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International Journal of Differential Equations
Volume 2010 (2010), Article ID 315421, 16 pages
http://dx.doi.org/10.1155/2010/315421
Research Article

On the Speed of Spread for Fractional Reaction-Diffusion Equations

Department of Mathematics, Georgetown University, Box 571233, Washington, DC 20057, USA

Received 12 August 2009; Revised 12 October 2009; Accepted 25 October 2009

Academic Editor: Om Agrawal

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 𝜕 𝑡 𝑢 + 𝐴 𝑢 = 𝑔 ( 𝑢 ) is discussed, where 𝐴 is a fractional differential operator on of order 𝛼 ( 0 , 2 ) , the 𝐶 1 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.