Rigorous Computation of Non-uniform Patterns for the 2-Dimensional Gray-Scott Reaction-Diffusion Equation

In this paper a method to rigorously compute several non trivial solutions of the Gray-Scott reaction-diffusion system defined on a 2-dimensional bounded domain is presented. It is proved existence, within rigorous bounds, of non uniform patterns significantly far from being a perturbation of the homogenous states. As a result, a non local diagram of families that bifurcate from the homogenous states is depicted, also showing coexistence of multiple solutions at the same parameter values. Combining analytical estimates and rigorous computations, the solutions are sought as fixed points of a operator in a suitable Banach space. To address the curse of dimensionality, a variation of the existing technique is presented, necessary to enable successful computations in reasonable time.