Quasi-linear diffusion equations with gradient terms and \(L^{1}\) data. (Q1428635)

The goal of this paper is to prove the existence of a generalized solution of the following quasi-linear parabolic problem \[ \begin{cases} u_t-\Delta_x u+ |u|^{\beta-2} u|\nabla u|^q= |u|^{\alpha- 2}&\text{in }\Omega\times (0,T),\\ u(x,t)= 0 &\text{on }\partial\Omega\times (0,T),\\ u(x,0)= u_0(x) &\text{in }\Omega,\end{cases} \] where \(\Omega\) is a bounded open set in \(\mathbb{R}^d\), \(1\leq q\leq 2\), \(0\leq p< q\) and \(\alpha,\beta> 1\). The existence result relies on a stability theorem with respect to the initial datum.