Boundedness in a quasilinear parabolic-parabolic Keller-Segel system   with subcritical sensitivity
  We consider the quasilinear parabolic-parabolic Keller-Segel system $$   u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v),   \qquad x\in\Omega, \ t>0,   v_t=\Delta v -v + u,   x\in\Omega, \ t>0, $$ under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset\R^n$ with $n\ge 2$. It is proved that if $\frac{S(u)}{D(u)}\le cu^{\alpha}$ 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 $\frac{S(u)}{D(u)} \ge cu^\alpha$ 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 $\io u_0=M$. 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). 
Youshan Tao and Michael Winkler
math.AP