On some class of parabolic equations with singular term

We give the existence of solutions of some nonlinear parabolic equations with a singular term of the form $$\begin{cases} u_t-\Delta_{p(x)} u=\frac{1}{u^\delta}\quad &\text{ in }(0,T)\times\Omega\\ u(0,x)=u_0(x)&\text{ in }\Omega\\ u(t,x)=0 &\text{ on } (0,T)\times\partial\Omega \end{cases}$$ where $\Omega$ is an open bounded domain with smooth boundary in $\mathbb{R}^N$ (with $N \geqslant 2$), $u_0$ is a nonnegative function in $L^2(\Omega)$, $\delta > 0$, $T > 0$, $p \in C(\bar{\Omega})$, and \( 2 \leqslant p(\cdot) \leqslant N\), The operator \[-\Delta_{p(x)} u = -\text{div}(|\nabla u|^{p(x)-2} \nabla u).\] is called p(x)-Laplacian, which becomes the \(p\)-Laplacian when \(p(x) \equiv p\) (a constant).\\ This case was investigated by Ida de Bonis and Maria Michaela Porzio in their study on singular parabolic problems.