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

Matthieu Alfaro|arXiv (Cornell University)|May 3, 2016

Stability and Controllability of Differential Equations26 references41 citations

TL;DR

This paper establishes that the Fujita exponent—determining blow-up vs. global existence in nonlocal diffusion equations—depends critically on the tail behavior of the dispersal kernel $ J $. For kernels with algebraic tails, the exponent transitions from heat-type ($ p_F = 2/N $) to fractional-type ($ p_F = \alpha/N - 1 $) depending on whether the second moment of $ J $ is finite, with applications to population dynamics showing a hair trigger effect when long-range dispersal dominates.

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 $\\partial \\_t u=J*u-u+u^{1+p}(1-u)$

Motivation & Objective

• To determine how the tail behavior of the dispersal kernel $ J $ affects the Fujita exponent in nonlocal reaction-diffusion equations.
• To analyze the transition between heat-type and fractional-type Fujita exponents based on the integrability of $ J $'s second moment.
• To establish conditions under which the hair trigger effect occurs in population dynamics models with long-range dispersal and weak Allee effects.
• To extend classical Fujita results from local (heat equation) and fractional diffusion to nonlocal diffusion with fat-tailed kernels.
• To provide a rigorous link between the decay of $ \widehat{J}(\xi) $ near $ \xi = 0 $ and the critical exponent for blow-up or extinction.

Proposed method

• Analyzes the nonlocal diffusion equation $ \partial_t u = J*u - u + u^{1+p} $ in $ \mathbb{R}^N $, focusing on the role of $ \widehat{J}(\xi) $ near $ \xi = 0 $.
• Uses asymptotic analysis of the Fourier transform of $ J $ to classify tail behaviors: compact support, exponential decay, or algebraic decay $ J(x) \sim |x|^{-\alpha} $.
• Applies comparison principles and subsolution constructions to prove blow-up or extinction, using a time-dependent barrier function $ \Phi(t,x) $.
• Estimates the decay of convolutions $ J^{*(k)} $ via the characteristic function $ \widehat{J}(\xi) $, leveraging $ 1 - \widehat{J}(\xi) \sim A|\xi|^\beta $ as $ \xi \to 0 $.
• Establishes the hair trigger effect by constructing a subsolution $ W(t,x) $ that grows from small initial data to near-1 values uniformly in compact sets.
• Uses the time-regularized solution $ \psi(T, \cdot) $ and estimates the tail integral $ \int_{|y| \geq m\tau^{1/\beta}} \psi(T,x-y) dy \leq C' T / \tau $ to control convergence.

Experimental results

Research questions

• RQ1 How does the tail behavior of the dispersal kernel $ J $ affect the Fujita exponent in nonlocal diffusion equations?
• RQ2 Under what conditions does the Fujita exponent transition from heat-type ($ p_F = 2/N $) to fractional-type ($ p_F = \alpha/N - 1 $) for algebraic tails?
• RQ3 What role does the finiteness of the second moment of $ J $ play in determining the critical exponent?
• RQ4 In population dynamics with long-range dispersal and weak Allee effects, when does the hair trigger effect occur?
• RQ5 Can the hair trigger effect be rigorously proven for nonlocal equations with fat-tailed kernels?

Key findings

• For compactly supported or exponentially decaying kernels, the Fujita exponent is $ p_F = \frac{2}{N} $, matching the classical heat equation.
• For algebraic tails $ J(x) \sim |x|^{-\alpha} $ with $ N < \alpha \leq N+2 $, the Fujita exponent is $ p_F = \frac{\alpha}{N} - 1 $, corresponding to a fractional diffusion regime.
• When $ \alpha > N+2 $, the second moment of $ J $ is finite, and the Fujita exponent reverts to the heat-type $ p_F = \frac{2}{N} $.
• The hair trigger effect holds for $ p < \frac{1}{2} \frac{\beta}{N} $, where $ \beta $ is the exponent in $ 1 - \widehat{J}(\xi) \sim A|\xi|^\beta $, ensuring solutions from small initial data reach near-1 values uniformly in compact sets.
• The proof relies on constructing a subsolution that grows from $ \varepsilon $ to $ 1 - 2\varepsilon $ in finite time, using time-regularized convolution estimates and tail decay control.
• The critical exponent depends on the local behavior of $ \widehat{J}(\xi) $ near $ \xi = 0 $, linking spectral properties to long-time dynamics.