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For a balanced bistable reaction-diffusion equation, an axisymmetric traveling front has been well known. This paper proves that an axially asymmetric traveling front with any positive speed does exist in a balanced bistable reaction-diffusion equation. Our method is as follows. We use a pyramidal traveling front for an unbalanced reaction-diffusion equation whose cross section has a major axis and a minor axis. Preserving the ratio of the major axis and a minor axis to be a constant and taking the balanced limit, we obtain a traveling front in a balanced bistable reaction-diffusion equation. This traveling front is monotone decreasing with respect to the traveling axis, and its cross section is a compact set with a major axis and a minor axis when the constant ratio is not 1.
Cite this article
Masaharu Taniguchi, Axially asymmetric traveling fronts in balanced bistable reaction-diffusion equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 7, pp. 1791–1816DOI 10.1016/J.ANIHPC.2019.05.001