Abstract
This paper considers the following coupled chemotaxis system:
with homogeneous Neumann boundary conditions in a bounded domain \(\Omega\subset\mathbb{R}^{N}\) (\(N\ge3\)) with smooth boundaries, where \(\chi_{1}\), \(\chi_{2}\), \(\mu_{1}\), \(\mu_{2}\), \(a_{1}\), \(a_{2}\), α, β and γ are positive. Based on the maximal Sobolev regularity, the existence of a unique global bounded classical solution of the problem is established under the assumption that both \(\mu_{1}\) and \(\mu_{2}\) are sufficiently large.
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1 Introduction
In this paper, we consider the higher dimension quasilinear fully parabolic two-species chemotaxis system
where \(\Omega\subset\mathbb{R}^{N}\) (\(N\ge3\)) is a bounded domain with smooth boundaries ∂Ω, and the constants \(\chi_{1}\), \(\chi_{2}\), \(\mu_{1}\), \(\mu_{2}\), \(a_{1}\), \(a_{2}\), α, β and γ are positive. The functions \(\phi,\psi\in C^{2}([0,\infty))\) satisfy
with \(k_{1},k_{2}>0\), \(s_{0}>1\), \(p,q\in\mathbb{R}\).
The system (1.1) arises in mathematical biology as a model for two biological species which move in the direction of higher concentration of a signal produced by themselves. Here, \(u=u(x,t)\) and \(v=v(x,t)\) represent the densities of the two populations, respectively, and \(w=w(x,t)\) denotes the concentration of the chemical.
There are many results about the one-species chemotaxis systems with logistic source when \(v\equiv0\) in (1.1), that is,
All solutions are global in time and remain bounded whenever \(n\le 2\) and \(\mu_{1} >0\) is arbitrary [1], or \(n\ge3\) and \(\mu_{1}>\mu_{0}\) with some constant \(\mu_{0}(\chi_{1})>0\) [2, 3]. Especially, the convexity of Ω which is required in [2] is unnecessary in [3].
As for two-species models without logistic-type growth restrictions, that is, when \(\mu_{1}=\mu_{2}=0\), the resulting system inherits some important properties from the original Keller-Segel model for single-species chemotaxis; see [4, 5] and the references therein. In particular, the striking phenomenon of finite-time blow-up, known to occur in both parabolic-elliptic and fully parabolic versions of the latter ([6, 7]), has also been detected in parabolic-parabolic-elliptic two-species systems ([8–12]).
Apart from the aforementioned system, a source of logistic type is included in (1.1). For the semilinear parabolic-parabolic-elliptic version of (1.1), in the case of weak competition when both \(a_{1}<1\) and \(a_{2}<1\), the large time behavior has been addressed in [13], and also in [14]. Here we point out that the smallness condition on the chemotactic strengths in [14] seems more natural than that in [13]. When \(a_{1}>1\) and \(0\le a_{2}<1\), it has been shown in [15] that the solution \((u,v,w)\) converges to \((0,1,\frac{\beta}{\gamma} )\) as \(t\rightarrow \infty\) under some assumptions on \(\chi_{1}\), \(\chi_{2}\), \(a_{1}\), \(a_{2}\). For the currently considered fully parabolic system (1.1), when \(\phi(u)\equiv u\), \(\psi(v)\equiv v\), the authors in [16] have proved that the system (1.1) possesses a global solution for \(n\le2\) and any positive constant \(\mu_{1}\), \(\mu_{2}\). For the case \(n\ge3\), the large time behavior has been obtained but there lacks a proof of the existence of a global solution. Especially, the authors in [17] proved that for the bounded convex domain Ω and \(\gamma\ge\frac{1}{2}\), the problem (1.1) possesses a global solution with large \(\mu_{1}\) and \(\mu_{2}\).
Our goal in this paper is to investigate the global existence and boundedness of solutions to (1.1). The main result of the present paper is the following theorem.
Theorem 1.1
Suppose that \(\Omega\subset\mathbb{R}^{N}\) (\(N\ge3\)) is a bounded domain with smooth boundaries, and ϕ and ψ satisfy (1.2). Then there is \(\mu_{0}>0\) such that if \(\max\{\mu_{1},\mu_{2}\}>\mu_{0}\), for each nonnegative \(u_{0}(x),v_{0}(x)\in C^{0}(\bar{\Omega})\) and \(w_{0}(x)\in W^{1,r}(\Omega)\) with \(r>N\), system (1.1) admits a unique classical solution \((u,v,w)\) such that
Moreover, \((u,v,w)\) is bounded in \(\Omega\times(0,\infty)\).
2 Preliminaries
The local existence of solutions to (1.1) can be addressed by methods involving standard parabolic regularity theory in a suitable fixed point approach.
Lemma 2.1
Suppose \(\Omega\subset\mathbb{R}^{N}\) (\(N\ge3\)) is a bounded domain with smooth boundaries, and ϕ and ψ satisfy (1.2); let \(r>N\). Then for each nonnegative \(u_{0}(x),v_{0}(x)\in C^{0}(\bar{\Omega})\) and \(w_{0}(x)\in W^{1,r}(\Omega)\), there exists \(T_{\mathrm{max}}\in(0,\infty ]\) and a uniquely determined triple \((u,v,w)\) of functions
which solves (1.1) classically in \(\Omega\times(0,T_{\mathrm{max}})\). Moreover, \(T_{\mathrm{max}}<\infty\) if and only if
Let us cite the following auxiliary statement from [3].
Lemma 2.2
Let \(r\in(1, \infty)\). Consider the following evolution equation:
For each \(w_{0}\in W^{2,r}(\Omega)\) (\(r>N\)) with \(\frac{\partial w_{0}}{\partial n}=0\) on ∂Ω and any \(u,v\in L^{r}((0,T); L^{r}(\Omega))\), there exists a unique solution
Moreover, there exists \(C_{r}>0\), such that if \(s_{0}\in[0,T)\), \(w(\cdot, s_{0})\in W^{2,r}(\Omega)\) (\(r>N\)) with \(\frac{\partial w(\cdot, s_{0})}{\partial n}=0\), then
Proof
Let \(h(x,s)=e^{\gamma s} w(x,s)\). We derive that h satisfies
Applying the maximal Sobolev regularity ([18], Theorem 3.1) to h, and using the Hölder inequality, we have
Consequently, for any \(s_{0}>0\), replacing \(v(t)\) by \(v(t+s_{0})\), we prove (2.3). □
The following lemma, which can be proved by applying Moser-type iteration techniques, which can be found in [19], will be used to prove global existence and boundedness:
Lemma 2.3
Let \(N\ge1\), and suppose that there exists \(k_{0}\ge1\) such that \(k_{0}>N/2\) and
Then \(T_{\max}=\infty\) and
3 Proof of Theorem 1.1
In this section, we prove our main result.
Lemma 3.1
Suppose \(\Omega\subset\mathbb{R}^{N}\) (\(N\ge3\)) is a bounded domain with smooth boundaries, \(\chi_{1}, \chi_{2}\in\mathbb{R}^{+}\). For any \(k>1\), \(\eta>0\) and \(s_{0}>0\), there exists \(\mu_{k, \eta}>0\) and \(C=C(k, \vert \Omega \vert , \mu_{1}, \mu_{2}, \chi_{1}, \chi_{2}, \eta, u_{0}, v_{0}, w_{0})>0\) such that if \(\min\{\mu_{1}, \mu_{2}\}>\mu_{k, \eta}\), then
for all \(t\in(s_{0},\infty)\).
Proof
We fix \(s_{0}\in(0,T_{\max})\) such that \(s_{0}\le1\). For any constant \(k>1\), we take \(u^{k-1}\) as a test function for the first equation in (1.1) and integrate by parts. Then we have
for all \(t\in(s_{0},T_{\max})\). Then Young’s inequality implies the following two inequalities for any \(\varepsilon>0\) (to be determined) and some constants \(c_{1}\) and \(c_{2}\):
and
where \(c_{1}=c_{1}(\mu_{1}, \varepsilon, k, \gamma)=\frac{1}{k} (1+\frac{1}{k} )^{-(k+1)}\varepsilon^{-k} (\mu_{1}+\frac{\gamma(k+1)}{k} )^{k+1}\) and \(c_{2}=\sup_{k>1}\frac{1}{k} (1+\frac{1}{k} )^{-(k+1)}<\infty\). By substituting (3.3) and (3.4) into (3.2), we find that
Similarly, for some constants \(c_{3}\) and \(c_{4}\), we have
Applying the variation-of-constants formula to the above inequalities shows that
and
for all \(t\in(s_{0},T_{\max})\), where
and
are independent of t. Now, we apply Lemma 2.2 to see that there is \(C_{k}>0\) such that
Put the inequalities (3.7) and (3.8) together and apply (3.9); then we arrive at
for all \(t\in(s_{0},T_{\max})\), with the constant \(c_{7}>0\) being independent of t.
Let \(\mu_{k,\eta}=\max\{\eta+c_{2}\eta^{-k}\chi_{1}^{k+1}C_{k}, \eta+c_{4}\eta^{-k}\chi_{2}^{k+1}C_{k}\}\), which is independent of ε. We can choose \(\varepsilon\in(0,\min\{\mu_{1},\mu_{2}\}-\mu_{k,\eta})\) such that
It entails
for all \(t\in(s_{0},T_{\max})\), with the constant \(c_{8}=c_{8}(\mu_{1}, \varepsilon, \eta, k, \gamma, w(s_{0}))\) being independent of t. This completes the proof. □
In order to prove Theorem 1.1, we should give an estimation for \((u,v.w)\) when \(t\in(0,s_{0})\). We know by Lemma 2.1 that \(u(\cdot,s_{0}), v(\cdot,s_{0}), w(\cdot,s_{0})\in C^{2}(\bar{\Omega})\) with \(\frac{\partial w(\cdot,s_{0})}{\partial n}=0\) on ∂Ω, so that we can pick \(M>0\) such that
Combining Lemma 3.1 with the estimates (3.12), we readily arrive at our main result.
Proof of Theorem 1.1
Let \(\mu_{0}=\inf_{\eta>0}\mu_{k_{0},\eta}\). We know by Lemma 3.1 and (3.12) that (2.6) holds when \(\min\{\mu_{1},\mu_{2}\}>\mu_{0}\), and hence (2.7) is true. Lemma (2.1) shows that \((u,v)\) is global. □
4 Conclusion
The paper considers a quasilinear fully parabolic two-species chemotaxis system of higher dimension. The existence of a unique global bounded classical solution of problem (1.1) is established under the assumption that the coefficients of the kinetic terms are large enough. We point out that the convexity of Ω and the assumption \(\gamma\ge\frac{1}{2}\), which are required in [17], are unnecessary in our theorem due to the technique used here. We also notice that the result of Theorem 1.1 is independent of the value of p and q in (1.1), and thus extends the result for the semilinear case.
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Acknowledgements
The authors would like to express sincere gratitude to the referees for their valuable suggestions and comments on the original manuscript. The first author is supported by NSF of China (11601140) and the Scientific Research Fund of the Hunan Provincial Education Department (16B047). The second author is supported by NSF of China (11501438).
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Zhou, S., Yang, C. Boundedness in a quasilinear fully parabolic two-species chemotaxis system of higher dimension. Bound Value Probl 2017, 115 (2017). https://doi.org/10.1186/s13661-017-0846-1
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DOI: https://doi.org/10.1186/s13661-017-0846-1