EXTINCTION AND NON-EXTINCTION OF SOLUTIONS TO A FAST DIFFUSIVE p-LAPLACE EQUATION WITH A NONLOCAL SOURCE

Abstract. In this paper, the authors establish the conditions for the ex-tinction of solutions, in finite time, of the fast diffusive p-Laplace equation ut = div(|∇u|p−2∇u) + a Ω uq(y, t)dy, 1 < p < 2, in a bounded domain Ω ⊂ RN with N ≥ 1. More precisely, it is shown that if q> p − 1, any solution vanishes in finite time when the initial datum or the coefficient a or the Lebesgue measure of the domain is small, and if 0 < q < p − 1, there exists a solution which is positive in Ω for all t> 0. For the critical case q = p − 1, whether the solutions vanish in finite time or not depends crucially on the value of aµ, where µ = Ω φp−1(x)dx and φ is the unique positive solution of the elliptic problem −div(|∇φ|p−2∇φ) = 1, x ∈ Ω; φ(x) = 0, x ∈ ∂Ω. This is a main difference between equations with local and nonlocal sources. 1