## [4.5.2016] Fujita blow up phenomena and hair trigger effect: the role of dispersal tails

Today ArXiv will be this paper of Matthieu Alfaro.

Firstly, a side interesting fact is the term “hair trigger effect”, which sounds cool to me  🙂

It can be translated naively as “adding just a small hair could change everything”. More precisely, there is a steady state that if you add “just a little bit of something” to it, then the trajectory goes differently and eventually ends up at something totally different. Until here it does sound familiar, doesn’t it? [Yes, to me it’s nothing else but instability of the steady state (in some sense)]

Let’s come back to the paper. The author studied a nonlinear nonlocal evolution problem in $\mathbb R^N$ of the following form

$u_t = J * u - u + u^{1+p}$.

This problem was studied previously with the assumption of compact support of the kernel $J$.

This work extends the result to the case that $J$ can have unbounded support, but eventually decays at infinity. There two typical cases are:

1. $J$ decays exponentially: $J(\xi) \approx e^{-k|\xi|}$; or
2. $J$ decays algebraically (fat tails): $J(\xi) \approx |\xi|^{-\alpha}$

Depending on these kinds of decay, the author established the Fujita type blow up for the considered equation. That is, depending on the polynomial $p$ (combining with the decay rate of $J$), the solution can either blow up for any nontrivial initial data or blow up for large data and exists globally (extinct) for small initial data.