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

$$\left\{\begin{array}{lll}u_t = \nabla \cdot \left(D(u) \nabla u\right) - \nabla \cdot \left(S(u) \nabla v\right) + f(u), &\quad x \in \Omega, \, t > 0,\\ v_t = \Delta v - v + u, &\quad x \in \Omega, \, t > 0\end{array}\right.$$

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

$$\alpha + 2\beta < \left\{\begin{array}{lll}\gamma - 1 + \frac{2}{n}, &\quad {\rm for} \, 1 \leq \gamma < 2,\\ \gamma - 1 + \frac{4}{n + 2}, &\quad {\rm for} \, \gamma \geq 2,\end{array}\right.$$

then for sufficiently smooth initial data, the problem possesses a unique global classical solution which is uniformly bounded.