A parabolic problem involving p(x)-Laplacian, a power and a singular nonlinearity

The purpose of this paper is to study nonlinear singular parabolic equations with p(x)-Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution: \(\begin{aligned} \frac{\partial u}{\partial t}-\Delta _{p(x)}u&=\lambda u^{q(x)-1} + u^{-\delta (x)}g+ f & \text {in}~Q_T,\\ u&= 0 & \text {on}~\Sigma _T,\\ u(0,\cdot )&=u_0(\cdot ) & \text {in}~\Omega \nonumber . \end{aligned}\) $$\begin{array}{ccc}\hfill \frac{\partial u}{\partial t}-{\mathrm{\Delta }}_{p(x)}u& =\lambda u^{q(x)-1}+u^{-\delta (x)}g+f\hfill & \hfill \text{in}{Q}_T,\\ \hfill u& =0\hfill & \hfill \text{on}{\mathrm{\Sigma }}_T,\\ \hfill u(0,·)& =u_0(·)\hfill & \hfill \text{in}\mathrm{\Omega }.\end{array}$$ Here \(Q_T=\Omega \times (0,T)\) $Q_T=\mathrm{\Omega }\times (0,T)$ , \(\Sigma _T=\partial \Omega \times (0,T)\) ${\mathrm{\Sigma }}_T=\partial \mathrm{\Omega }\times (0,T)$ , \(\Omega \) $\mathrm{\Omega }$ is a bounded domain in \(\mathbb {R}^N\) ${\mathbb{R}}^N$ ( \(N\ge 2\) $N\ge 2$ ) with Lipschitz continuous boundary \(\partial \Omega \) $\partial \mathrm{\Omega }$ , \(\lambda \in (0,\infty )\) $\lambda \in (0,\infty )$ , \(f\in L^1(Q_T)\) $f\in L^1(Q_T)$ , \(g\in L^\infty (\Omega )\) $g\in L^{\infty }(\mathrm{\Omega })$ , \(u_0\in L^r(\Omega )\) $u_0\in L^r(\mathrm{\Omega })$ with \(r\ge 2\) $r\ge 2$ , \(\delta :\overline{\Omega }\rightarrow (0,\infty )\) $\delta :\overline{\mathrm{\Omega }}\to (0,\infty )$ is continuous, and \(p,q\in C(\overline{\Omega })\) $p,q\in C(\overline{\mathrm{\Omega }})$ with \(\underset{x\in \overline{\Omega }}{\max }~p(x)<N\) $\underset{x\in \overline{\mathrm{\Omega }}}{max}p(x)<N$ , \(q(\cdot )<p^*(\cdot )\) $q(·)<p^{\ast }(·)$ .

The article is distinguished into two cases according to the choice of f with different range of parameters \(p(\cdot )\) $p(·)$ , \(q(\cdot )\) $q(·)$ .