| Contents |
| In this talk, I will present recent joint contributions with P. Takac and B. Bougherara about the following quasilinear and singular parabolic equation:
$$ \left\{
\begin{array}{ll}
\partial_t u - \Delta_p u = u^{-\delta} + f(x, u,\nabla u) \ \text { in } (0,T)\times \Omega =Q_T, \\
u =0 \ \text { on } (0,T ) \times \partial \Omega, \ u > 0 \text { in } Q_T, \\
u(0,x) =u_0\geq 0 \text { in } \Omega,
\end{array}
\right.$$
where $\Omega$ stands for a regular bounded domain of $\mathbb{R}^N$, $2\leq p$, $T > 0$ and $u_{0} \in L^{r}(\Omega)$ with $r\geq 2$ large enough. The nonlinear term $f : \Omega \times \mathbb{R} \times \mathbb{R}^{N} \to \mathbb{R}$ is
a Caratheodory function satisfying addtional growth conditions. I will present existence
results to (P). In the case $\delta < 2+ \frac{1}{p-1}$, we give further results : uniqueness of the solution and regularity results. Proofs used some estimates based on logarithmic Sobolev inequalities to get ultracontractivity of the associated semi-group. |
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