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Journal of Applied Mathematics
Volume 2013, Article ID 391056, 9 pages
http://dx.doi.org/10.1155/2013/391056
Research Article

Global Existence and Convergence Rates for the Strong Solutions in to the 3D Chemotaxis Model

1Department of Primary Education, Hunan National Vocational College, Yueyang, Hunan 414006, China
2Department of Mathematics, Hunan Institute of Science and Technology, Yueyang, Hunan 414006, China
3The Hubei Key Laboratory of Mathematical Physics, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Received 5 September 2013; Accepted 7 November 2013

Academic Editor: K. S. Govinder

Copyright © 2013 Weijun Xie 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

We are concerned with a 3D chemotaxis model arising from biology, which is a coupled hyperbolic-parabolic system. We prove the global existence of a strong solution when -norm of the initial perturbation around a constant state is sufficiently small. Moreover, if additionally, -norm of the initial perturbation is bounded; the optimal convergence rates are also obtained for such a solution. The proofs are obtained by combining spectral analysis with energy methods.

1. Introduction

In this paper, we investigate global existence and optimal convergence rates of strong solutions to the following 3D chemotaxis model: with initial data where is a positive constant. The system (1) is one of the models describing the chemotaxis phenomenon in biology and is closely related to the following system: which is motivated by biological considerations and numerical computations carried out by Othmer and Stevens in [1] and Levine and Sleeman in [2]. Here, denotes the particle density and is the concentration of chemicals. is the diffusion rate of particles. The function is commonly referred to as the chemotactic potential and denotes the chemical kinetics. Depending on the specific modeling goals, the kinetic function has a wide variability. In this paper, we consider a class of nonlinear kinetic functions : where is a positive constant and is a smooth function satisfying for all under consideration.

In fact, as in [3, 4], let with being a positive constant and is defined in (4); the system (3) can be rewritten in the following form: Furthermore, by setting we can rewrite the system (6) as Finally, for positive constants , , and to be determined below, let , , , and ; then the system (8) becomes If we choose that is, then it is easy to see that and satisfy If we replace the variables by , (12) is exactly (1).

To go directly to the theme of this paper, we now only review some former results which are closely related. For the one-dimensional version of the chemotaxis model (1), the existence and asymptotic behavior of smooth solutions have been studied by several authors. When the kinetic function is linear, that is, with (>0) and (≥0) being given constants, the corresponding system reads as follows: The initial boundary value problem and Cauchy problem for the system (13) were considered by [3, 5] and [4], respectively. In [3], they considered the initial boundary value problem for the system (13). When is sufficiently small, they proved the global existence of smooth solutions to the system (13). The authors in [5] generalized the results of [3] to the arbitrarily large initial data case. In [4], the authors obtained the global existence of smooth solutions to the Cauchy problem for the system (13) with large initial data. Recently, the authors in [68] extended the results of [35] to the nonlinear kinetic function case, respectively. For high dimensions, the global well-posedness of smooth small solution to (1) with was investigated in [5, 9] for initial-boundary value problem and Cauchy problem, respectively. For other related results, such as nonlinear stability of waves in one dimension and so on, please refer to [68, 1031] and references therein.

However, to our knowledge, so far there is no result on the optimal convergence rates of the strong solutions to the Cauchy problem (1)-(2). The main motivation of this paper is to give a positive answer to this question. In particular, we prove the global existence of a strong solution when -norm of the initial perturbation around a constant state is sufficiently small. Moreover, if in addition, -norm of the initial perturbation is bounded, the optimal convergence rates are also obtained for such a solution. The proofs are based on energy methods and spectral analysis which have been developed in [3235] and references therein.

Before stating our main results, we explain the notations and conventions used throughout this paper. We denote positive constants by . Moreover, the character “” may differ in different places.    denotes the usual Lebesgue space with the norm    denotes the usual th-order Sobolev space with the norm where .

Now, we are ready to state our main results.

Theorem 1. Assume that and is sufficiently small; then there exists a unique globally strong solution of the Cauchy problem (1)-(2) such that for any , satisfying where is a positive constant independent of .
Moreover, if, in addition, is bounded, then there is a positive constant independent of such that for any , the solution has the following decay properties:

Remark 2. As compared to the classic results in [9, 17, 30], where smallness conditions on -norm of the initial data were proposed, we are able to prove the global existence and convergence rates for the strong solutions to the Cauchy problem under only that -norm of the initial data is sufficiently small.

Remark 3. Similar ideas can be applied to study the initial boundary value problem. This will be reported in a forthcoming paper.

Remark 4. Here and in what follows, denotes the vector with and represents each component of the vector belonging to the function space .

The proofs of Theorem 1 are based on energy methods and spectral analysis. The key point for the proof of global existence is to obtain some a priori estimates independent of . Then, we can combine spectral analysis with the uniform a priori estimates to prove our desired decay estimates (17)–(20).

The rest of this paper is organized as follows. In Section 2, we reformulate the problem. In Section 3, we give some elementary facts on the decay-in-time estimates on for the linearized system. Sections 4 and 5 are dedicated to prove Theorem 1.

2. Reformulated System

In this section, we will first reformulate the problem. Set

Taking change of variables and linearizing the system around , we can reformulate the Cauchy problem (1)-(2) as where Here and in the sequel, for the notational simplicity, we still denote the reformulated variables by .

To prove the global existence of a solution to (22), 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 [4, 3638].

Proposition 5 (local existence). Assume that . Then, there exists a sufficiently small positive constant depending only on such that the Cauchy problem (22) admits a unique solution satisfying

Proposition 6 (a priori estimate). Let and . Assume that the Cauchy problem (22) has a solution on for some in the same function class as in Proposition 5. Then, there exist a small constant and a constant , which are independent of T, such that if then for any , it holds that Moreover, if, in addition, is bounded, then there is a positive constant independent of such that for any , the solution has the following decay properties:

Theorem 1 follows from Propositions 5 and 6 and standard continuity arguments. The proof of Proposition 6 will be given in Sections 4 and 5.

3. Linear Decay Estimates

In this section, we consider the Cauchy problem for the linearized equations corresponding to (22)1-(22)2: The solutions of the linear system (31) can be expressed as Here, is the Green's matrix for the system (31).

To derive the large time behavior of the solutions, we first give an explicit expression for the Fourier transform of Green's matrix .

Lemma 7. The Fourier transform of Green's matrix for the linear system (31) is given by where The representation above holds for .

Proof. The proof is in spirit of Hoff and Zumbrun [39, 40]. Applying the Fourier transform to the system (31), we have From (35)-(36), we obtain the following initial value problem for : The defined in (34) are exactly the eigenvalues of the ODE (37); hence for , The initial conditions in (37) give Substituting (39) into (38), by a straightforward computation, we obtain To compute , we substitute (40) into (35) to obtain
Hence, (40)-(41) give (33) and the proof of Lemma 7 is completed.

To derive the long-time decay rate of solutions in -framework, we need to verify the approximation of Green's function for both lower frequency and high frequency. In terms of the definition of the eigenvalues (34), we are able to obtain that it holds for that where

Next, we deal with the high frequency . From the definitions of the eigenvalues (34), we can analyze the eigenvalues for . Indeed, we have the leading orders of the eigenvalues for as This approximation gives the leading order terms of the elements of Green's function as follows: where and are some positive constants.

With the help of the formula (33) for the Fourier transform of Green's matrix and the asymptotical analysis on its elements, we are able to establish the time decay rate. Indeed, we have the time decay rate of the global solution to the Cauchy problem for the linear problem (31) as follows.

Lemma 8. Assume that is the solution of the linear problem (31) with the initial data ; then for .

Proof. From (33), (42), and (45), we have with being a constant here and below, and where here and below denotes a small but fixed constant.
Therefore, to prove (46), we only need to deal with the case for since the case for can be proved similarly. By virtue of the pointwise estimate (47), the Parseval theorem, and Hausdorff-Young's inequality, for , we have Thus, (49) gives (46) immediately, and the proof of Lemma 8 is completed.

4. A Priori Estimates

Throughout this section and the next section, we assume that all conditions of Proposition 6 are satisfied. Moreover, we make a priori assumption: where is a sufficiently small positive constant.

In the following, a series of lemmas on the energy estimates are given. Firstly, we will obtain the lower-order energy estimate for in the following lemma.

Lemma 9. There exists a positive constant , which is sufficiently large and independent of , such that for any .

Proof. Multiplying (22)1-(22)2 by , respectively, then summing up and integrating the resultant equation, we obtain Applying mean value theorem, Hölder’s inequality, and Sobolev’s inequality, it is clear that the two terms on the right-hand side of (52) can be estimated as follows: Combining (52) with (53) and using the fact that is sufficiently small, we have
Next, we will estimate . Multiplying (22)2 by and integrating the resulting equation over , we have where from (22)1, the first term on the right-hand side can be written as Then, it follows from (55)-(56), (23), Hölder’s inequality, and Young’s inequality that This together with the fact that is sufficiently small implies that Taking the curl for (22)1 and noting that , we have Since , we have from (59) that Combining (58) with (60) yields
Finally, multiplying (54) by which is suitably large and adding it to (61), one has (51) since is sufficiently small. This completes the proof of Lemma 9.

The following lemma is concerned with the higher-order energy estimate on .

Lemma 10. There exists a positive constant , which is sufficiently large and independent of , such that for any .

Proof. Applying to (22)1-(22)2 and multiplying them by , , respectively, and then integrating them over , we obtain Next, we estimate the terms and one by one. To begin with, by using (23), (50), Hölder’s inequality, Cauchy’s inequality, and Sobolev’s inequality, we can estimate the term as follows: Using similar arguments, we also have the following estimate for the term : Substituting (64) and (65) into (63) and noting that is sufficiently small, we have For higher-order derivatives of , we can apply the similar arguments used in obtaining (66) to get Combining (66) and (67) and using the fact that is sufficiently small, we have
Next, we estimate . To begin with, applying to (22)2 and then multiplying by , we have from Cauchy’s inequality that By integrating by parts several times, we estimate the first term on the right-hand side of (69) as follows: From (23), (50), Hölder’s inequality, Cauchy’s inequality, and Sobolev’s inequality, we have Substituting (70) and (71) into (69) yields Due to (60), we have Substituting (73) into (72) and using the fact that is sufficiently small, we obtain
Finally, since is sufficiently small, multiplying (68) by a suitably large positive constant and adding it to (74) give (62), and this completes the proof of Lemma 10.

5. Global Existence and Convergence Rate

In this section, we devote ourselves to prove Proposition 6. To begin with, we give the following lemma which is concerned with the a priori decay-in-time estimates on .

Lemma 11. Let , then for any .

Proof. From Duhamel's principle and Lemma 8, we have From (23), (50), Hölder’s inequality, and Sobolev’s inequality, we have Therefore, (75) follows from (76) and (77) immediately, and this completes the proof of Lemma 11.

Now, we are in position to prove Proposition 6.

5.1. Proof of Proposition 6

We will do it in two steps.

Step  1. Since is suitably small, from Lemmas 9 and 10, we can choose a suitably large positive constant such that for any . Notice that the expression under in (78) is equivalent to . Hence, by integrating (78) directly in time, we obtain (26).

Step  2. Define the temporal energy functional for any , where it should be mentioned that is equivalent to since is sufficiently large.

By virtue of Lemma 10, we have Adding to both sides of (80) yields where is a positive constant independent of . As [7, 17], we define Notice that Then, from (75) and (83), we have which together with Gronwall's inequality and (81) implies Since is nondecreasing, we have from (85) that for any , which together with the fact that is small enough implying that Thus, (82) and (87) yield which together with Sobolev’s inequality gives Therefore, we have proved (27) and (28).

Next, we turn to prove (29). To begin with, by using Sobolev's inequality and (27), we have On the other hand, applying Duhamel's principle and Lemma 8 again, we have from (26) and (28) that for any .

Hence, by the interpolation, it holds that for any , for any , where . Thus, we have proved (29). Therefore, (30) follows from (22)-(23) and (27)–(29) and this completes the proof of Proposition 6.

Acknowledgments

The second author was partially supported by the National Natural Science Foundation of China nos. 11301172 and 11226170, China Postdoctoral Science Foundation funded project no. 2012M511640, and Hunan Provincial Natural Science Foundation of China no. 13JJ4095.

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