Traveling wave solutions to some reaction diffusion equations with fractional Laplacians
- 528 Downloads
We show the nonexistence of traveling wave solutions in the combustion model with fractional Laplacian \(\displaystyle (-\Delta )^s\) when \(\displaystyle s\in (0,1/2]\). Our method can be used to give a direct and simple proof of the nonexistence of traveling fronts for the usual Fisher-KPP nonlinearity. Also we prove the existence and nonexistence of traveling wave solutions for different ranges of the fractional power \(s\) for the generalized Fisher–KPP type model.
Mathematics Subject Classificationprimary 35B32 35C07 35J20 35R09 35R11 45G05 47G10
This work was partially supported by a grant from the Simons Foundation (Award # 199305) and a NSF IPA award. The authors would also like to thank the anonymous referee for helpful suggesions for the revision of the manuscript.
- 2.Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and hamiltonian estimates. ArXiv (2010)Google Scholar
- 6.Gui, C., Zhao, M.: Traveling Wave Solutions of Allen–Cahn Equation with a Fractional Laplacian, Ann. I. H. Poincaré-AN (2014). doi: 10.1016/j.anihpc.2014.03.005
- 7.Kolmogorov, A., Petrovskii, I., Piskunov, N.: A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. Bull. Moscow Univ. Math. Ser. A 1, 1–25 (1937)Google Scholar
- 8.Landkof, N.S.: Foundations of Modern Potential Theory. In: Doohovskoy, A.P., (ed) Die Grundlehren der mathematischen Wissenschaften, Band 180. Translated from the Russian. Springer, New York, (1972)Google Scholar
- 9.Mellet, A., Roquejoffre, J., Sire, Y.: Existence and asymptotics of fronts in non local combustion models. Arxiv (2010)Google Scholar