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Global existence and boundedness in a quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type
Boundary Value Problems volume 2016, Article number: 9 (2016)
Abstract
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.
1 Introduction
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)
\(u\in L^{\infty} ((0,T);L^{\infty}(\Omega) )\) and \(D(u)\nabla u \in L^{2}_{\mathrm{loc}} ((0,T);L^{2} (\Omega) )\),
-
(2)
\(v\in L^{\infty} ((0,T);W^{1,\infty}(\Omega) )\) and \(w\in L^{\infty} ((0,T);W^{1,\infty}(\Omega) )\),
-
(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).
2 Preliminaries
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. □
3 Global bounded classical solutions in the case of nondegenerate diffusion
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). □
4 Global bounded weak solutions in the case of degenerate diffusion
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. □
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Acknowledgements
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).
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Wang, Y. Global existence and boundedness in a quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type. Bound Value Probl 2016, 9 (2016). https://doi.org/10.1186/s13661-016-0518-6
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DOI: https://doi.org/10.1186/s13661-016-0518-6