标题： 非局部异质反应扩散系统的图灵不稳定性：基于计算机辅助证明的方法 显示英文标题

标题： Turing instability for nonlocal heterogeneous reaction-diffusion systems: A computer-assisted proof approach

摘要： 本文提供了一个计算机辅助证明，用于反应扩散系统中由异质非局部性诱导的图灵不稳定性。由于异质性和非局部性，线性傅里叶分析导致\textit{强耦合的}无限维微分系统。通过引入适当的基变换以及无穷矩阵的格尔希戈林圆盘定理，我们首先证明了对于充分大的$N$，所有$N$-th 格什戈林圆盘完全位于左半平面。对于剩下的有限多个圆盘，一个计算机辅助证明表明，如果非局部项的强度$\delta$足够大，则恰好存在一个具有正实部的特征值，从而证明了图灵不稳定性。此外，通过对该特征值作为$\delta$函数的详细研究，我们得到了一个尖锐的阈值$\delta^*$，它是图灵不稳定性分支点。 显示英文摘要

摘要： This paper provides a computer-assisted proof for the Turing instability induced by heterogeneous nonlocality in reaction-diffusion systems. Due to the heterogeneity and nonlocality, the linear Fourier analysis gives rise to \textit{strongly coupled} infinite differential systems. By introducing suitable changes of basis as well as the Gershgorin disks theorem for infinite matrices, we first show that all $N$-th Gershgorin disks lie completely on the left half-plane for sufficiently large $N$. For the remaining finitely many disks, a computer-assisted proof shows that if the intensity $\delta$ of the nonlocal term is large enough, there is precisely one eigenvalue with positive real part, which proves the Turing instability. Moreover, by detailed study of this eigenvalue as a function of $\delta$, we obtain a sharp threshold $\delta^*$ which is the bifurcation point for Turing instability.