On the Speed of Spread for Fractional Reaction-Diffusion Equations - Hans Engler (International Journal of Differential Equations)
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.