Traveling waves for a bistable equation with nonlocal-diffusion

Abstract

We consider a single component reaction-diffusion equation in one dimension with bistable nonlinearity and a nonlocal space-fractional diffusion operator of Riesz-Feller type. Our main result shows the existence, uniqueness (up to translations) and stability of a traveling wave solution connecting two stable homogeneous steady states. In particular, we provide an extension to classical results on traveling wave solutions involving local diffusion. This extension to evolution equations with Riesz-Feller operators requires several technical steps. These steps are based upon an integral representation for Riesz-Feller operators, a comparison principle, regularity theory for space-fractional diffusion equations, and control of the far-field behavior.

References in corpus (2)

• The fundamental solution of the space-time fractional diffusion equation
• Phasefield theory for fractional diffusion-reaction equations and applications

Cited by in corpus (8)

• Fractional Dissipative PDEs
• Asymptotic stability of traveling wave solutions for nonlocal viscous conservation laws with explicit decay rates
• Metastable Speeds in the Fractional Allen-Cahn Equation
• Bistable reaction equations with doubly nonlinear diffusion
• Traveling wave dynamics for Allen-Cahn equations with strong irreversibility
• Vanishing viscosity limit of a conservation law regularised by a Riesz-Feller operator
• Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem
• A central limit theorem for the spatial Lambda Fleming-Viot process with selection