Uniqueness from pointwise observations in a multi-parameter inverse problem

Michel Cristofol; Jimmy Garnier; François Hamel; Lionel Roques

doi:10.3934/cpaa.2012.11.173

Article Contents

2012, Volume 11, Issue 1: 173-188. Doi: 10.3934/cpaa.2012.11.173

This issue Previous Article Approximation of nonlinear parabolic equations using a family of conformal and non-conformal schemes Next Article An effective design method to produce stationary chemical reaction-diffusion patterns

Uniqueness from pointwise observations in a multi-parameter inverse problem

1.
Aix-Marseille Université, LATP, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20
2.
UR 546 Biostatistique et Processus Spatiaux, INRA, F-84000 Avignon, France, and Aix-Marseille Université, LATP, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20
3.
Aix-Marseille Université & Institut Universitaire de France, LATP, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20
4.
UR 546 Biostatistique et Processus Spatiaux, INRA, F-84000 Avignon

Received: February 28, 2010

Revised: October 31, 2010

Published: December 2011

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Abstract

In this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coefficients of one-dimensional reaction-diffusion equations. Such reaction-diffusion equations include the classical model of Kolmogorov, Petrovsky and Piskunov as well as more sophisticated models from biology. When the reaction term contains an unknown polynomial part of degree $N,$ with non-constant coefficients $\mu_k(x),$ our result gives a sufficient condition for the uniqueness of the determination of this polynomial part. This sufficient condition only involves pointwise measurements of the solution $u$ of the reaction-diffusion equation and of its spatial derivative $\partial u / \partial x$ at a single point $x_0,$ during a time interval $(0,\varepsilon).$ In addition to this uniqueness result, we give several counter-examples to uniqueness, which emphasize the optimality of our assumptions. Finally, in the particular cases $N=2$ and $N=3,$ we show that such pointwise measurements can allow an efficient numerical determination of the unknown polynomial reaction term.

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Mathematics Subject Classification: 65M32, 35K20, 35K55, 35K57.

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