Nonlinear Degenerate Parabolic Equations with a Singular Nonlinearity

Abstract

In this paper, we study the existence and regularity results for some parabolic equations with degenerate coercivity, and a singular right-hand side. The model problem is

$$ \left \{ \textstyle\begin{array}{l@{\quad }l} \frac{\partial u}{\partial t}-\text{div} \left ( \frac{\left (1+\vert \nabla u\vert ^{-\Lambda }\right )\vert \nabla u\vert ^{p-2}\nabla u}{(1+\vert u\vert )^{\theta }} \right )=\frac{f}{(e^{u}-1)^{\gamma }} & \text{in}\;\;Q_{T}, \\ u(x,0)=0 & \text{on}\;\; \Omega , \\ u =0 & \text{on}\;\; \partial Q_{T}, \end{array}\displaystyle \right . $$

(0.1)

where $\Omega$ is a bounded open subset of $\mathbb{R}^{N}$ $N\geq 2$, $T>0$, $\Lambda \in [0,p-1)$, $f$ is a non-negative function belonging to $L^{m}(Q_{T})$, $Q_{T}=\Omega \times (0,T)$, $\partial Q_{T}=\partial \Omega \times (0,T)$, $0\leq \theta < p-1+\frac{p}{N}+\gamma (1+\frac{p}{N})$ and $0\leq \gamma < p-1$.