[Reconstruction simultanée de deux coefficients dans une équation de réaction–diffusion non-linéaire]
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.
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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@article{CRMATH_2012__350_9-10_469_0,
author = {Cristofol, Michel and Roques, Lionel},
title = {An inverse problem involving two coefficients in a nonlinear reaction{\textendash}diffusion equation},
journal = {Comptes Rendus. Math\'ematique},
pages = {469--473},
publisher = {Elsevier},
volume = {350},
number = {9-10},
year = {2012},
doi = {10.1016/j.crma.2012.04.019},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2012.04.019/}
}
TY - JOUR AU - Cristofol, Michel AU - Roques, Lionel TI - An inverse problem involving two coefficients in a nonlinear reaction–diffusion equation JO - Comptes Rendus. Mathématique PY - 2012 SP - 469 EP - 473 VL - 350 IS - 9-10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2012.04.019/ DO - 10.1016/j.crma.2012.04.019 LA - en ID - CRMATH_2012__350_9-10_469_0 ER -
%0 Journal Article %A Cristofol, Michel %A Roques, Lionel %T An inverse problem involving two coefficients in a nonlinear reaction–diffusion equation %J Comptes Rendus. Mathématique %D 2012 %P 469-473 %V 350 %N 9-10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2012.04.019/ %R 10.1016/j.crma.2012.04.019 %G en %F CRMATH_2012__350_9-10_469_0
Cristofol, Michel; Roques, Lionel. An inverse problem involving two coefficients in a nonlinear reaction–diffusion equation. Comptes Rendus. Mathématique, Tome 350 (2012) no. 9-10, pp. 469-473. doi : 10.1016/j.crma.2012.04.019. http://www.numdam.org/articles/10.1016/j.crma.2012.04.019/
[1] Uniqueness from pointwise observation in a multi-parameter inverse problem, Commun. Pure Appl. Anal., Volume 11 (2012) no. 1, pp. 173-188
[2] 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] Some stability estimates in determining sources and coefficients, J. Inverse Ill-Posed Probl., Volume 14 (2006), pp. 355-373
[5] Homogenization and influence of fragmentation in a biological invasion model, Discrete Contin. Dyn. Syst., Volume 25 (2009), pp. 321-342
[6] Global Carleman inequalities for parabolic systems and application to controllability, SIAM J. Control Optim., Volume 45 (2006), pp. 1395-1446
[7] The wave of advance of advantageous genes, Ann. Eugenics, Volume 7 (1937), pp. 335-369
[8] Partial Differential Equations of Parabolic Type, Prentice–Hall, Englewood Cliffs, NJ, 1964
[9] 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] 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] Modelling the impact of an invasive insect via reaction–diffusion, Math. Biosci., Volume 216 (2008) no. 1, pp. 47-55
[12] On population resilience to external perturbations, SIAM J. Appl. Math., Volume 68 (2007) no. 1, pp. 133-153
[13] On the determination of the nonlinearity from localized measurements in a reaction–diffusion equation, Nonlinearity, Volume 23 (2010), pp. 675-686
[14] Mathematical analysis of the optimal habitat configurations for species persistence, Math. Biosci., Volume 210 (2007) no. 1, pp. 34-59
[15] 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] Species persistence decreases with habitat fragmentation: an analysis in periodic stochastic environments, J. Math. Biol., Volume 55 (2007) no. 2, pp. 189-205
[17] Random dispersal in theoretical populations, Biometrika, Volume 38 (1951), pp. 196-218
[18] Traveling periodic-waves in heterogeneous environments, Theoret. Population Biology, Volume 30 (1986) no. 1, pp. 143-160
[19] Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Problems, Volume 17 (2001), pp. 1181-1202
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