Journal of Dynamics and Differential Equations

, Volume 18, Issue 4, pp 841–861 | Cite as

Entire Solutions with Merging Fronts to Reaction–Diffusion Equations


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≈ − ∞.


reaction–diffusion equation entire solution traveling front wave bistable nonlinearity merging fronts 

Ams 2000 Subject Classification

35K57 35B05 35B40 


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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

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

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