Global existence, asymptotic behavior, and pattern formation driven by the parametrization of a nonlocal Fisher-KPP problem

Jing Li; Li Chen; Christina Surulescu

doi:10.1016/j.jde.2020.06.039

paper

Global existence, asymptotic behavior, and pattern formation driven by the parametrization of a nonlocal Fisher-KPP problem

arXiv:1909.07934 · doi:10.1016/j.jde.2020.06.039

Abstract

The global boundedness and the hair trigger effect of solutions for the nonlinear nonlocal reaction-diffusion equation \begin{align*} u_t=Δu+μu^α(1-κJ*u^β),\quad\hbox{in} \;\mathbb R^N\times(0,\infty),\; N\geq 1 \end{align*} with $α \geq 1$ , $β, μ, κ > 0$ and $u (x, 0) = u_{0} (x)$ are investigated. Under appropriate assumptions on $J$ , it is proved that for any nonnegative and bounded initial condition, if $α \in [1, α^{*})$ with $α^{*} = 1 + β$ for $N = 1, 2$ and $α^{*} = 1 + \frac{2 β}{N}$ for $N > 2$ , then the problem has a global bounded classical solution. Under further assumptions on the initial datum, the solutions satisfying $0 \leq u (x, t) \leq κ^{- \frac{1}{β}}$ for any $(x, t) \in R^{N} \times [0, + \infty)$ are shown to converge to $κ^{- \frac{1}{β}}$ uniformly on any compact subset of $R^{N}$ , which is known as the hair trigger effect. 1D numerical simulations of the above nonlocal reaction-diffusion equation are performed and the effect of several combinations of parameters and convolution kernels on the solution behavior is investigated. The results motivate a discussion about some conjectures arising from this model and further issues to be studied in this context. A formal deduction of the model from a mesoscopic formulation is provided as well.

44 pages, 5 figures, 27 plots

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On the stability of the state 1 in the non-local Fisher-KPP equation in bounded domains

Global existence, asymptotic behavior, and pattern formation driven by the parametrization of a nonlocal Fisher-KPP problem

Abstract

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