Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model

Zhi-An Wang; Kun Zhao

doi:10.3934/cpaa.2013.12.3027

Article Contents

2013, Volume 12, Issue 6: 3027-3046. Doi: 10.3934/cpaa.2013.12.3027

This issue Previous Article On symmetry results for elliptic equations with convex nonlinearities Next Article Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results

Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model

Zhi-An Wang^1, and
Kun Zhao^2,

1.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
2.
Department of Mathematics, Tulane University, New Orleans, LA 70118

Received: March 31, 2012

Revised: October 31, 2013

Published: October 2013

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Abstract

In the first part of this paper, we investigate the qualitative behavior of classical solutions for a one-dimensional parabolic system derived from a repulsive chemotaxis model on bounded domains. It is shown that classical solutions to the initial-boundary value problem exist globally in time for large data and converge to constant equilibrium states exponentially in time. The results indicate that repulsive chemotaxis exhibits a strong tendency against pattern formation. In the second part, we study diffusion limit and convergence rate of the model toward a non-diffusive problem studied in [11]. It is shown that when the chemical diffusion coefficient $\varepsilon$ tends to zero, the solution is convergent in $L^{\infty}$-norm with respect to $\varepsilon$ at order $O(\varepsilon)$.

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Mathematics Subject Classification: Primary: 35K55, 35K57, 35K45, 35K50; Secondary: 92C15, 92C17, 92B99.

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