Boundedness in a quasilinear parabolic-parabolic       Keller-Segel system with the sensitivity $v^{-1}S(u)$

Kentarou  Fujie; Chihiro  Nishiyama; Tomomi Yokota

doi:10.3934/proc.2015.0464

Article Contents

2015, Volume 2015: 464-472. Doi: 10.3934/proc.2015.0464 open access

This issue Previous Article Well-posedness for a class of nonlinear degenerate parabolic equations Next Article Remark on a semirelativistic equation in the energy space

Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with the sensitivity $v^{-1}S(u)$

1.
Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601

Received: June 30, 2014

Revised: October 31, 2014

Published: October 2015

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Abstract

This paper is concerned with global existence and boundedness of classical solutions to the quasilinear fully parabolic Keller-Segel system $u_t = \nabla \cdot(D(u)\nabla u) -\nabla \cdot (v^{-1}S(u)\nabla v)$, $v_t= \Delta v-v+u$. In [7,4], global existence and boundedness were established in the system without $v^{-1}$. In this paper the signal-dependent sensitivity $v^{-1}$ is taken into account via the Weber-Fechner law. A uniform-in-time estimate for $v$ obtained in [2] defeats the singularity of $v^{-1}$.

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Mathematics Subject Classification: Primary: 35B35, 35B45; Secondary: 92C17.

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References

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