Fujita blow up phenomena and hair trigger effect: The role of dispersal tails

Abstract

We consider the nonlocal diffusion equation ${\partial }_{t}u=J\ast u-u+u^{1+p}$ in the whole of ${\mathbb{R}}^N$. We prove that the Fujita exponent dramatically depends on the behavior of the Fourier transform of the kernel $J$ near the origin, which is linked to the tails of $J$. In particular, for compactly supported or exponentially bounded kernels, the Fujita exponent is the same as that of the nonlinear Heat equation ${\partial }_{t}u=\mathrm{\Delta }u+u^{1+p}$. On the other hand, for kernels with algebraic tails, the Fujita exponent is either of the Heat type or of some related Fractional type, depending on the finiteness of the second moment of $J$. As an application of the result in population dynamics models, we discuss the hair trigger effect for ${\partial }_{t}u=J\ast u-u+u^{1+p}(1-u)$.