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Febuary 2020 Convergence of solutions for Fisher–KPP equation with a free boundary condition
Jingjing Cai, Yuan Chai, Qiong Liu
Rocky Mountain J. Math. 50(1): 55-68 (Febuary 2020). DOI: 10.1216/rmj.2020.50.55

Abstract

We consider a free boundary problem: ut=uxx+f(t,u) (g(t)<x<h(t)) with free boundary conditions h(t)=ux(t,h(t))α(t) and g(t)=ux(t,g(t))+α(t), where α(t) is a positive T-periodic function, f(t,u) is a Fisher–KPP type of nonlinearity and T-periodic in t. Such a problem can model the spreading of a biological or chemical species in time-periodic environment, where free boundaries mimic the spreading fronts of the species. We mainly study the convergence of bounded solutions. There is a T-periodic function α0(t) which plays a key role in the dynamics. More precisely:

(i) When 0<α<α0, we obtain a trichotomy result:

  1. Spreading, i.e., h(t),g(t)+ and u(t,)P(t) as t, where P(t) is the periodic solution of the ODE ut=f(t,u).

  2. Vanishing, i.e., limt𝒯h(t)= limt𝒯g(t) and limt𝒯maxg(t)xh(t)u(t,x)=0, where 𝒯 is some positive constant.

  3. Transition, i.e., 0< limth(t),limtg(t)<+, 0< limt[h(t)g(t)]<+ and u(t,)V(t,), where V is a T-periodic solution with compact support.

(ii) In the case αα0, vanishing happens for any solution.

Citation

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Jingjing Cai. Yuan Chai. Qiong Liu. "Convergence of solutions for Fisher–KPP equation with a free boundary condition." Rocky Mountain J. Math. 50 (1) 55 - 68, Febuary 2020. https://doi.org/10.1216/rmj.2020.50.55

Information

Received: 26 April 2019; Revised: 20 August 2019; Accepted: 12 September 2019; Published: Febuary 2020
First available in Project Euclid: 30 April 2020

zbMATH: 07201554
MathSciNet: MR4092544
Digital Object Identifier: 10.1216/rmj.2020.50.55

Subjects:
Primary: 35B40, 35K20, 35K55, 35R35

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.50 • No. 1 • Febuary 2020
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