Entire Solutions with Merging Fronts to Reaction–Diffusion Equations
Y. Morita, and H. Ninomiya. Journal of Dynamics and Differential Equations, 18 (4):
841--861(October 2006)
Abstract
We deal with a reaction–diffusion equation u t = u xx + f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ1(x + c 1 t) (c 1 < 0) and ψ2(x + c 2 t) (c 2 > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all $$(x, t) R^2$$. We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ1(x + c 1 t) and ψ2(x + c 2 t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c 1, we show the existence of an entire solution which behaves as ψ1( − x + c 1 t) in $$xın(-ınfty, (c_1+c)t/2$$ and φ(x + ct) in $$xın(c_1+c)t/2,ınfty)$$ for t≈ − ∞.
%0 Journal Article
%1 yoshihisa2006entire
%A Morita, Yoshihisa
%A Ninomiya, Hirokazu
%D 2006
%J Journal of Dynamics and Differential Equations
%K Fisher-KPP entire_solution transient_dynamics reaction-diffusion travelling_wave
%N 4
%P 841--861
%T Entire Solutions with Merging Fronts to Reaction–Diffusion Equations
%U http://dx.doi.org/10.1007/s10884-006-9046-x
%V 18
%X We deal with a reaction–diffusion equation u t = u xx + f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ1(x + c 1 t) (c 1 < 0) and ψ2(x + c 2 t) (c 2 > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all $$(x, t) R^2$$. We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ1(x + c 1 t) and ψ2(x + c 2 t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c 1, we show the existence of an entire solution which behaves as ψ1( − x + c 1 t) in $$xın(-ınfty, (c_1+c)t/2$$ and φ(x + ct) in $$xın(c_1+c)t/2,ınfty)$$ for t≈ − ∞.
@article{yoshihisa2006entire,
abstract = {We deal with a reaction–diffusion equation u t = u xx + f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ1(x + c 1 t) (c 1 < 0) and ψ2(x + c 2 t) (c 2 > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all $$(x, t) \in \mathbb{R}^{2}$$. We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ1(x + c 1 t) and ψ2(x + c 2 t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c 1, we show the existence of an entire solution which behaves as ψ1( − x + c 1 t) in $$x\in(-\infty, (c_1+c)t/2]$$ and φ(x + ct) in $$x\in[(c_1+c)t/2,\infty)$$ for t≈ − ∞.},
added-at = {2009-11-12T04:06:36.000+0100},
author = {Morita, Yoshihisa and Ninomiya, Hirokazu},
biburl = {https://www.bibsonomy.org/bibtex/2c0e00e10b4480046795da559f8faaa20/peter.ralph},
description = {SpringerLink - Journal Article},
interhash = {66e08fea71e8ea79213d968624293307},
intrahash = {c0e00e10b4480046795da559f8faaa20},
journal = {Journal of Dynamics and Differential Equations},
keywords = {Fisher-KPP entire_solution transient_dynamics reaction-diffusion travelling_wave},
month = {#oct#},
number = 4,
pages = {841--861},
timestamp = {2013-09-12T22:23:01.000+0200},
title = {Entire Solutions with Merging Fronts to Reaction–Diffusion Equations},
url = {http://dx.doi.org/10.1007/s10884-006-9046-x},
volume = 18,
year = 2006
}