[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.

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About baotangquoc

Lecturer School of Applied Mathematics and Informatics Hanoi University of Science and Technology No 1, Dai Co Viet Street, Hanoi
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