On the Speed of Spread for Fractional Reaction-Diffusion Equations

TL;DR

This paper investigates the speed at which solutions to fractional reaction-diffusion equations spread, revealing conditions for unbounded or finite speed based on the reaction term's properties and fractional order.

Contribution

It introduces new traveling wave solutions and establishes conditions determining whether solutions spread with finite or unbounded speed in fractional reaction-diffusion equations.

Findings

Solutions spread unboundedly if g satisfies certain growth near 0 for lpha > 1.

Solutions spread with finite speed if g'() < 0.

Traveling wave solutions are constructed for analysis.

Abstract

The fractional reaction diffusion equation u_t + Au = g(u) is discussed, where A is a fractional differential operator on the real line with order \alpha between 0 and 2, the C^1 function g vanishes at 0 and 1, and either g is non-negative on (0,1) or g < 0 near 0. In the case of non-negative g, it is shown that solutions with initial support on the positive half axis spread into the left half axis with unbounded speed if g satisfies some weak growth condition near 0 in the case \alpha > 1, or if g is merely positive on a sufficiently large interval near 1 in the case \alpha < 1. On the other hand, it shown that solutions spread with finite speed if g'(0) < 0. The proofs use comparison arguments and a new family of traveling wave solutions for this class of problems.