Existence, uniqueness and asymptotic behavior of solutions for a singular parabolic equation

Abstract

In this paper, we are concerned with a singular parabolic equation ∂ v ∂ t − Δ v = f ( x , t ) − μ | ∇ v | 2 v in a smooth bounded domain Ω ⊂ R N subject to zero Dirichlet boundary condition and initial condition φ ⩾ 0 . Under the assumptions on μ , φ and f ( x , t ) , some existence and uniqueness results are obtained by applying parabolic regularization method and the sub-supersolutions method. We also discuss the asymptotic behaviors of solutions in the sense of L 2 ( 0 , T ; W 0 1 , 2 ( Ω ) ) and L ∞ ( 0 , T ; L 2 ( Ω ) ) norms as μ → 0 or μ → ∞ . As a byproduct we obtain the existence of solutions for some problems which blow up on the boundary.