Detection of dead cores for reaction-diffusion equations with a non-smooth nonlinearity

Abstract

This paper studies a class of nonlinear elliptic PDEs, arising for stationary reaction-diffusion models. The non-smooth nonlinearity gives rise to dead cores, that is, subdomains where the solution of the PDE vanishes. The paper gives a solid foundation for the numerical solution of the problem, including proper extensions of known maximum–minimum principles, Céa lemma, and convergence estimation of the FEM for locally Hölder continuous operators. Based on these, we finally detect dead cores numerically in various typical situations.