Existence and asymptotic behavior for a singular parabolic equation

Abstract:

We prove global existence of nonnegative solutions to the singular parabolic equation $u_t -\Delta u + \chi _{ \{ u>0 \} } ( -u^{-\beta } + \lambda f(u) )=0$ in a smooth bounded domain $\Omega \subset \mathbb {R}^N$ with zero Dirichlet boundary condition and initial condition $u_0 \in C(\Omega )$, $u_0 \geq 0$. In some cases we are also able to treat $u_0 \in L^\infty (\Omega )$. Then we show that if the stationary problem admits no solution which is positive a.e., then the solutions of the parabolic problem must vanish in finite time, a phenomenon called “quenching”. We also establish a converse of this fact and study the solutions with a positive initial condition that leads to uniqueness on an appropriate class of functions.