Global Existence and Asymptotic Behavior of Solutions to the Hyperbolic Keller-Segel Equation with a Logistic Source
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).
KeywordsPartially dissipative hyperbolic system Global existence Keller-Segel equation
Mathematics Subject Classification35Q92 35L60
M. Chae was supported by NRF-2015R1C1A2A01054919.
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