Existence and nonexistence of global solutions of some nonlocal degenerate parabolic equations

Abstract

This paper investigates the global existence and nonexistence of positive solutions of the nonlinear degenerate parabolic equation μt = f (μu) (Δμ + a ʃ ωμ dx) with a homogeneous Dirichlet boundary condition. It is proved that there exists no global positive solution if and only if ʃ 1/(sf(s)) ds < ∞ and ʃ ωϕ(x) dx >/a , where ϕ( x) is the unique positive solution of the linear elliptic problem −Δ ϕ(x) = 1, x ∈ Ω; ϕ(x) = 0, x ∈ ∂ω