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

作者：Sun Ningkui^*

来源：Nonlinear Analysis: Real World Applications , 2015, 24: 98-107.

DOI：10.1016/j.nonrwa.2015.01.007

• Reaction-diffusion equation
• Free boundary problem
• Asymptotic behavior
• Sharp threshold

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.

• 出版日期2015-8
• 单位同济大学