An inverse problem involving two coefficients in a nonlinear reaction–diffusion equation

Abstract

This Note deals with a uniqueness and stability result for a nonlinear reaction–diffusion equation with heterogeneous coefficients, which arises as a model of population dynamics in heterogeneous environments. We obtain a Lipschitz stability inequality which implies that two non-constant coefficients of the equation, which can be respectively interpreted as intrinsic growth rate and intraspecific competition coefficients, are uniquely determined by the knowledge of the solution on the whole domain at two times t0 and t1 and on a subdomain during a time interval which contains t0 and t1. This inequality can be used to reconstruct the coefficients of the equation using only partial measurements of its solution.