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

TL;DR

This paper investigates how the tail behavior of the dispersal kernel in a nonlocal diffusion equation influences the critical Fujita exponent, revealing a dependence on the kernel's Fourier transform and tail properties, with implications for population dynamics.

Contribution

It establishes a link between the kernel's tail behavior and the Fujita exponent, extending understanding of nonlocal diffusion equations and their critical thresholds.

Findings

Fujita exponent depends on the Fourier transform of the kernel near zero.

For compactly supported or exponential kernels, the exponent matches the nonlinear heat equation.

Algebraic tails lead to Fujita exponents of heat or fractional type, influenced by the second moment.

Abstract

We consider the nonlocal diffusion equation $\partial \_t u=J*u-u+u^{1+p}$ in the whole of $\R ^N$. 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 $\partial \_tu=\Delta u+u^{1+p}$. 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 $\partial \_t u=J*u-u+u^{1+p}(1-u)$