Nonlinear Degenerate Parabolic Equations with a Singular Nonlinearity

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 0.1 $\{\begin{array}{ll}\frac{\partial u}{\partial t}-\text{div}\left(\frac{(1+|\mathrm{\nabla }u{|}^{-\mathrm{\Lambda }})|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u}{{(1+|u|)}^{\theta }}\right)=\frac{f}{{(e^u-1)}^{\gamma }}& \text{in}{Q}_T,\\ u(x,0)=0& \text{on}\mathrm{\Omega },\\ u=0& \text{on}\partial Q_T,\end{array}$ \( \left \{ \textstyle\begin{array}{l@{\quad }l} \frac{\partial u}{\partial t}-\text{div} \left ( \frac{\left (1+\vert \nabla u\vert ^{-\Lambda }\right )\vert \nabla u\vert ^{p-2}\nabla u}{(1+\vert u\vert )^{\theta }} \right )=\frac{f}{(e^{u}-1)^{\gamma }} & \text{in}\;\;Q_{T}, \\ u(x,0)=0 & \text{on}\;\; \Omega , \\ u =0 & \text{on}\;\; \partial Q_{T}, \end{array}\displaystyle \right . \) where $\mathrm{\Omega }$ $\Omega $ is a bounded open subset of ${\mathbb{R}}^N$ $\mathbb{R}^{N}$ $N\ge 2$ $N\geq 2$ , $T>0$ $T>0$ , $\mathrm{\Lambda }\in [0,p-1)$ $\Lambda \in [0,p-1)$ , $f$ $f$ is a non-negative function belonging to $L^m(Q_T)$ $L^{m}(Q_{T})$ , $Q_T=\mathrm{\Omega }\times (0,T)$ $Q_{T}=\Omega \times (0,T)$ , $\partial Q_T=\partial \mathrm{\Omega }\times (0,T)$ $\partial Q_{T}=\partial \Omega \times (0,T)$ , $0\le \theta <p-1+\frac{p}{N}+\gamma (1+\frac{p}{N})$ $0\leq \theta < p-1+\frac{p}{N}+\gamma (1+\frac{p}{N})$ and $0\le \gamma <p-1$ $0\leq \gamma < p-1$ .