On the logarithmic Keller-Segel-Fisher/KPP system

Yanni Zeng; Kun Zhao

doi:10.3934/dcds.2019220

Article Contents

2019, Volume 39, Issue 9: 5365-5402. Doi: 10.3934/dcds.2019220

This issue Previous Article Converse theorem on a global contraction metric for a periodic orbit Next Article Stochastic homogenization for a diffusion-reaction model

On the logarithmic Keller-Segel-Fisher/KPP system

Yanni Zeng^1, and
Kun Zhao^2, ,

1.
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA
2.
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA

^* Corresponding author: Kun Zhao
^* Corresponding author: Kun Zhao

Received: September 30, 2018

Revised: January 31, 2019

Published: May 2019

Y. Zeng was partially supported by the Simons Foundation grant 244905. K. Zhao was partially supported by the Simons Foundation grant 413028

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Abstract

We consider the Cauchy problem of a Keller-Segel type chemotaxis model with logarithmic sensitivity and logistic growth. We study the global well-posedness, long-time behavior, vanishing coefficient limit and decay rate of solutions in $ \mathbb{R} $. By utilizing energy methods, we show that for any given classical initial datum which is a perturbation around a constant equilibrium state with finite energy (not small), there exists a unique global-in-time solution to the Cauchy problem, and the solution converges to the constant equilibrium state, as time goes to infinity. Under the same initial condition, it is shown that the solution with positive chemical diffusion coefficient converges to the solution with zero chemical diffusion coefficient, as the coefficient goes to zero. Furthermore, for a slightly smaller class of initial data, we identify the algebraic decay rates of the solution to the constant equilibrium state by employing time-weighted energy estimates.

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Mathematics Subject Classification: Primary: 35Q92; Secondary: 35K45, 35K57.

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