# American Institute of Mathematical Sciences

2015, 2015(special): 464-472. doi: 10.3934/proc.2015.0464

## 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, Japan, Japan

Received  July 2014 Revised  November 2014 Published  November 2015

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 $v^{-1}$.
Citation: Kentarou Fujie, Chihiro Nishiyama, Tomomi Yokota. Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with the sensitivity $v^{-1}S(u)$. Conference Publications, 2015, 2015 (special) : 464-472. doi: 10.3934/proc.2015.0464
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