Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source

Abstract

This paper deals with the Neumann boundary value problem for the system $$u_t=\nabla \cdot \left(D(u)\nabla u\right)-\nabla \cdot \left(S(u)\nabla v\right)+f(u),x\in \Omega ,t>0$$ $$v_t=\Delta v-v+u,x\in \Omega ,t>0$$ in a smooth bounded domain $\Omega \subset {\mathbb{R}}^n$ $(n\ge 1)$, where the functions $D(u)$ and $S(u)$ are supposed to be smooth satisfying $D(u)\ge M{u}^{-\alpha }$ and $S(u)\le M{u}^{\beta }$ with $M>0$, $\alpha \in \mathbb{R}$ and $\beta \in \mathbb{R}$ for all $u\ge 1$, and the logistic source $f(u)$ is smooth fulfilling $f(0)\ge 0$ as well as $f(u)\le a-\mu u^{\gamma }$ with $a\ge 0$, $\mu >0$ and $\gamma \ge 1$ for all $u\ge 0$. It is shown that if $\alpha +2\beta <\gamma -1+\frac{2}{n}$, for $1\le \gamma <2$ and $\alpha +2\beta <\gamma -1+\frac{4}{n+2}$, for $\gamma \ge 2$, then for sufficiently smooth initial data the problem possesses a unique global classical solution which is uniformly bounded.

10 pages

References in corpus (1)

• New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models