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Article Dans Une Revue Communications on Pure and Applied Analysis Année : 2012

Uniqueness from pointwise observations in a multi-parameter inverse problem

Résumé

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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Dates et versions

hal-00596238 , version 1 (26-05-2011)

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Michel Cristofol, Jimmy Garnier, Francois Hamel, Lionel Roques. Uniqueness from pointwise observations in a multi-parameter inverse problem. Communications on Pure and Applied Analysis, 2012, 11 (1), pp.173--188. ⟨10.3934/cpaa.2012.11.173⟩. ⟨hal-00596238⟩
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