Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Entire solutions of reaction—diffusion equations with balanced bistable nonlinearities

Xinfu Chena1, Jong-Shenq Guoa2 and Hirokazu Ninomiyaa3

a1 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA (

a2 Department of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan (

a3 Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu 520-2194, Japan (


This paper deals with entire solutions of a bistable reaction—diffusion equation for which the speed of the travelling wave connecting two constant stable equilibria is zero. Entire solutions which behave as two travelling fronts approaching, with super-slow speeds, from opposite directions and annihilating in a finite time are constructed by using a quasi-invariant manifold approach. Such solutions are shown to be unique up to space and time translations.

(Received December 08 2004)

(Accepted December 13 2005)