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Journal of Applied Mathematics

Volume 2014, Article ID 186125, 7 pages

http://dx.doi.org/10.1155/2014/186125
Research Article

The Global Existence of Solutions in Time for a Chemotaxis Model with Two Chemicals

1Department of Basic Courses, Beijing Union University, Beijing 100101, China

2School of Sciences, Zhejiang A&F University, Hangzhou, Zhejiang 311300, China

Received 6 April 2014; Revised 21 May 2014; Accepted 27 May 2014; Published 16 June 2014

Academic Editor: Zhidong Teng

Copyright © 2014 Qian Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper concerns the uniform boundedness and global existence of solutions in time for the chemotaxis model with two chemicals. We prove the system has global existence of solutions in time for any dimension .

1. Introduction and Statement of Main Result

Chemotaxis is the influence of chemical substances in the environment on the movement of mobile species. Keller and Segel [1] proposed the general chemotaxis system where is the density function of cells (e.g., Dictyostelium discoideum) that are attracted by a chemical substance (e.g., cAMP) produced by themselves and the movement towards a higher concentration of the chemical substance, whose concentration function is . and are the random diffusion rates of cells and the chemical, respectively; represents the chemotactic flux of cells and is positive for positive and and is called the sensitivity function; and is the creation rate of the chemical, while is the degradation rate of the chemical.

The simplest case of (1) is that ,   , and are all positive constants, , and with being a positive constant. This was called by Childress and Perkus the “minimal model.” When dimension , solutions exist globally; see [2]. For , global existence depends on a threshold: when the initial mass lies below the threshold solutions exist globally, while above the threshold solutions blow up in finite time; these results were derived by various authors; see the review article [3, 4]. Many authors have analyzed system (1) for several variants, such as global existence, blow up solutions, and many other results; see [35].

Painter et al. [6] proposed a chemotaxis model with two chemicals. They considered a Turing system [7] as a mechanism for providing spatially heterogeneous chemical distributions to which a cell population chemotactically responds. That was the following model: where is a cell population, and are chemicals, and the cell population responds chemotactically to both chemical species. and are the chemotactic sensitivity functions and and define the chemical kinetics. ,   , and are taken as constants.

A special form of system (2) is as follows: where is a bounded domain. The kinetics of (3) are described by a simplified model for the glycolysis reaction [8, 9]. As we know, the global existence of solutions in time of two species such as (1) is investigated by many authors; however, the global existence of solutions in time of three species is studied little. In this paper, we study the global existence of solutions in time of (3); by applying analysis semigroup and energy method we will prove that system (3) has global solutions in time in any dimension .

We state the main result of this paper as follows.

Theorem 1. For any , satisfying ,   ,   on , (3) has a unique positive global solution such that

2. The Proof of Theorem 1

Theorem 2. For any ,   satisfying ,   ,   on , one has the following conclusions. (i)(3) has a unique solution on with satisfying (ii)Moreover, if, for small , , is bounded; then ; that is, has global existence, and for any , .

Proof. Equation (3) can be written as where Since the eigenvalues of are positive, (6) is normally parabolic; then (i) follows from [10]. Note that (6) is also a “triangular system,” so in virtue of [11], we complete the proof of Theorem 2.

In the following, we always assume ,   satisfying , ,   on .

Lemma 3. For any dimension , any solution of (3) has the following estimate: where and depends only on , , , and .

Proof. Integrating the second equation of (3) over , we have Integrating (9) with respect to , we get Multiplying the second equation of (3) by and integrating with respect to over , we get Integrating (11) with respect to and together with (10), we obtain

Lemma 4. For any dimension and small constants , any solution of (3) has the following estimate: where and is a constant depending on ,   ,   , , and .

Proof. Multiplying the second equation of (3) by and integrating with respect to over , we get Here (14) implies Define , by Hölder’s inequality; (15) can be written as Gagliardo-Nirenberg inequality implies In virtue of (17) and Young’s inequality, we have with and .

Taking suitable such that , we obtain with .

In view of (16)–(19), we get with and .

By Poincaré inequality, there exists a constant depending on , , such that we have It follows from (20) and (22) that After simple calculation, we obtain with , , and .

Since and , we have with .

Substituting ,   ,   into (25) yields Without loss of generality, we can assume Taking and letting of (28), together with Lemma 3, we have

With the notation we have the following Lemma 5.

Lemma 5. For any dimension , has the following estimate: where and is a constant depending on ,   ,    , , ,   , and .

Proof. In view of (3), satisfies the following equation: let , with domain ; then generates a linear analysis semigroup on satisfying and for . Taking , the fractional space with ; taking we get .

In virtue of (32), we obtain

By Lemma 4 and (30), one has where is a constant depending on ,   ,   , , ,   , and .

Similar as the proof of Lemma 5, we can prove where is a constant depending on ,   ,   , , ,   , , and .

Lemma 5 and (35) yield where is a constant depending on ,    ,    ,    ,  , and .

Lemma 6. For any dimension , any solution of (3) has the following estimate: where and is a constant depending on , , , , , , , , , and .

Proof. Integrating the first equation of (3) with respect to over and together with the boundary condition, we get In the following, we will use the inequality as follows: with depending only on and .

Multiplying the first equation of (3) by and integrating with respect to over imply (39) and Hölder’s inequality yield For , by (41), we have Integrating (42) with respect to over , we obtain Let then we get Taking ,   , we get For and (38) one gets where is a constant depending on ,   ,  , , ,   , ,   ,   , and . Now by Theorem 2(ii) and Lemma 3–Lemma 6, we have proved the Theorem 1.

For , ,   , ,   , and , we have the numerical simulation solutions of (3) as shown in Figure 1.

fig1
Figure 1

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is partially supported by the NNSF of China (71373023), the Beijing Natural Science Foundation (1132003, 1122016, KZ201310028030, KM201311417006, and KM201210017008), and the Zhejiang A&F University Talent Program (2013FR078).

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