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{align*} \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{align*} Here $Q_T=\Omega\times(0,T)$, $\Sigma_T=\partial\Omega\times(0,T)$, $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz continuous boundary $\partial\Omega$, $\lambda\in(0,\infty)$, $f\in L^1(Q_T)$, $g\in L^\infty(\Omega)$, $u_0\in L^r(\Omega)$ with $r\geq 2$, $\delta:\overline{\Omega}\rightarrow(0,\infty)$ is continuous, and $p,q\in C(\overline{\Omega})$ with $\underset{x\in\overline{\Omega}}{\max}~p(x)<N$, $q(\cdot)<p^*(\cdot)$. The article is distinguished into two cases according to the choice of $f$ with different range of parameters $p(\cdot)$, $q(\cdot)$.