Universal bounds for quasilinear parabolic equations with convection

We prove a universal bound, independent of the initial data, for all global nonnegative solutions of the Dirichlet problem of the quasilinear parabolic equation with convection $u_t = \Delta u^m +a\cdot \nabla u^q+ u^p$ in $\Omega\times (0,\infty)$, where $\Omega$ is a smoothly bounded domain in $\mathbf{R^N}$, $a \in \mathbf {R^N}$, $ 1 \le m < p$ <$m+2/(N+1)$ and $(m+1)/2 \le q < (m+p)/2$ (or $q = (m+p)/2$ and $|a|$ is small enough). The universal bound can be obtained by showing that any solution $u$ in $\Omega\times(0,T)$ satisfies the estimate $ \||u(t)\||_{L^{\infty}(\Omega)} \le C(p,m,q,|a|, \Omega,\alpha,T)t^{-\alpha}$ in $ 0 $<$t \le T/2 $ for $\alpha $>$ (N+1 )/[(m-1)(N+1)+2]$, which describes the initial blow-up rates of solutions.