Bounded Solutions for Anisotropic Degenerate Parabolic Problems with Singular Term

Abstract Note: Please see pdf for full abstract with equations. In this paper, we study the existence result of bounded solutions to a nonlinear anisotropic parabolic equations with degenerate coercivity, and the singular term in the right hand side. The model problem is \begin{equation*} \left\{\begin{array}{ll} \frac{\partial u}{\partial t}-\sum_{i=1}^{N}D_{i} \left(\frac{u^{p_{i}-1}(1+D u)^{-1}D u+\vert D u\vert^{p_{i}-2}D u}{(1+\vert u\vert)^{\theta}}\right)=\frac{f}{u^{\gamma}} & \hbox{in}\;\;Q, \\ u(x,0)=0 & \hbox{on}\;\; \Omega,\\ u =0 & \hbox{on}\;\; \Gamma, \end{array} \right. \end{equation*} where $\Omega$ is a bounded open subset of $\mathbb{R}^{N}$ $N\geq2$, $T>0$, $2\leq p_{i} \frac{N}{\overline{p}}+1$ ($\overline{p}$ defined in \eqref{0001}) and $Q=\Omega\times(0,T)$. The main idea in the proof is based on a Stampacchia's lemma with a good choice of test function that enables us to obtain a priori estimates. Mathematics Subject Classification (2010). 35K65; 35K55; 35B45; 35B65.