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

AbstractIn this paper, we are concerned with a singular parabolic equation ∂v∂t−Δv=f(x,t)−μ|∇v|2v in a smooth bounded domain Ω⊂RN 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 L2(0,T;W01,2(Ω)) and L∞(0,T;L2(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