Global analysis of smooth solutions to a hyperbolic-parabolic coupled system

doi:10.1007/s11464-013-0331-9

Front Math Chin

2013, Vol. 8

Issue (6) : 1437-1460 https://doi.org/10.1007/s11464-013-0331-9

RESEARCH ARTICLE

Global analysis of smooth solutions to a hyperbolic-parabolic coupled system

Yinghui ZHANG^1,2(

), Haiying DENG³, Mingbao SUN¹

1. Department of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China; 2. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China; 3. Department of Mathematics, Hunan First Normal College, Changsha 410205, China

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Abstract

We investigate a model arising from biology, which is a hyperbolicparabolic coupled system. First, we prove the global existence and asymptotic behavior of smooth solutions to the Cauchy problem without any smallness assumption on the initial data. Second, if the H^s∩L¹-norm of initial data is sufficiently small, we also establish decay rates of the global smooth solutions. In particular, the optimal L² decay rate of the solution and the almost optimal L² decay rate of the first-order derivatives of the solution are obtained. These results are obtained by constructing a new nonnegative convex entropy and combining spectral analysis with energy methods.

Keywords Global analysis hyperbolic-parabolic system decay rate convex entropy

Corresponding Author(s): ZHANG Yinghui,Email:zhangyinghui0910@126.com

Issue Date: 01 December 2013

Cite this article:

Yinghui ZHANG,Haiying DENG,Mingbao SUN. Global analysis of smooth solutions to a hyperbolic-parabolic coupled system[J]. Front Math Chin, 2013, 8(6): 1437-1460.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-013-0331-9
https://academic.hep.com.cn/fmc/EN/Y2013/V8/I6/1437

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