Global existence and finite-time blow-up for a class of nonlocal parabolic problems

Abstract

An analysis of the nonlocal parabolic equation and its associated steady state equation, (S) with Dirichlet boundary conditions is given, assuming ω ⊂ ℝn is a smooth bounded domain, δ > 0, p ≥ 0, and f is positive and locally Lipschitz continuous. Existence-nonexistence results are proven for (S) when f(u) = eu or e-u and ω is a ball or star-shaped domain. For f(u) ≥ c > 0, n = 1, and p ≥ 1, we prove that (P) has a global bounded solution for any nonnegative initial data u0(x) and any δ > 0. For f(u) = eu, n = 2, ω = B1(0), u0(x) radially symmetric, nonnegative, if p > 1, (P) has a unique, globally bounded solution for any δ > 0. If p = 1, 0 < δ < 8π, (P) again has a bounded global solution. For f(u) = eu, n = 1 or 2, if p < 1 and δ > δ* where δ* is critical value for existence-nonexistence for (S), then the solution u of (P) blows up in finite time.