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• W72827188 abstract "We consider the traveling fronts of the reaction diffusion equation: ut + (−∆)u = f(u), in R× R, for f ∈ C(R). Namely, the solution to the following equation:  (−∆)u(x) + cu′(x) = f(u(x)), ∀x ∈ R lim x→−∞ u(x) = 0, lim x→∞ u(x) = 1 (0.0.1) where c is the speed of propagation of the front and the operator (−∆) denotes the fractional power of the Laplacian in one dimension with 0 < s < 1. Recall the fractional Laplacian is defined as follows: (−∆)u(x) = C1,s(P.V.) ∫ R u(x)− u(y) |x− y|1+2s dy, where (P.V.) stands for Cauchy principal value and C1,s = 2sΓ((1 + 2s)/2) π1/2Γ(1− s) . We show the nonexistence of traveling fronts in the combustion model with fracii tional Laplacian (−∆) when s ∈ (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 waves solutions for different ranges of the fractional power s for the generalized Fisher-KPP type model. When considering the Allen-Cahn type nonlinearity, we show the approach of the solution to the traveling front for a large range of initial value problems. Traveling Fronts to Reaction Diffusion Equations with Fractional Laplacians" @default.
• W72827188 created "2016-06-24" @default.
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• W72827188 date "2014-01-01" @default.
• W72827188 modified "2023-09-27" @default.
• W72827188 title "Traveling Fronts to Reaction Diffusion Equations with Fractional Laplacians" @default.
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