Existence and asymptotic behavior for a singular parabolic equation

Abstract

We prove global existence of nonnegative solutions to the singular parabolic equation u t − Δ u + χ { u > 0 } ( − u − β + λ f ( u ) ) = 0 u_t -\Delta u + \chi _{ \{ u>0 \} } ( -u^{-\beta } + \lambda f(u) )=0 in a smooth bounded domain Ω ⊂ R N \Omega \subset \mathbb {R}^N with zero Dirichlet boundary condition and initial condition u 0 ∈ C ( Ω ) u_0 \in C(\Omega ) , u 0 ≥ 0 u_0 \geq 0 . In some cases we are also able to treat u 0 ∈ L ∞ ( Ω ) 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.