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

Michel Cristofol; Lionel Roques

doi:10.1016/j.crma.2012.04.019

Partial Differential Equations

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]

Présenté par : the Editorial Board

Michel Cristofol ¹ ; Lionel Roques ²

¹ Laboratoire dʼanalyse topologie probabilités, CNRS UMR 6632, universités dʼAix-Marseille, 13453 Marseille cedex 13, France
² UR 546 Biostatistique et processus spatiaux, INRA, 84000 Avignon, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 469-473.

Résumé
Abstract

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 $t_{0}$ et $t_{1}$ , ainsi que sa connaissance sur un sous-domaine durant un intervalle de temps contenant $t_{0}$ et $t_{1}$ , détermine de façon unique deux coefficients hétérogènes de lʼéquation.

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.

Reçu le : 2009-09-12
Accepté le : 2012-04-25
Publié le : 2012-05-01

DOI : 10.1016/j.crma.2012.04.019

Affiliations des auteurs :

Michel Cristofol ¹ ; Lionel Roques ²

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     title = {An inverse problem involving two coefficients in a nonlinear reaction{\textendash}diffusion equation},
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     language = {en},
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Michel Cristofol; Lionel Roques. An inverse problem involving two coefficients in a nonlinear reaction–diffusion equation. Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 469-473. doi : 10.1016/j.crma.2012.04.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.04.019/

Bibliographie
Cité par

[1] M. Cristofol; J. Garnier; F. Hamel; L. Roques Uniqueness from pointwise observation in a multi-parameter inverse problem, Commun. Pure Appl. Anal., Volume 11 (2012) no. 1, pp. 173-188

[2] M. Cristofol; L. Roques Biological invasions: Deriving the regions at risk from partial measurements, Math. Biosci., Volume 215 (2008) no. 2, pp. 158-166

[3] M. Cristofol, L. Roques, Simultaneous reconstruction of two coefficients in a nonlinear parabolic equation, in preparation, 2012.

[4] M. Choulli; M. Yamamoto Some stability estimates in determining sources and coefficients, J. Inverse Ill-Posed Probl., Volume 14 (2006), pp. 355-373

[5] M. El Smaily; F. Hamel; L. Roques Homogenization and influence of fragmentation in a biological invasion model, Discrete Contin. Dyn. Syst., Volume 25 (2009), pp. 321-342

[6] E. Fernández-Cara; S. Guerrero Global Carleman inequalities for parabolic systems and application to controllability, SIAM J. Control Optim., Volume 45 (2006), pp. 1395-1446

[7] R.A. Fisher The wave of advance of advantageous genes, Ann. Eugenics, Volume 7 (1937), pp. 335-369

[8] A. Friedman Partial Differential Equations of Parabolic Type, Prentice–Hall, Englewood Cliffs, NJ, 1964

[9] M.V. Klibanov Global uniqueness of a multidimensional inverse problem for a nonlinear parabolic equation by a Carleman estimate, Inverse Problems, Volume 20 (2004), pp. 1003-1032

[10] A.N. Kolmogorov; I.G. Petrovsky; N.S. Piskunov Étude de lʼéquation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bulletin de lʼUniversité dʼÉtat de Moscou, Série Internationale A, Volume 1 (1937), pp. 1-26

[11] L. Roques; M.-A. Auger-Rozenberg; A. Roques Modelling the impact of an invasive insect via reaction–diffusion, Math. Biosci., Volume 216 (2008) no. 1, pp. 47-55

[12] L. Roques; M.D. Chekroun On population resilience to external perturbations, SIAM J. Appl. Math., Volume 68 (2007) no. 1, pp. 133-153

[13] L. Roques; M. Cristofol On the determination of the nonlinearity from localized measurements in a reaction–diffusion equation, Nonlinearity, Volume 23 (2010), pp. 675-686

[14] L. Roques; F. Hamel Mathematical analysis of the optimal habitat configurations for species persistence, Math. Biosci., Volume 210 (2007) no. 1, pp. 34-59

[15] L. Roques; A. Roques; H. Berestycki; A. Kretzschmar A population facing climate change: joint influences of Allee effects and environmental boundary geometry, Population Ecology, Volume 50 (2008) no. 2, pp. 215-225

[16] L. Roques; R.S. Stoica Species persistence decreases with habitat fragmentation: an analysis in periodic stochastic environments, J. Math. Biol., Volume 55 (2007) no. 2, pp. 189-205

[17] J.G. Skellam Random dispersal in theoretical populations, Biometrika, Volume 38 (1951), pp. 196-218

[18] N. Shigesada; K. Kawasaki; E. Teramoto Traveling periodic-waves in heterogeneous environments, Theoret. Population Biology, Volume 30 (1986) no. 1, pp. 143-160

[19] M. Yamamoto; J. Zou Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Problems, Volume 17 (2001), pp. 1181-1202

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