Global existence and blow-up of solutions for parabolic equations involving the Laplacian under nonlinear boundary conditions

Abstract

This paper is concerned with the existence and blow-up of solutions to the following linear parabolic equation: ut−Δu+u = 0 in Ω×(0, T) , under nonlinear boundary condition in a bounded domain Ω ⊂ Rn , n ≥ 1, with smooth boundary. We obtain a threshold result for the global existence of solutions, next we shall prove the existence time T of solution is finite when the initial energy satisfies certain condition.