Some inverse stability results for the bistable reaction–diffusion equation using Carleman inequalities

Boulakia, Muriel; Grandmont, Céline; Osses, Axel

doi:10.1016/j.crma.2009.03.022

Partial Differential Equations

Some inverse stability results for the bistable reaction–diffusion equation using Carleman inequalities

Presented by: Gilles Lebeau

Boulakia, Muriel ¹; Grandmont, Céline ²; Osses, Axel ³

¹ Université Pierre et Marie Curie-Paris 6, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
² INRIA, Projet REO, Rocquencourt, BP 105, 78153 Le Chesnay cedex, France
³ Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS) FCFM, U. de Chile, Casilla 170/3, correo 3, Santiago, Chile

Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 619-622.

Abstract
Résumé

We consider the bistable equation $v_{t} - Δ v = f (v, x)$ , $f (v, x) = a (x) v (1 - v) (v - α (x))$ with homogeneous Neumann boundary conditions in a bounded domain $Ω \subset R^{3}$ with regular boundary. For this equation, we prove Lipschitz stability for the inverse problem of recovering parameters a and α from measurements of v in $(0, T) \times ω$ , where ω is an arbitrary nonempty open subset of Ω and measurements of $v (t_{0})$ in the whole domain Ω at some positive time $t_{0}$ such that $0 < t_{0} < T$ . The result is based in some suitable global Carleman estimate for the nonlinear problem.

Dans un domaine $Ω \subset R^{3}$ borné de frontière régulière, nous considérons l'équation bistable $v_{t} - Δ v = f (v, x)$ , $f (v, x) = a (x) v (1 - v) (v - α (x))$ complétée par des conditions de Neumann homogène au bord. Pour cette équation, nous prouvons un résultat de stabilité lipschitzienne pour le problème inverse qui consiste à identifier les paramètres a et α à partir de mesures de v sur $(0, T) \times ω$ , où $ω \subset Ω$ est un ouvert non vide quelconque et des mesures de $v (t_{0})$ dans tout le domaine Ω avec $t_{0}$ tel que $0 < t_{0} < T$ . Le résultat est basé sur une inegalité de Carleman globale pour le problème non linéaire.

Received: 2008-07-15
Accepted: 2009-03-24
Published online: 2009-04-16

DOI: 10.1016/j.crma.2009.03.022

Author's affiliations:

Boulakia, Muriel ¹; Grandmont, Céline ²; Osses, Axel ³

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     title = {Some inverse stability results for the bistable reaction{\textendash}diffusion equation using {Carleman} inequalities},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {619--622},
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Boulakia, Muriel; Grandmont, Céline; Osses, Axel. Some inverse stability results for the bistable reaction–diffusion equation using Carleman inequalities. Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 619-622. doi : 10.1016/j.crma.2009.03.022. http://www.numdam.org/articles/10.1016/j.crma.2009.03.022/

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