Gradient estimates and symmetrization for Fisher-KPP front propagation\n with fractional diffusion

2015/02/22 by Jean‐Michel Roquejoffre, Roquejoffre, Jean-Michel, Andrei Tarfulea +1
Mathematics · Medicine · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1502.06304

openalex publication_date 2015/02/22 · openalex created_date 2022/10/20 · openalex updated_date 2026/07/28

Abstract

In this paper, we study gradient decay estimates for solutions to the\nmulti-dimensional Fisher-KPP equation with fractional diffusion. It is known\nthat this equation exhibits exponentially advancing level sets with strong\nqualitative upper and lower bounds on the solution. However, little has been\nshown concerning the gradient of the solution. We prove that, under mild\nconditions on the initial data, the first and second derivatives of the\nsolution obey a comparative exponential decay in time. We then use this\nestimate to prove a symmetrization result, which shows that the reaction front\nflattens and quantifiably circularizes, losing its initial structure.\n

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• The Fisher-KPP equation with nonlinear fractional diffusion
• Propagating front solutions in a time-fractional Fisher-KPP equation
• Propagation in Fisher-KPP type equations with fractional diffusion in periodic media
• Transition between linear and exponential propagation in Fisher-KPP type reaction-diffusion equations