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

Michel Cristofol (1, 2, 3, 4) , Lionel Roques (5)

1 I2M - Institut de Mathématiques de Marseille
2 AMU - Aix Marseille Université
3 CNRS - Centre National de la Recherche Scientifique
4 ECM - École Centrale de Marseille
5 BioSP - Biostatistique et Processus Spatiaux

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 $t_0$ and $t_1$ and on a subdomain during a time interval which contains $t_0$ and $t_1$. This inequality can be used to reconstruct the coefficients of the equation using only partial measurements of its solution.

Domaines: Equations aux dérivées partielles [math.AP]

Michel Cristofol, Lionel Roques. An inverse problem involving two coefficients in a nonlinear reaction-diffusion equation. 2010. ⟨hal-00592268⟩