Global existence and exponential stability for the strong solutions in \(H^{2}\) to the 3-D chemotaxis model
- Yinghui Zhang^{1}Email author and
- Weijun Xie^{2}
Received: 5 April 2015
Accepted: 15 June 2015
Published: 10 July 2015
Abstract
We prove the global existence of a unique strong solution to the initial boundary value problem for the 3-D chemotaxis model on a bounded domain with slip boundary condition when the initial perturbation is small in \(H^{2}\). Moreover, based on energy methods, we also prove that the strong solution converges to a steady state exponentially fast in time.
Keywords
1 Introduction
Direct applications of (1.2) include two aspects: (1) the modeling of haptotaxis, where cells move toward an increasing concentration of immobilized signals such as surface or matrix-bound adhesive molecules; (2) the initiation of angiogenesis, which is a vital process in the growth and development of granulation tissue and wound healing and is a fundamental step in the transition of tumors from a dormant to a malignant state. A comprehensive qualitative and numerical analysis of (1.2) was provided in [2]. In particular, explicit solutions describing and predicting aggregation, blowup, and collapse were constructed in one-dimensional space, based on special choices of initial data and the method of matched asymptotic expansion. The results were generalized by Yang et al. [3]. More discussions on model (1.2) can be found in [4, 5].
However, to our knowledge, so far there has been no result on global existence and asymptotic behavior of the strong solutions to the initial boundary value problem (1.1), (1.10). The main motivation of this paper is to give a positive answer to this question. In particular, we prove the global existence and exponential stability of a strong solution when the initial perturbation is small in \(H^{2}\). The proofs are based on energy methods which have been developed in [35–39] and the references therein.
Now, we are in a position to state the main results.
Theorem 1.1
Remark 1.2
As compared to the classic results in [8, 17, 30, 36], where smallness conditions on the \(H^{3}\)-norm of the initial data were proposed, we are able to prove the global existence and exponential stability for the strong solutions to the initial boundary problem under only the \(H^{2}\)-norm of the initial data is sufficiently small.
Finally, let us explain on some of the main difficulties and techniques involved in the process. First, by noting that we consider the \(H^{2}\) case, it is nontrivial to follow the framework of [11] directly, where the global existence and exponential decay rates of strong solutions in \(H^{3}\) for system (1.1) with \(f(u)=u\) are obtained. In fact, the main idea in [11] is to reduce the total energy of the solution to those of the lower order spatial derivatives and temporal derivatives of u, together with the div and curl of v. However, this method does not work in our \(H^{2}\) case. One main observation in this paper is that the total energy of the solution is equivalent to the sum of \(H^{1}\)-norm of \(\nabla\cdot\mathbf{v}\) and \(L^{2}\)-norm of Δu. With this in hand, we can make full use of the dissipation structure of the system and deal with nonlinear terms and boundary terms carefully to close the energy estimates of solutions. Second, compared to [11], we need to make careful energy estimates on nonlinear terms arising from the nonlinearity of \(f(u)\) (see (3.16), (3.19), (3.24), (3.30), and (3.35)).
The rest of this paper is devoted to proving Theorem 1.1. In Section 2, we reformulate the problem. In Section 3, we deduce the a priori estimate of the solutions and complete the proof of Theorem 1.1.
2 Reformulated system
To prove the global existence of a solution to (2.1), we will combine the local existence result together with a priori estimates. To begin with, we state the following local existence, the proof of which can be found in [40].
Proposition 2.1
(Local existence)
Proposition 2.2
(A priori estimate)
Theorem 1.1 follows from Propositions 2.1-2.2 and standard continuity arguments. The proof of Proposition 2.2 will be given in Section 3.
3 Proof of Proposition 2.2
The proof of Proposition 2.2 is based on several steps of careful energy estimates which are stated as a sequence of lemmas. First we recall some inequalities of Sobolev type (see [41]).
Lemma 3.1
As in [11], the following lemma (see [42]) plays an important role in our proofs, which gives the estimate of ∇v by \(\nabla\cdot\mathbf{v}\) and \(\nabla\times \mathbf{v}\).
Lemma 3.2
Lemma 3.3
Proof
The proof of the first inequality in (3.4) is trivial. Therefore, we have completed the proof of Lemma 3.2. □
Lemma 3.2 reduced the estimates of \(\mathbb{E}(t)\) to those for \(\mathbb{G}(t)\). Our next goal is to deduce the estimates of \(\mathbb{G}(t)\).
Lemma 3.4
Proof
We will prove Lemma 3.3 in five steps.
Declarations
Acknowledgements
The first author was partially supported by the National Natural Science Foundation of China #11301172, #11226170, Hunan Provincial Natural Science Foundation of China #13JJ4095, and the Scientific Research Fund of Hunan Provincial Education Department #14B077. The second author was supported by the Scientific Research Fund of Hunan Provincial Education Department #14C0536.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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