A Singular Parabolic Equation: Existence, Stabilization

Abstract

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(0,x) =u_0(x)\;\text{in}\Omega, {aligned}. {equation} % where $\Omega$ is an open bounded domain with smooth boundary in $\R^{\rm N}$, $1 < p< \infty$, $0<\delta$ and $T>0$. 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<\lambda_1(\Omega), {equation} % (where $\lambda_1(\Omega)$ is the first eigenvalue of $-\Delta_p$ in $\Omega$ with homogeneous Dirichlet boundary conditions) and $u_0\in L^\infty(\Omega)\cap W^{1,p}_0(\Omega)$, satisfying a cone condition defined below. Then, for any $\delta\in (0,2+\frac{1}{p-1})$, we prove the existence and the uniqueness of a weak solution $u \in{\bf V}(Q_T)$ to $({\rm P_t})$. Furthermore, $u\in C([0,T], W^{1,p}_0(\Omega))$ and the restriction $\delta<2+\frac{1}{p-1}$ is sharp. Finally, in the last section we analyse the case $p=2$. Using the interpolation spaces theory and the semigroup theory, we prove the existence and the uniqueness of weak solutions to $({\rm P}_t)$ for any $\delta>0$ 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 $\delta<3$.