Journal of Applied Mathematics

Volume 2013 (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

Weijun Xie,¹ Yinghui Zhang,2,3 Yuandong Xiao,² and Wei Wei²

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 [6–8] extended the results of [3–5] 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 [6–8, 10–31] 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 [32–35] 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, 36–38].

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.

References

H. G. Othmer and A. Stevens, “Aggregation, blowup, and collapse: the ABCs of taxis in reinforced random walks,” SIAM Journal on Applied Mathematics, vol. 57, no. 4, pp. 1044–1081, 1997.

H. A. Levine and B. D. Sleeman, “A system of reaction diffusion equations arising in the theory of reinforced random walks,” SIAM Journal on Applied Mathematics, vol. 57, no. 3, pp. 683–730, 1997.

M. Zhang and C. Zhu, “Global existence of solutions to a hyperbolic-parabolic system,” Proceedings of the American Mathematical Society, vol. 135, no. 4, pp. 1017–1027, 2007.

J. Guo, J. Xiao, H. Zhao, and C. Zhu, “Global solutions to a hyperbolic-parabolic coupled system with large initial data,” Acta Mathematica Scientia. Series B, vol. 29, no. 3, pp. 629–641, 2009.

T. Li, R. H. Pan, and K. Zhao, “On a hybrid type chemotaxis model on bounded domains with large data,” SIAM Journal on Applied Mathematics, vol. 72, no. 1, pp. 417–443, 2012.

Y. H. Zhang, Z. Tan, B. S. Lai, and M. B. Sun, “Global analysis of smooth solutions to a generalized hyperbolic-parabolic system modeling chemotaxis,” Chinese Annals of Mathematics. Series A, vol. 33, no. 1, pp. 27–38, 2012.

Y. H. Zhang, Z. Tan, and M.-B. Sun, “Global existence and asymptotic behavior of smooth solutions to a coupled hyperbolic-parabolic system,” Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 465–482, 2013.

Y. H. Zhang, Z. Tan, and M. B. Sun, “Global smooth solutions to a coupled hyperbolic-parabolic system,” Chinese Annals of Mathematics. Series A, vol. 34, no. 1, pp. 29–46, 2013.

D. Li, T. Li, and K. Zhao, “On a hyperbolic-parabolic system modeling chemotaxis,” Mathematical Models and Methods in Applied Sciences, vol. 21, no. 8, pp. 1631–1650, 2011.

L. Corrias, B. Perthame, and H. Zaag, “A chemotaxis model motivated by angiogenesis,” Comptes Rendus de l'Académie des Sciences. Série I, vol. 336, no. 2, pp. 141–146, 2003.

L. Corrias, B. Perthame, and H. Zaag, “Global solutions of some chemotaxis and angiogenesis systems in high space dimensions,” Milan Journal of Mathematics, vol. 72, pp. 1–28, 2004.

S. Gueron and N. Liron, “A model of herd grazing as a travelling wave, chemotaxis and stability,” Journal of Mathematical Biology, vol. 27, no. 5, pp. 595–608, 1989.

D. Horstmann and A. Stevens, “A constructive approach to traveling waves in chemotaxis,” Journal of Nonlinear Science, vol. 14, no. 1, pp. 1–25, 2004.

E. F. Keller and L. A. Segel, “Traveling bands of chemotactic bacteria: a theoretical analysis,” Journal of Theoretical Biology, vol. 30, no. 2, pp. 235–248, 1971.

R. Lui and Z. A. Wang, “Traveling wave solutions from microscopic to macroscopic chemotaxis models,” Journal of Mathematical Biology, vol. 61, no. 5, pp. 739–761, 2010.

T. Nagai and T. Ikeda, “Traveling waves in a chemotactic model,” Journal of Mathematical Biology, vol. 30, no. 2, pp. 169–184, 1991.

R. Duan, A. Lorz, and P. Markowich, “Global solutions to the coupled chemotaxis-fluid equations,” Communications in Partial Differential Equations, vol. 35, no. 9, pp. 1635–1673, 2010.

D. Horstmann and M. Winkler, “Boundedness vs. blow-up in a chemotaxis system,” Journal of Differential Equations, vol. 215, no. 1, pp. 52–107, 2005.

Y. Yang, H. Chen, and W. Liu, “On existence of global solutions and blow-up to a system of the reaction-diffusion equations modelling chemotaxis,” SIAM Journal on Mathematical Analysis, vol. 33, no. 4, pp. 763–785, 2001.

R. Duan and H. Ma, “Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity,” Indiana University Mathematics Journal, vol. 57, no. 5, pp. 2299–2319, 2008.

Y. H. Zhang, H. Deng, and M. Sun, “Global analysis of smooth solutions to a hyperbolic-parabolic coupled system,” Frontiers of Mathematics in China, vol. 8, no. 6, pp. 1437–1460, 2013.

T. Li and Z.-A. Wang, “Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,” SIAM Journal on Applied Mathematics, vol. 70, no. 5, pp. 1522–1541, 2009.

T. Li and Z.-A. Wang, “Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis,” Journal of Differential Equations, vol. 250, no. 3, pp. 1310–1333, 2011.

H. Peng, L. Ruan, and C. Zhu, “Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis,” Kinetic and Related Models, vol. 5, no. 3, pp. 563–581, 2012.

Z. Chen, “Asymptotic stability of strong rarefaction waves for the compressible fluid models of Korteweg type,” Journal of Mathematical Analysis and Applications, vol. 394, no. 1, pp. 438–448, 2012.

Z. Ge and J. Yan, “Analysis of multiscale finite element method for the stationary Navier-Stokes equations,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 385–394, 2012.

Z. Tan, H. Wang, and J. Xu, “Global existence and optimal decay rate for the strong solutions to the compressible fluid models of Korteweg type,” Journal of Mathematical Analysis and Applications, vol. 390, no. 1, pp. 181–187, 2012.

Y. Li, “Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force,” Journal of Mathematical Analysis and Applications, vol. 388, no. 2, pp. 1218–1232, 2012.

Z. Tan and G. Wu, “Global existence for the non-isentropic compressible Navier-Stokes-Poisson system in three and higher dimensions,” Nonlinear Analysis: Real World Applications, vol. 13, no. 2, pp. 650–664, 2012.

Y. Li and X. Yang, “Global existence and asymptotic behavior of the solutions to the three-dimensional bipolar Euler-Poisson systems,” Journal of Differential Equations, vol. 252, no. 1, pp. 768–791, 2012.

L. Tebou, “Stabilization of some coupled hyperbolic/parabolic equations,” Discrete and Continuous Dynamical Systems. Series B, vol. 14, no. 4, pp. 1601–1620, 2010.

R. Duan, H. Liu, S. Ukai, and T. Yang, “Optimal - convergence rates for the compressible Navier-Stokes equations with potential force,” Journal of Differential Equations, vol. 238, no. 1, pp. 220–233, 2007.

R. Duan, S. Ukai, T. Yang, and H. Zhao, “Optimal convergence rates for the compressible Navier-Stokes equations with potential forces,” Mathematical Models and Methods in Applied Sciences, vol. 17, no. 5, pp. 737–758, 2007.

R. Duan, S. Ukai, and T. Yang, “A combination of energy method and spectral analysis for study of equations of gas motion,” Frontiers of Mathematics in China, vol. 4, no. 2, pp. 253–282, 2009.

R. Duan, S. Ukai, T. Yang, and H. Zhao, “Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications,” Communications in Mathematical Physics, vol. 277, no. 1, pp. 189–236, 2008.

S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics [Doctoral thesis], Kyoto University, 1983.

S. Kató, “On local and global existence theorems for a nonautonomous differential equation in a Banach space,” Funkcialaj Ekvacioj, vol. 19, no. 3, pp. 279–286, 1976.

T. Nishida, “Nonlinear hyperbolic equations and related topics in fluid dynamics,” Publicacions Matemàtiques, vol. 128, pp. 1053–1068, 1978.

D. Hoff and K. Zumbrun, “Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow,” Indiana University Mathematics Journal, vol. 44, no. 2, pp. 603–676, 1995.

D. Hoff and K. Zumbrun, “Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves,” Zeitschrift für Angewandte Mathematik und Physik, vol. 48, no. 4, pp. 597–614, 1997.