Bounded solutions in anisotropic degenerate parabolic problems with a singular term

Abstract

In this paper, we study the existence of bounded solutions for a nonlinear anisotropic parabolic equation with degenerate coercivity and a singular term on the right-hand side. The model problem considered is as follows

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial u}{\partial t}-\sum _{i=1}^{N}D_{i} \left( \frac{u^{p_{i}-1}(1+D u)^{-1}D u+\vert D u\vert ^{p_{i}-2}D u}{(1+\vert u\vert )^{\theta }}\right) =\frac{f}{u^{\gamma }} & \hbox {in}\;\;Q, \\ u(x,0)=0 & \hbox {on}\;\; \Omega ,\\ u =0 & \hbox {on}\;\; \Gamma , \end{array} \right. \end{aligned}$$

where \(\Omega \) is a bounded open subset of \(\mathbb {R}^{N}\) \(N\ge 2\), \(T>0\), \(2\le p_{i}<N\) for every \(i=1,\ldots ,N\), \(\theta ,\gamma \ge 0\), \(0\le f\in L^{m}(Q)\) with \(m>\frac{N}{\overline{p}}+1\) (\(\overline{p}\) defined in (2.1)) and \(Q=\Omega \times (0,T)\). The main idea in the proof is based on Stampacchia’s lemma, which allows us to obtain a priori estimates by making a suitable choice of a test function.