Journal of Dynamics and Differential Equations

, Volume 18, Issue 4, pp 841–861

Entire Solutions with Merging Fronts to Reaction–Diffusion Equations


DOI: 10.1007/s10884-006-9046-x

Cite this article as:
Morita, Y. & Ninomiya, H. J Dyn Diff Equat (2006) 18: 841. doi:10.1007/s10884-006-9046-x

We deal with a reaction–diffusion equation ut = uxx + 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 + c1t) (c1 < 0) and ψ2(x + c2t) (c2 > 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 + c1t) and ψ2(x + c2t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c >  − c1, we show the existence of an entire solution which behaves as ψ1( − x + c1t) in \(x\in(-\infty, (c_1+c)t/2]\) and φ(x + ct) in \(x\in[(c_1+c)t/2,\infty)\) for t≈ − ∞.


reaction–diffusion equationentire solutiontraveling front wavebistable nonlinearitymerging fronts

Ams 2000 Subject Classification


Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of Applied Mathematics and InformaticsRyukoku UniversitySeta OtsuJapan