Abstract for Gabriela Jaramillo

Gabriela Jaramillo (University of Houston)

Thursday, March 30, 2023

Analysis and simulation of a nonlocal one-dimensional Gray-Scott model

Abstract: The Gray-Scott model is a set of reaction-diffusion equations that describes chemical systems far from equilibrium. Interest in this model stems from its ability to generate spatio-temporal structures, including pulses, spots, stripes, and self-replicating patterns. In this talk we consider an extension of this model in which the spread of the different chemicals is assumed to be nonlocal, and can thus be represented by a convolution term. In particular, we focus on the case of strictly positive, symmetric, L^1 convolution kernels that have a finite second moment. Modeling the equations on a finite interval we define nonlocal analogues of Dirichlet and Neumann boundary conditions, and prove the existence of small-time weak solutions for the corresponding system. We then use this result to develop a finite element numerical scheme that helps us explore the effect of nonlocal diffusion on the formation of pulse solutions.