Existence and nonexistence of global positive solutions for degenerate parabolic equations in exterior domains

Abstract

This article deals with the degenerate parabolic equations in exterior domains and with inhomogeneous Dirichlet boundary conditions. We obtain that p c = (σ + m) n/( n-σ-2) is its critical exponent provided max{-1,[(1- m) n–2]/( n+1)} < σ < n – 2. This critical exponent is not the same as that for the corresponding equations with the boundary value 0, but is more closely tied to the critical exponent of the elliptic type degenerate equations. Furthermore, we demonstrate that if max{1, σ + m} < p ≤ p c , then every positive solution of the equations blows up in finite time; whereas for p > p c , the equations admit global positive solutions for some boundary values and initial data. Meantime, we also demonstrate that its positive solutions blow up in finite time provided n ≤ σ + 2.