Abstract
We construct traveling waves of the fractional bistable equation by approximating the fractional Laplacian \({(D^{2})^{\alpha}, \alpha \in (0, 1)}\), with operators \({J \ast u - (\int_{R} J)u}\), where J is nonsingular. Since the resulting approximating equations are known to have traveling waves, the solutions are obtained by passing to the limit. This provides an answer to the statement (about existence and properties) “This construction will be achieved in a future work” before Assumption 2 in Imbert and Souganidis [6]. With a modification of a part of the argument, we also get the existence of traveling waves for the ignition nonlinearity in the case \({\alpha \in (1/2, 1)}\).
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Chmaj, A. Existence of traveling waves in the fractional bistable equation. Arch. Math. 100, 473–480 (2013). https://doi.org/10.1007/s00013-013-0511-6
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DOI: https://doi.org/10.1007/s00013-013-0511-6