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

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

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