Bounded solutions in anisotropic degenerate parabolic problems with a singular term

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 \(\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial u}{\partial t}-\sum _{i=1}^{N}D_{i} \left( \frac{u^{p_{i}-1}(1+D u)^{-1}D u+\vert D u\vert ^{p_{i}-2}D u}{(1+\vert u\vert )^{\theta }}\right) =\frac{f}{u^{\gamma }} & \hbox {in}\;\;Q, \\ u(x,0)=0 & \hbox {on}\;\; \Omega ,\\ u =0 & \hbox {on}\;\; \Gamma , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\frac{\partial u}{\partial t}-{\sum }_{i=1}^N{D}_i\left(\frac{u^{p_i-1}{(1+Du)}^{-1}Du+{|Du|}^{p_i-2}Du}{{(1+|u|)}^{\theta }}\right)=\frac{f}{u^{\gamma }}\hfill & \text{in}Q,\hfill \\ u(x,0)=0\hfill & \text{on}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on}\mathrm{\Gamma },\hfill \end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a bounded open subset of \(\mathbb {R}^{N}\) ${\mathbb{R}}^N$ \(N\ge 2\) $N\ge 2$ , \(T>0\) $T>0$ , \(2\le p_{i}<N\) $2\le p_i<N$ for every \(i=1,\ldots ,N\) $i=1,\dots ,N$ , \(\theta ,\gamma \ge 0\) $\theta ,\gamma \ge 0$ , \(0\le f\in L^{m}(Q)\) $0\le f\in L^m(Q)$ with \(m>\frac{N}{\overline{p}}+1\) $m>\frac{N}{\overline{p}}+1$ ( \(\overline{p}\) $\overline{p}$ defined in (2.1)) and \(Q=\Omega \times (0,T)\) $Q=\mathrm{\Omega }\times (0,T)$ . 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.