This paper deals with the Cauchy problem of a nonlocal bistable reaction–diffusion equation $\frac{\partial u}{\partial t}=\Delta u+\mu u^2(1-\kappa J\ast u)-\gamma u,(x,t)\in {\mathbb{R}}^N\times (0,\infty )$with $N\le 2$, $\mu ,\kappa ,\gamma >0$ and $u(x,0)=u_0\left(x\right)$. Under appropriate assumptions on $J$, it is proved that for any nonnegative and bounded initial condition, this problem admits a global bounded classical solution for $N=1$, while for $N=2$, global bounded classical solution exists for large $\kappa$ values. Moreover, for small $\mu$ values and small initial data, the solution is shown to converge to 0 exponentially or locally uniformly as $t\to \infty$, which is referred as the Allee effect.

中文翻译：

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种群动态中非局部双稳态反应扩散方程的全局有界性和Allee效应

本文讨论了一个非局部双稳态反应扩散方程的柯西问题 $\frac{\partial ü}{\partial Ť}=\Delta ü+\mu {ü}^2（\mathrm{1个}-\kappa Ĵ\ast ü）-\gamma ü，（X，Ť）\in {\mathbb{[R}}^{ñ}\times （0，\infty ）$和 $ñ\le 2$， $\mu ，\kappa ，\gamma >0$ 和 $ü（X，0）={ü}_0（X）$。在适当的假设下$Ĵ$，证明了对于任何非负且有界的初始条件，该问题都允许一个全局有界经典解 $ñ=\mathrm{1个}$，而 $ñ=2$，对于大型 $\kappa$价值观。而且，对于小$\mu$ 值和小的初始数据，解显示为以指数形式或局部均匀收敛到0 $Ť\to \infty$，即Allee效应。