Year: 2026
Type: preprint
Abstract: We study a PDE-constrained variational problem for the optimal spatial distribution of a catalytic material reacting with a substrate in a biological or materials system. With a given catalyst concentration, the substrate concentration is determined by a reaction-diffusion equation. The reaction functional for the catalyst concentrations is defined to be the total reaction output, which is the product of catalyst and substrate concentrations. Both the linear and Michaelis-Menten reaction kinetics are considered. We construct reaction-maximizing sequences through the localization of catalysts using barrier functions and show the non-existence of a global maximizer. We also derive the first and second variations of the reaction functional. By constructing a competing catalyst concentration, we show the non-existence of a local maximizer. Based on our analysis, we propose a regularized variational model and prove the existence of its global maximizer.
Source: arXiv (Cornell University)
Authors: Khan, Mohammad Yasir Feroz, Li, Bo
Institution: University of California San Diego
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7222044336