MATHEMATICA BOHEMICA, Vol. 139, No. 4, pp. 639-647, 2014

# Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity

## Kentarou Fujie, Tomomi Yokota

Kentarou Fujie, Tomomi Yokota, Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan, e-mail: kentarou.fujie@gmail.com, yokota@rs.kagu.tus.ac.jp

Abstract: This paper deals with parabolic-elliptic chemotaxis systems with the sensitivity function $\chi(v)$ and the growth term $f(u)$ under homogeneous Neumann boundary conditions in a smooth bounded domain. Here it is assumed that $0< \chi(v)\leq{{\chi}_0}/{v^k}$ $(k\geq1$, ${\chi}_0>0)$ and $\lambda_1-\mu_1 u \leq f(u)\leq\lambda_2-\mu_2 u$ $(\lambda_1,\lambda_2,\mu_1,\mu_2>0)$. It is shown that if $\chi_0$ is sufficiently small, then the system has a unique global-in-time classical solution that is uniformly bounded. This boundedness result is a generalization of a recent result by K. Fujie, M. Winkler, T. Yokota.

Keywords: chemotaxis; global existence; boundedness

Classification (MSC 2010): 35B40, 35K60

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