\) satisfies (1.3) and (1.4). Suppose that

Then there exists at least one nonnegative global weak solution \((u,v,w)\) to system (1.2). Moreover, \((u,v,w)\) satisfies

where \(C>0\) is a constant independent of t.

The rest of this paper is organized as follows. In Section 2, we first prove the local existence and uniqueness of a solution to system (1.2) and then give mass estimates. In Section 3, we give some fundamental estimates for the solution \((u,v,w)\) to system (1.2) and then prove Theorem 1.1. In Section 4, we establish the existence of global bounded weak solutions to system (1.2).

In this section, we first state the local well-posedness of system (1.2) and then give the mass estimates.

Lemma 2.1

Assume that \(u_{0}\in W^{1,\infty}(\Omega)\) is a nonnegative function and D satisfies (1.3), (1.4), and (1.5). Then there exist \(T_{\max}\in(0,\infty]\) and a unique triple \((u, v,w)\) of nonnegative functions from \(C^{0} (\overline{\Omega }\times[0,T_{\max}) )\cap C^{2,1} (\overline{\Omega} \times(0, T_{\max}) )\) solving (1.2) classically in \(\Omega\times (0,T_{\max})\). Moreover,

Proof

(i) Existence. Let \(T\in(0,1)\), which is specified below. We define

which is a bounded closed convex subset of space \(\mathcal{X}:=C^{0} (\bar{\Omega} \times[0,T] ) \).

For any given \(\tilde{u}\in S_{T}\), there exists a unique \((v,w)\) such that v and w solve the following elliptic equations

and

respectively. Then we can find a unique u solving the following parabolic equation:

Thus, we can introduce a mapping \(\Phi: \tilde{u}(\in\mathcal {S}_{T})\longmapsto u \) by defining \(\Phi(\tilde{u})=u\).

We next show that Φ has a fixed point for T sufficiently small. The elliptic regularity [37], Theorem 8.34, implies that (2.2) admits a unique solution \(v(\cdot,t)\in C^{1+\theta}(\Omega)\) for some \(\theta\in(0,1)\). Similarly, (2.3) also possesses a unique solution \(w(\cdot,t)\in C^{1+\theta}(\Omega)\). Moreover, the Sobolev embedding theorem and the \(L^{p}\) estimates yield that

and

with \(p>n\) and some constants \(C_{1}>0\) and \(C_{2}>0\). It then follows from [38], Theorem 6.1, that \(u\in C^{\theta,\frac{\theta}{2}} (\Omega \times(0,T) )\) with

for some \(\theta\in(0,1)\) and \(C_{3}>0\), where \(C_{3}\) depends on \(\min_{0\leq s\leq R}D(s)\), \(\|\nabla v\|_{L^{\infty}((0,T);C^{\theta}(\bar {\Omega}))}\) and \(\|\nabla w\|_{L^{\infty}((0,T);C^{\theta}(\bar{\Omega }))}\). Thus, we obtain

Hence if we take \(T<(\frac{1}{C_{3}})^{\frac{2}{\theta}} \), then

which implies that \(u\in\mathcal{S}_{T} \). Then we conclude that \(\Phi (\mathcal{S}_{T})\subset\mathcal{S}_{T}\) and \(\Phi(\mathcal{S}_{T})\) is compact in \(\mathcal{S}_{T}\) by (2.5). Moreover, we can easily deduce that Φ is a continuous operator. Thus, the Schauder fixed point theorem gives that there exists at least one fixed point \(u\in \mathcal{S}_{T} \) of Φ.

(ii) Regularity and nonnegativity. By the elliptic regularity theory we see that \(v(\cdot,t)\in C^{2+\theta }(\bar{\Omega})\) and \(w(\cdot,t)\in C^{2+\theta}(\bar{\Omega})\). It then follows from (2.5) that \(v(x,t)\in C^{2+\theta,\frac{\theta }{2}} (\bar{\Omega}\times[\eta,T] )\) and \(w(x,t)\in C^{2+\theta ,\frac{\theta}{2}} (\bar{\Omega}\times[\eta,T] )\) for all \(\eta\in (0,T)\). The parabolic regularity theory [38], Theorem 6.1, entails that

We may prolong the solution to the interval \([0,T_{\max})\) with either \(T_{\max}=\infty\) or \(T_{\max}<\infty\), where, in the latter case,

Finally, the parabolic and elliptic comparison principles ensure the nonnegativity of u, v, and w.

(iii) Uniqueness. The proof for the uniqueness of solutions to system (1.1) is inspired by a method in [23]. We suppose that \((u_{1},v_{1},w_{1})\) and \((u_{2},v_{2},w_{2})\) are two classical solutions to system (1.2) in \(\Omega\times(0,T)\) with the same initial data. Fix \(T_{1}\in(0,T)\).

It is clear that \(v_{1}-v_{2}\) satisfies the equation

Thus, we differentiate (2.7) on t and then take \(v_{1}-v_{2}\) as a test function to have

for any \(t\in(0,T_{1})\), where \(A(s)=\int_{0} ^{s}D(s)\,ds\). For the first term on the right-hand side of (2.8), we obtain from the mean value theorem and the Young inequality that

for some positive constant \(C_{4}\in [D(s_{1}),D(s_{2}) ]\), where

For the second integral on the right-hand side of (2.8), we can use the Hölder’s inequality to have

Notice that \(| u_{2}|\leq C_{5}\), \(|\nabla v_{1}|\leq C_{6}\), and \(|\nabla w_{1}|\leq C_{7}\) with some positive constants \(C_{6}\), \(C_{5}\), and \(C_{7}\) in \(\Omega\times(0, T_{1})\). Thus,

Inserting (2.11) into (2.10) and using Young’s inequality, we obtain

Similarly, we can conclude that

To estimate the last integral in (2.13), we notice that \(w_{1}-w_{2}\) satisfies the equation

Taking \(w_{1}-w_{2}\) as a test function, we obtain

which, together with Young’s inequality, yields that

Thus, the last term in (2.13) can be estimated as

Summarily, combining (2.8), (2.9), (2.12), and (2.15), we obtain

By Gronwall’s inequality we derive that \(v_{1}=v_{2} \) and \(u_{1}=u_{2}\) in \(\Omega\times(0,T_{1})\). By (2.14) we also have \(w_{1}=w_{2}\) in \(\Omega\times(0,T_{1})\). Hence, \(v_{1}=v_{2} \), \(u_{1}=u_{2}\), and \(w_{1}=w_{2}\) in \(\Omega\times(0,T) \) due to the arbitrariness of \(T_{1}\in(0,T)\). This implies the uniqueness of solutions. □

The following lemma deals with the mass identities.

Lemma 2.2

Let the assumptions in Lemma 2.1 hold. Then the classical solution \((u,v,w)\) of (1.2) fulfills

Moreover, we have

provided that \(u_{0}>0\).

Proof

We integrate each equation of (1.2) with respect to \(x \in\Omega\) and then obtain

for all \(t\in(0,T_{\max})\). It is clear that (2.16)-(2.18) hold. By the maximum principle, we obtain the positivity (2.19) of u. □

In this section, we mainly investigate the existence of global bounded classical solutions to system (1.2) with nondegenerate diffusion. We first consider the case that the repulsion prevails over the attraction (i.e., \(\xi\gamma-\chi\alpha>0\)).

Lemma 3.1

Assume that \(\xi\gamma-\chi\alpha>0\). Suppose that \(u_{0}\in W^{1,\infty }(\Omega)\) is a nonnegative function and D satisfies (1.3), (1.4), and (1.5). Then, for any \(p>\frac{n}{2}\), there exists a constant \(C>0\) independent of t such that the solution \((u,v,w)\) of (1.2) fulfills

Proof

We multiply the first equation in (1.2) by \(u^{p-1}\) and integrate by parts over Ω to have

for all \(t\in(0,T_{\max})\). Thus, from the second and third equations in (1.2) we obtain

which, together with \(v\geq0\), yields that

By \(\xi\gamma-\chi\alpha>0\) and Young’s inequality we deduce that

where \(C_{1}:=\xi\delta\frac{p-1}{p+1} [\frac{2\xi\delta p}{(\xi\gamma -\chi\alpha)(p+1)} ]^{p}\). Substituting (3.4) into (3.3) yields

Following a similar procedure as in [33], we go to estimate the term \(\int_{\Omega}w^{p+1}\,dx\). Here we give a sketch for completeness. Since w solves

where \(\delta>0\) and \(\gamma>0\), we can apply \(L^{p}\) estimates [39, 40] on (3.6) to obtain

with some constant \(C_{2}>0\). Then by the Gagliardo-Nirenberg interpolation inequality [41] and the \(L^{1}\) estimates of w (Lemma 2.2) we find that

with some constants \(C_{3}>0\) and \(C_{4}>0\), where

Since \(p>\frac{n}{2}\), it is easy to check that \(\theta\in(0,1)\) and \((p+1)\theta< p\). Hence, using Young’s inequality twice, we have

Substituting (3.9) into (3.5), we obtain

Then by taking

we have

where \(C_{6}:=C_{1} C_{5}\). By Young’s inequality again, we obtain

for \(t\in(0,T_{\max})\). Thus, we conclude that

where \(C_{7}:=C_{6}+\frac{|\Omega|}{p+1} [\frac{4p}{(\xi\gamma-\chi \alpha)(p^{2}-1)} ]^{p}\). By Gronwall’s inequality we have

which implies the desired uniform estimates. □

Next, considering the case that the attraction dominates over the repulsion, we can deduce a similar uniform estimate under the assumption of \(m>2-\frac{2}{n}\). We will show that the stronger diffusion plays a key role in deducing such a uniform bound.

Lemma 3.2

Assume that \(\xi\gamma-\chi\alpha\le0\) and \(m>2-\frac{2}{n}\). Suppose that \(u_{0}\in W^{1,\infty}(\Omega)\) is a nonnegative function and D satisfies (1.3), (1.4), and (1.5). Then, for any \(p>\frac{n}{2}\), there exists a constant \(C>0\) independent of t such that the solution \((u,v,w)\) of system (1.2) fulfills

Proof

Combining (3.2) with (1.4), we derive

for all \(t\in(0,T_{\max})\). This, along with \(v\geq0\), yields

By Young’s inequality we obtain

where \(C_{1}:=(p-1)(\chi\alpha-\xi\gamma+\xi\delta)\) and \(C_{2}:=(p-1)\xi \delta\). Similarly to the deduction of (3.8) in Lemma 3.1, we find that there exist some constants \(C_{3}>0\) and \(C_{4}>0\) such that

where \((p+1)\theta=\frac{np^{2}}{np+2p-n}\in(0,p)\) by \(p>\frac{n}{2}\). Using Young’s inequality twice yields

where \(C_{5}:=C_{4}(|\Omega|+2)\). Hence, inserting (3.15) into (3.14), we obtain

where \(C_{6}:=C_{1}+C_{2}C_{4}\) and \(C_{7}:=C_{2}C_{5}\). Since \(p>\frac{n}{2}\) and \(m\geq1\), we have \(p>\frac{(2-m)n}{2}>\frac{2n-mn-2}{2}\) and then obtain

By the Gagliardo-Nirenberg inequality we derive that there exists \(C_{8}>0\) such that

where

and

Since \(m>2-\frac{2}{n}\), we have

Thus, we use Young’s inequality to derive

where

Inserting (3.18) into (3.16) yields

where \(C_{11}:=C_{7}+C_{10}\). Since \(p>\frac{n}{2}\) and \(m\geq1\), it is easy to check that

By using the Gagliardo-Nirenberg inequality again, we can find a constant \(C_{12}>0\) such that

where

and

Since \(m>2-\frac{2}{n}\), we find that \(\frac{2 \theta_{2} p}{p+m-1}=\frac{p-1}{\frac{1}{n}-\frac{1}{2}+\frac{p+m-1}{2}} <\frac{p-1}{\frac{1}{n}-\frac{1}{2}+\frac{p+2-\frac{2}{n}-1}{2}}=\frac {2(p-1)}{p}<2 \). Thus, using Young’s inequality yields that

where

Substituting (3.21) into (3.19) yields that

where \(C_{15}:=C_{11}+C_{14}\). Thus, using Gronwall’s inequality, we have

which implies the desired uniform \(L^{p}\) estimates. □

We now turn to the existence of global bounded classical solutions.

Proof of Theorem 1.1

According to the \(L^{p}\) estimates of w (see (3.7)), we obtain from Lemmas 3.1 and 3.2 that

with some positive constant \(C_{1}\). Then, by choosing \(p>n\), from the Sobolev embedding theorem we can derive that there exists a constant \(C_{2}>0\) such that

Similarly, there exists a constant \(C_{3}>0\) such that

With the aid of Lemmas 3.1 and 3.2 and using Lemma A.1 in [20] (see also [42]), we can conclude that there exists a positive constant \(C_{4}>0\) such that

which, together with the extensibility criterion (2.1), implies that \(T_{\max}=+\infty\). Thus, \((u,v,w)\) is a global bounded classical solution to system (1.2). □

In this section, we consider system (1.2) with degenerate diffusion (i.e., \(D(u)\geq0\) for all \(u\geq0\)). We first consider the following regularized system with nondegenerate diffusion for \(\varepsilon\in(0,1)\), which satisfies all the formal arguments:

where \(D_{\varepsilon}\) is defined by

Thus, \(D_{\varepsilon}\) satisfies (1.3), (1.4), and (1.5). The following proposition is a direct consequence of Theorem 1.1.

Proposition 4.1

Let \(\varepsilon\in(0,1)\), and let \(u_{0}\in W^{1,\infty}(\Omega)\) be a nonnegative function. Suppose that \(\xi\gamma-\chi\alpha>0\) or \(\xi\gamma-\chi\alpha\le0\) and \(m>2-\frac{2}{n}\). Then system (4.1) admits a unique global bounded classical solution \((u_{\varepsilon}, v_{\varepsilon}, w_{\varepsilon})\).

Next, we go to find some estimates to \((u_{\varepsilon}, v_{\varepsilon}, w_{\varepsilon})\), which are independent of ε and used to obtain some convergence properties. By taking \(\varepsilon\to0\) we will establish the existence of global bounded weak solutions. The following two lemmas based on the ideas in [18] are used to prove the existence of the limit function of \(\nabla\int^{u_{\varepsilon}+\varepsilon}_{0} D(z)\,dz\).

Lemma 4.1

Let \(T>0\), and let the assumptions in Proposition 4.1 hold. Let \((u_{\varepsilon},v_{\varepsilon},w_{\varepsilon})\) be a solution to system (4.1) on \((0,T)\). Then

where \(C_{1}\) is a positive constant independent of ε.

Proof

Taking \(u_{\varepsilon}\) as a test function on the first equation in (4.1) and integrating it over \(\Omega\times(0,T)\), we derive

It then follows from the second and third equations in (4.1) that

From Proposition 4.1 we obtain that there exist some positive constants \(c_{1}\), \(c_{2}\), \(c_{3}\), \(c_{4}\), and \(c_{5}\) independent of ε such that

Therefore, we have

which yields the desired estimate

where \(C_{1}:=\frac{1}{2}c_{1}^{2} (\chi\beta c_{2}+(\xi\gamma+\chi\alpha )c_{1}+\xi\delta c_{3} )|\Omega|\). □

Lemma 4.2

Let \(T>0\), and let the assumptions in Proposition 4.1 hold. Let \((u_{\varepsilon},v_{\varepsilon},w_{\varepsilon})\) be a solution to system (4.1) on \((0,T)\). Then

where \(C_{2}\) is a positive constant independent of ε.

Proof

We multiply the first equation in (4.1) by \(\frac{d}{dt}\int _{0}^{u_{\varepsilon}+\varepsilon}D(z)\,dz\) and then integrate it over Ω to obtain

where we used \(\frac{d}{dt}\int_{0}^{u_{\varepsilon}+\varepsilon}D(z)\,dz =D^{\frac{1}{2}}(u_{\varepsilon}+\varepsilon)\frac{d}{dt}\int _{0}^{u_{\varepsilon}+\varepsilon}D^{\frac{1}{2}}(z)\,dz\). By Young’s inequality we obtain

Since \(\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}< c_{1}\) and \(D\in C^{2} ([0,\infty) )\), we have \(\|D(u_{\varepsilon}+\varepsilon)\|_{L^{\infty}(\Omega)}< c_{\infty}\) with some constant \(c_{\infty}>0\). Thus, by Young’s inequality and (4.3) we derive

Similarly,

Substituting the last two inequalities into (4.5), we obtain

Setting \(C_{\max}:=4\max \{ (\chi^{2} c_{4} ^{2} + \xi^{2} c_{5} ^{2} ), [2c_{1} ^{2} c_{\infty}\chi^{2}(\alpha^{2} c_{1} ^{2} +\beta^{2} c_{2} ^{2})|\Omega |+2c_{1} ^{2} c_{\infty}\xi^{2}(\gamma^{2} c_{1} ^{2} +\delta c_{3} ^{2})|\Omega| ] \}\) yields

Multiplying (4.6) by t and integrating it over \((0, T)\), we obtain

By (4.2) the integrals on the right-hand side of (4.7) can be estimated as

Then substituting (4.8) into (4.7) and using (4.2) again, we have

where \(C_{2}:=\max \{\frac{1}{2}c_{\infty}\|u_{0}\|^{2}_{L^{2}(\Omega)}, C_{\max}C_{1}, [\frac{1}{2}C_{\max}\|u_{0}\|^{2}_{L^{2}(\Omega)}+C_{\max }+c_{\infty}C_{1} ] \} \). By taking the supremum with respect to t on \((0, T)\) we complete the proof of (4.4). □

We now prove Theorem 1.2. Our method is also partially inspired by [18].

Proof of Theorem 1.2

For any given \(T>0\), we have \(\| u_{\varepsilon}\|_{L^{\infty} (0,T; L^{p}(\Omega) )}< C\) (\(p\in[1,\infty]\)), where C is a positive constant independent of T and ε. Then there exist a subsequence \(\{u_{\varepsilon_{j}} \}_{j\in\mathbb{N}}\) and a function \(u\in L^{\infty} (0,T; L^{p}(\Omega) )\) such that

for any \(p\in[1,\infty]\), where \(\varepsilon_{j}\rightarrow0\) as \(j\rightarrow\infty\). By using \(D\in C^{2} ([0,\infty) )\) and \(\| u_{\varepsilon}(t)\|_{L^{\infty}(\Omega)}< c_{1}\) again, from Lemma 4.1 we deduce that \(\int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac {1}{2}}(z)\,dz\) is bounded in \(L^{2} (0,T; H^{1}(\Omega) ) \). Hence, there exist a subsequence (still denoted by \(\{u_{\varepsilon_{j}}\} _{j\in\mathbb{N}}\)) and a function \(\vartheta\in L^{2} (0,T; H^{1}(\Omega) )\) such that

On the other hand, by letting \(\tau>0\), from Lemma 4.2 we have

which implies that \(\int_{0}^{u_{\varepsilon}+\varepsilon}D^{\frac {1}{2}}(z)\,dz\) is bounded in \(H^{1} (\tau,T; L^{2}(\Omega) )\) (in particular, it is bounded in \(H^{1} (\tau,T; H^{-1}(\Omega) )\). Thus, by the Aubin-Lions lemma there exists a subsequence (still denoted by \(\{u_{\varepsilon_{j}}\}_{j\in\mathbb{N}}\)) such that

Set \(f(r):=\int_{0} ^{r} D^{\frac{1}{2}}(z)\,dz\). We see that \(f(r)\) is a strictly increasing and continuous function. Thus, the inverse function \(f^{-1}(r)\) of f exists and is continuous. Moreover, we can obtain that

Since \(\tau>0\) is arbitrary, we deduce from (4.10) and (4.11) that

Since \(\|v_{\varepsilon}(t)\|_{W^{1,\infty}(\Omega)}< c_{2}+c_{4}\), there exist a subsequence \(\{v_{\varepsilon_{j}}\}_{j \in\mathbb{N}}\) (hereafter, we still denote the subscript of the subsequence by \(\{ v_{\varepsilon_{j}}\}_{j\in\mathbb{N}}\) for simplicity) and functions v such that

Similarly, there exist subsequence \(\{w_{\varepsilon_{j}}\}_{n\in\mathbb {N}}\) and functions w such that

due to \(\|w_{\varepsilon}(t)\|_{W^{1,\infty}(\Omega)}< c_{3}+c_{5}\).

For any given \(T\in(0,\infty)\), we take \(\varphi\in C_{0} ^{\infty} (\Omega\times[0,T) )\). Then multiplying the first, second, and third equations in (4.1) by φ and integrating those on \(\Omega \times(0,T)\) we see that

Noting that \(D^{\frac{1}{2}}(u_{\varepsilon}+\varepsilon)|\nabla\varphi |\leq c^{\frac{1}{2}}_{\infty}\|\nabla\varphi\|_{L^{\infty}}\) and thus \(D^{\frac{1}{2}}(u_{\varepsilon}+\varepsilon)|\nabla\varphi|\in L^{2} (0,T; L^{2}(\Omega) )\), we see from (4.11) that

which, together with (4.10) and (4.12), yields that

as \(j\to\infty\). Similarly, since

by (4.11), from (4.13) and (4.14) we see that

as \(j\to\infty\). Summarily, by collecting (4.9), (4.13), (4.14), (4.16), and (4.17), from (4.15) we obtain that

upon letting \(j\rightarrow\infty\). Hence, \((u,v,w)\) is a global weak solution to system (1.2). Moreover, we deduce from (4.9), (4.13), (4.14), and Theorem 1.1 that

which implies the uniform boundedness of \((u,v,w)\). Thus, we complete the proof of Theorem 1.2. □

The author would like to express his gratitude to Professor Zhaoyin Xiang for his helpful comments. The author is also grateful to the anonymous reviewers for their detailed comments and useful suggestions, which greatly improved the paper. This work is partially supported by the Natural Science Project of Sichuan Province Department of Education (No. 16ZB0075), the Youth Research and Innovation of SWPU (No. 2013XJZT004), the Regional Science Foundation of China (No. 61362029) and the Key Project of Higher Education of Ningxia (No. NGY2013004).

Affiliations

1. School of Sciences, Southwest Petroleum University, Chengdu, 610500, China

   • Yilong Wang

Corresponding author

Correspondence to Yilong Wang.

Competing interests

The author declares that they have no competing interests.

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