The Stability of Spot Patterns for the Brusselator Reaction-Diffusion System in Two Space Dimensions: Periodic and Finite Domain Settings

In this thesis, we asymptotically construct steady-state localized spot solutions to the Brusselator reaction-diffusion system in the semi-strong interaction regime characterized by an asymptotically large diffusivity ratio. We consider two distinct settings: a periodic pattern of localized spots in R2 concentrating at lattice points of a Bravais lattice, and multi-spot solutions that concentrate around some discrete points inside a finite domain. We use the method of matched asymptotic expansions, Floquet-Bloch theory, and the study of certain nonlocal eigenvalue problems to perform a linear stability analysis of these patterns. This analysis leads to a two-term approximation for a certain stability threshold characterized by a zero-eigenvalue crossing. Numerical results for the stability threshold are obtained, and compared with various approximations. For the periodic problem, a key feature for the determination of the stability threshold is to use an Ewald summation method to derive an explicit expression for the regular part of the Bloch Green function. Moreover, such an expression allows for the identification of the lattice that offers the optimum stability threshold. For the finite domain problem, we implement our asymptotic theory by calculating the stability threshold for an N -spot pattern where the spots are equidistantly spaced on a circular ring that is concentric within the unit disk.