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Comptes Rendus Mathématique
Volume 350, n° 9-10
pages 469-473 (mai 2012)
Doi : 10.1016/j.crma.2012.04.019
Received : 12 September 2009 ;  accepted : 25 April 2012
An inverse problem involving two coefficients in a nonlinear reaction–diffusion equation
Reconstruction simultanée de deux coefficients dans une équation de réaction–diffusion non-linéaire
 

Michel Cristofol a , Lionel Roques b
a Laboratoire dʼanalyse topologie probabilités, CNRS UMR 6632, universités dʼAix-Marseille, 13453 Marseille cedex 13, France 
b UR 546 Biostatistique et processus spatiaux, INRA, 84000 Avignon, France 

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

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Résumé

Dans cette Note, nous présentons un résultat dʼunicité et de stabilité pour une équation de réaction–diffusion non linéaire et à coefficients hétérogènes, intervenant notamment dans des modèles de dynamique des populations. Nous établissons une inégalité du type Lipschitz impliquant que la connaissance de la solution de lʼéquation sur tout le domaine dʼétude à des temps   et  , ainsi que sa connaissance sur un sous-domaine durant un intervalle de temps contenant   et  , détermine de façon unique deux coefficients hétérogènes de lʼéquation.

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