Uniqueness from pointwise observations in a multi-parameter inverse problem

Abstract

In this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coeffcients 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 coeffcients μk(x); our result gives a suffcient condition for the uniqueness of the determination of this polynomial part. This suffcient condition only involves pointwise measurements of the solution u of the reaction-diffusion equation and of its spatial derivative ℓu=ℓx at a single point x0; during a time interval (0; ε): 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 effcient numerical determination of the unknown polynomial reaction term.