Shape differentiation of a steady-state reaction-diffusion problem arising in Chemical Engineering: the case of non-smooth kinetic with dead core

TL;DR

This paper extends shape differentiation techniques for steady-state reaction-diffusion problems in chemical engineering to cases with less smooth nonlinearities, including Lipschitz continuous and blow-up nonlinearities, with applications to root-type cases.

Contribution

It demonstrates the existence of Gateaux shape derivatives under minimal regularity assumptions and defines derivatives for nonlinearities with blow-up behavior.

Findings

Gateaux derivatives exist for Lipschitz continuous nonlinearities.

Defined shape derivatives for nonlinearities with blow-up.

Applied results to root-type nonlinearities.

Abstract

In this paper we consider an extension of the results in shape differentiation of semilinear equations with smooth nonlinearity presented in J.I. D\'iaz and D. G\'omez-Castro: An Application of Shape Differentiation to the Effectiveness of a Steady State Reaction-Diffusion Problem Arising in Chemical Engineering. Electron. J. Differ. Equations in 2015 to the case in which the nonlinearities might be less smooth. Namely we will show that Gateaux shape derivatives exists when the nonlinearity is only Lipschitz continuous, and we will give a definition of the derivative when the nonlinearity has a blow up. In this direction, we will study the case of root-type nonlinearities.