Boundedness in a fully parabolic chemotaxis system with signal-dependent sensitivity and logistic term

Abstract

This paper deals with the chemotaxis system with signal-dependent sensitivity and logistic term \begin{align*}    &u_t=\Delta u - \nabla \cdot (u \chi(v)\nabla v)     + \mu u(1-u), %  \quad   \\    &v_t=\Delta v + u - v \end{align*}in $\Omega\times (0,\infty)$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$ ($n\ge 2$) with smooth boundary, $\mu > 0$ is a constant and $\chi$ is a function generalizing \begin{align*}  \chi(s) = \frac{K}{(1+s)^2} \quad (K>0,\ s>0). \end{align*}In the case that $\mu=0$ global existence and boundedness were established under some conditions (\cite{Mizukami-Yokota_02}); however, conditions for global existence and boundedness in the above system have not been studied. The purpose of this paper is to construct conditions for global existence and boundedness in the above system.