On the Speed of Spread for Fractional Reaction-Diffusion Equations

The fractional reaction diffusion equation 𠜕 ð ‘¡ ð ‘¢ + ð ´ ð ‘¢ = ð ‘” ( ð ‘¢ ) is discussed, where ð ´ is a fractional differential operator on â„ of order ð ›¼ ∈ ( 0 , 2 ) , the ð ¶ 1 function ð ‘” vanishes at ð œ = 0 and ð œ = 1 , and either ð ‘” ≥ 0 on ( 0 , 1 ) or ð ‘” < 0 near ð œ = 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 ð ‘” ( ð œ ) satisfies some weak growth condition near ð œ = 0 in the case ð ›¼ > 1 , or if ð ‘” is merely positive on a sufficiently large interval near ð œ = 1 in the case ð ›¼ < 1 . On the other hand, it shown that solutions spread with finite speed if ð ‘” î…ž ( 0 ) < 0 . The proofs use comparison arguments and a suitable family of travelling wave solutions.