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

Global Existence and Asymptotic Behavior of Solutions to the Hyperbolic Keller-Segel Equation... 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). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

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

Acta Applicandae Mathematicae , Volume 158 (1) – Apr 10, 2018

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References (26)

Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media B.V., part of Springer Nature
Subject
Mathematics; Computational Mathematics and Numerical Analysis; Applications of Mathematics; Partial Differential Equations; Probability Theory and Stochastic Processes; Calculus of Variations and Optimal Control; Optimization
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-018-0180-3
Publisher site
See Article on Publisher Site

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).

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Apr 10, 2018

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