Turing instability for nonlocal heterogeneous reaction-diffusion systems: A computer-assisted proof approach
Journal:
arXiv
Published Date:
Apr 7, 2025
Abstract
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.
