A Singular Parabolic Equation: Existence, Stabilization

We investigate the following quasilinear parabolic and singular equation, {equation} \tag{{\rm P$_t$}} \{{aligned} & u_t-\Delta_p u =\frac{1}{u^\delta}+f(x,u)\;\text{in}\,(0,T)\times\Omega, & u =0\,\text{on} \;(0,T)\times\partial\Omega,\quad u>0 \text{in}\, (0,T)\times\Omega, u\text{in}\Omega, {aligned}. {equation} % where $\Omega$ is an open bounded domain with smooth boundary in $\R^{\rm N}$, $1 0$. We assume that $(x,s)\in\Omega\times\R^+\to f(x,s)$ is a bounded below Caratheodory function, locally Lipschitz with respect to $s$ uniformly in $x\in\Omega$ and asymptotically sub-homogeneous, i.e. % {equation} \label{sublineargrowth} 0 \leq\displaystyle\lim_{t\to +\infty}\frac{f(x,t)}{t^{p-1}}=\alpha_f 0$ in $C([0,T], L^2(\Omega))\cap L^\infty(Q_T)$ and under suitable assumptions on the initial data we give additional regularity results. Finally, we describe their asymptotic behaviour in $L^\infty(\Omega)\cap H^1_0(\Omega)$ when $\delta<3$.