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

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 R u q(y,t)dy, 1 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 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 = Rp−1 (x)dx andis the unique positive solution of the elliptic problem −div(|∇�| p−2 ∇�) = 1, x ∈ ; �(x) = 0, x ∈ @. This is a main difference between equations with local and nonlocal sources.