Turing Instability for Nonlocal Heterogeneous Reaction-Diffusion Systems: A Computer-Assisted Proof Approach

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 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 δ of the nonlocal term is large enough, there is precisely one eigenvalue with positive real part, which proves the Turing instability. Moreover, by a detailed study of this eigenvalue as a function of δ, we obtain a sharp threshold δ∗ which is the bifurcation point for Turing instability.