Existence of positive solutions to nonlinear singular parabolic equations with Hardy potential

Abstract

In this paper, we study existence and regularity results for the solution to a nonlinear singular parabolic problems involving Hardy potential

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \frac{\partial u}{\partial t}-{\text {div}}a(x,t,\nabla u)-\mu \frac{u^{p-1}}{|x|^{p}}=\frac{f(x,t)}{u^{\gamma }} &{}\text{ in }&{}\,\, \Omega \times (0,T),\\ u=0 &{}\text{ on } &{}\,\, \partial \Omega \times (0,T),\\ u(x,0)=u_{0}(x) &{}\text{ in } &{}\,\, \Omega , \end{array} \right. \end{aligned}$$

where $\Omega$ is a bounded open subset on ${\mathbb {R}}^{N}, N\ge 3, 0\in \Omega$ and $\gamma >0,\; 2\le p<N,$ $0<T<+\infty ,\; \mu >0,$ $0\le f\in L^{m}(Q),\; m\ge 1$ and $u_{0}\in L^{\infty }(\Omega )$ satisfies

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