The global boundedness and the hair trigger effect of solutions for the nonlinear nonlocal reaction-diffusion equation \begin{align*} \pd ut=\Delta u+\mu u^\alpha(1-\kappa J*u^\beta),\quad\hbox{in} \;\mathbb R^N\times(0,\infty),\; N\geq 1 \end{align*} with $\alpha,\beta\geq1$, $\mu,\kappa>0$ 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 $\alpha\in[1,\alpha^*)\cup[1,\frac{1+\beta}{2}]$ with $\alpha^*=1+\beta$ for $N=1$ and $\alpha^*=1+\frac{2\beta}{N}$ for $N\geq 2$, then the problem has a global bounded classical solution. Under further assumptions on the initial datum, the solutions satisfying $0\leq u(x,t)\leq\kappa^{-\frac1\beta}$ for any $(x,t)\in\mathbb R^N\times[0,+\infty)$ are shown to converge to $\kappa^{-\frac1\beta}$ 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. Namely, for the supercritical case, the unboundedness of the solution is numerically tested. For the subcritical case, i.e. the case for which the boundedness of solutions has been proved, the hair trigger effect is revealed for small $\kappa$ values. For relatively large $\kappa$, take $\kappa=1$ for example, the hair trigger effect is numerically confirmed for small $\mu$'s, while for relatively large $\mu$ values, different patterns appear with different choices of convolution kernels. These 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.

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由非局部 Fisher-KPP 问题的参数化驱动的全局有界性、头发触发效应和模式形成

非线性非局部反应扩散方程解的全局有界性和毛发触发效应 \begin{align*} \pd ut=\Delta u+\mu u^\alpha(1-\kappa J*u^\beta)， \quad\hbox{in} \;\mathbb R^N\times(0,\infty),\; N\geq 1 \end{align*} 与 $\alpha,\beta\geq1$, $\mu,\kappa>0$ 和 $u(x,0)=u_0(x)$ 进行了研究。在 $J$ 的适当假设下，证明对于任何非负有界初始条件，如果 $\alpha\in[1,\alpha^*)\cup[1,\frac{1+\beta}{2} ]$ 与 $\alpha^*=1+\beta$ 对于 $N=1$ 和 $\alpha^*=1+\frac{2\beta}{N}$ 对于 $N\geq 2$，那么问题具有全局有界经典解。在对初始数据的进一步假设下，对于任何 $(x,t)\in\mathbb R^N\times，满足 $0\leq u(x,t)\leq\kappa^{-\frac1\beta}$ 的解[0, +\infty)$ 在 $\mathbb R^N$ 的任何紧致子集上一致收敛到 $\kappa^{-\frac1\beta}$，这被称为头发触发效应。执行上述非局部反应扩散方程的一维数值模拟，并研究了参数和卷积核的几种组合对求解行为的影响。即，对于超临界情况，对解的无界性进行了数值检验。对于亚临界情况，即解的有界性已被证明的情况，对于小的 $\kappa$ 值会显示毛发触发效应。对于较大的 $\kappa$，以 $\kappa=1$ 为例，对于较小的 $\mu$ 的头发触发效果是通过数值确认的，而对于较大的 $\mu$ 值，不同的模式出现不同的选择卷积核。这些激发了对该模型产生的一些猜想以及在此背景下要研究的进一步问题的讨论。还提供了从细观公式中对模型的正式推导。