Asymptotic behavior of solutions of a degenerate Fisher-KPP equation with free boundaries

摘要

The aim of this paper is to study the asymptotic behavior of solutions of a degenerate Fisher-KPP equation u(t) = u(xx) u(P) (1 - u) (p > 0) in the domain {(t, x) is an element of R-2 : t >= 0, x is an element of [g (t), h(t)]}, where g(t) and h(t) are two free boundaries. For p > 1 we obtain trichotomy result: spreading ([g(t), h(t)] R and u(t,.) -> 1 locally uniformly in R), vanishing (h(t) - g(t) < infinity and u(t,.) -> 0 uniformly in [g(t), h(t)]), and virtual vanishing ([g(t), h(t)] : R and u(t,.) > 0 uniformly in [g(t), h(t)]). For 0 < p < 1 we deduce that spreading can only happen, that is, 1 is the global attractor for all positive solutions. When spreading happens, we prove that the asymptotic spreading speed is continuous and strictly decreasing in p. (C) 2015 Elsevier Ltd. All rights reserved.