On the existence and regularity of solutions to singular parabolic p-Laplacian equations with absorption term

Abstract

In this paper, we study the existence and regularity results for nonlinear and singular parabolic problem with absorption term whose model is the following

$$\begin{aligned} \left\{ \begin{array}{lll} u_{t}-\text{ div }((a(x,t)+|u|^{q})|\nabla u|^{p-2}\nabla u)=\frac{f(x,t)}{u^{\gamma }} &{} \text{ in } &{} \varOmega \times (0,T),\\ u(x,t)=0 &{} \text{ on } &{} \partial \varOmega \times (0,T),\\ u(x,0)=u_{0}(x) &{} \text{ in } &{} \varOmega , \end{array} \right. \end{aligned}$$

with $\gamma>0, q>0, p>2$, and $\varOmega$ is a bounded open subset of ${\mathbb {R}}^{N}, (N\ge 3), 0<T<+\infty$, a(x, t) is a measurable bounded function, f is a nonnegative function belonging to$L^{m_{1}}(0,T; L^{m_{2}}(\varOmega ))$ with  $m_{1}\ge 1,\, m_{2}\ge 1$ and $u_{0}\in L^{\infty }(\varOmega )$ such that

$$\begin{aligned} \forall \omega \subset \subset \varOmega ,\, \exists \; d_{\omega }>0:\, u_{0}\ge d_{\omega }\;\;\; \text{ in }\,\; \omega . \end{aligned}$$