[02184] Oscillatory Translational Instability of Localized Spot Patterns in the Schnakenberg Reaction-Diffusion System in Defected 3D Domains

Session Time & Room : 1C (Aug.21, 13:20-15:00) @G501

Type : Contributed Talk

Abstract : For a two-component reaction-diffusion system in a bounded $3D$ domain, we investigate oscillatory instabilities of $N$-spot equilibrium. An $N$-spot equilibrium consists of localized spots in which the activator concentration is exponentially small everywhere except localized regions. In the stability analysis, we consider the translation mode and obtain the eigenvalue $\lambda$ is $\mathcal{O}(\varepsilon^2)$, which is the same order as the spot dynamics, while $\tau $ is $\mathcal{O}(\varepsilon^{-3})$. As a result, the system which contains the behavior of $\lambda$ and $\tau \lambda$ falls into the $\mathcal{O}(\varepsilon^2)$ correction. We later find that stability of these solutions is governed by a $3N \times 3N$ nonlinear matrix eigenvalue problem. Entries of the $3N \times 3N$ matrix involves terms calculated from certain Green’s function that contains information about the domain’s geometry. In the nonlinear matrix eigenvalue system, the most unstable eigenvalue decides the oscillation frequency at onset while the corresponding eigenvector determines the mode of spot oscillations. Further, we demonstrate the impact of various types of localized heterogeneity on this instability. An example of localized domain defects that we consider is to analyze the effect of perturbing the system by removing a small ball in the domain, which therefore allows a leakage of the chemical species out of the domain. Perturbation techniques is employed to compute Green’s function of near-spherical and near-cubic domains to gain analytic insight into how domain geometry select the dominant mode of oscillation. We show full solutions of the $3$-$D$ Schnakenberg PDE to confirm our asymptotic results.