Asymptotic behavior of solutions of a degenerate Fisher–KPP equation with free boundaries. (August 2015)

• Record Type: Journal Article
• Authors: Sun, Ningkui
• Is Part Of: Nonlinear analysis. Volume 24(2015)
• Page Start: 98
• Page End: 107
• Publication Date: 2015-08
• Subjects: Reaction–diffusion equation -- Free boundary problem -- Asymptotic behavior -- Sharp threshold; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2015.01.007
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: The aim of this paper is to study the asymptotic behavior of solutions of a degenerate Fisher–KPP equation u t = u x x + u p ( 1 − u ) ( p > 0 ) in the domain { ( t, x ) ∈ R 2 : t ≥ 0, x ∈ [ 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 ) < ∞ 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 .