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Global Existence and Asymptotic Behavior of Solutions to the Hyperbolic Keller-Segel Equation with a Logistic Source

  • Myeongju Chae
Article

Abstract

In this paper we consider a hyperbolic Keller-Segel system with a logistic source in two dimension. We show the system has a global smooth solution upon small perturbation around a constant equilibrium and the solution satisfies a dissipative energy inequality. To do this we find a convex entropy functional and a compensating matrix, which transforms the partially dissipative system into a uniformly dissipative one. Those two ingredients were crucial for the study of a partially dissipative hyperbolic system (Hanouzet and Natalini in Arch. Ration. Mech. Anal. 169(2):89–117, 2003; Kawashima in Ph.D. Thesis, Kyoto University, 1983; Yong in Arch. Ration. Mech. Anal. 172(2):247–266, 2004).

Keywords

Partially dissipative hyperbolic system Global existence Keller-Segel equation 

Mathematics Subject Classification

35Q92 35L60 

Notes

Acknowledgements

M. Chae was supported by NRF-2015R1C1A2A01054919.

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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsHankyong National UniversityAnseong-siRepublic of Korea

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