On the Speed of Spread for Fractional Reaction-Diffusion Equations

Abstract

The fractional reaction diffusion equation ${\partial }_{t}u+Au=g\left(u\right)$ is discussed, where $A$ is a fractional differential operator on $ℝ$ of order $\alpha \in \left(0,2\right)$, the $C^1$ function $g$ vanishes at $\zeta =0$ and $\zeta =1$, and either $g\ge 0$ on $\left(0,1\right)$ or near $\zeta =0$. 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 $g\left(\zeta \right)$ satisfies some weak growth condition near $\zeta =0$ in the case $\alpha >1$, or if $g$ is merely positive on a sufficiently large interval near $\zeta =1$ in the case . On the other hand, it shown that solutions spread with finite speed if . The proofs use comparison arguments and a suitable family of travelling wave solutions.