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

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 $α\geq1$, $β,μ,κ>0$ 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 $α^*=1+\frac{2β}{N}$ for $N>2$, then the problem has a global bounded classical solution. Under further assumptions on the initial datum, the solutions satisfying $0\leq u(x,t)\leqκ^{-\frac1β}$ for any $(x,t)\in\mathbb R^N\times[0,+\infty)$ are shown to converge to $κ^{-\frac1β}$ uniformly on any compact subset of $\mathbb R^N$, 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.