Copyright © 2010 Hans Engler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The fractional reaction diffusion equation ∂tu+Au=g(u) is discussed, where A is a fractional differential operator on ℝ of order α∈(0,2), the C1 function g vanishes at ζ=0 and ζ=1, and either g≥0 on (0,1) or g<0 near ζ=0. In the case of nonnegative 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(ζ) satisfies some weak growth condition near ζ=0 in the case α>1, or if g is merely positive on a sufficiently large interval near ζ=1 in the case α<1. On the other hand, it shown that solutions spread with finite speed if g′(0)<0. The proofs use comparison arguments and a suitable family of travelling wave solutions.