Comptes Rendus
Partial Differential Equations
An inverse problem involving two coefficients in a nonlinear reaction–diffusion equation
Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 469-473.

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.

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 t0 et t1, ainsi que sa connaissance sur un sous-domaine durant un intervalle de temps contenant t0 et t1, détermine de façon unique deux coefficients hétérogènes de lʼéquation.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.04.019

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
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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/

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[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.

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[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

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