Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source (1504.01293v1)

Abstract: This paper deals with the Neumann boundary value problem for the system $$u_t=\nabla\cdot\left(D(u)\nabla u\right)-\nabla\cdot\left(S(u)\nabla v\right)+f(u) ,\quad x\in\Omega,\ t>0$$ $$v_t=\Delta v-v+u,\quad x\in\Omega,\ t>0$$ in a smooth bounded domain $\Omega\subset\mathbb{R}^n$ $(n\geq1)$, where the functions $D(u)$ and $S(u)$ are supposed to be smooth satisfying $D(u)\geq Mu^{-\alpha}$ and $S(u)\leq Mu^{\beta}$ with $M>0$, $\alpha\in\mathbb{R}$ and $\beta\in\mathbb{R}$ for all $u\geq1$, and the logistic source $f(u)$ is smooth fulfilling $f(0)\geq0$ as well as $f(u)\leq a-\mu u^{\gamma}$ with $a\geq0$, $\mu>0$ and $\gamma\geq1$ for all $u\geq0$. It is shown that if $\alpha+2\beta<\gamma-1+\frac{2}{n}$, for $1\leq\gamma<2$ and $\alpha+2\beta<\gamma-1+\frac{4}{n+2}$, for $\gamma\geq2$, then for sufficiently smooth initial data the problem possesses a unique global classical solution which is uniformly bounded.