Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity

We consider the quasilinear parabolic-parabolic Keller-Segel system $$u_t=\mathrm{\nabla }\cdot (D(u)\mathrm{\nabla }u)-\mathrm{\nabla }\cdot (S(u)\mathrm{\nabla }v),x\in \mathrm{\Omega },t>0,v_t=\mathrm{\Delta }v-v+u,x\in \mathrm{\Omega },t>0,$$ under homogeneous Neumann boundary conditions in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with $n\ge 2$. It is proved that if $\frac{S(u)}{D(u)}\le c{u}^{\alpha }$ with $\alpha <\frac{2}{n}$ and some constant $c>0$ for all $u>1$ and some further technical conditions are fulfilled, then the classical solutions to the above system are uniformly-in-time bounded. This boundedness result is optimal according to a recent result by the second author ({\em Math. Meth. Appl. Sci.} {\bf 33} (2010), 12-24), which says that if $\frac{S(u)}{D(u)}\ge c{u}^{\alpha }$ for $u>1$ with $c>0$ and some $\alpha >\frac{2}{n}$, then for each mass $M>0$ there exist blow-up solutions with mass $\text{io}{u}_0=M$. In addition, this paper also proves a general boundedness result for quasilinear non-uniformly parabolic equations by modifying the iterative technique of Moser-Alikakos (Alikakos, {\em Comm. PDE} {\bf 4} (1979), 827-868).