Coefficient identification in parabolic equations with final data

Faouzi Triki (1)

1 EDP - Equations aux Dérivées Partielles

In this work we determine the second-order coefficient in a parabolic equation from the knowledge of a single final data. Under assumptions on the concentration of eigenvalues of the associated elliptic operator, and the initial state, we show the uniqueness of solution, and we derive a Lipschitz stability estimate for the inversion when the final time is large enough. The Lipschitz stability constant grows exponentially with respect to the final time, which makes the inversion ill-posed. The proof of the stability estimate is based on a spectral decomposition of the solution to the parabolic equation in terms of the eigenfunctions of the associated elliptic operator, and an ad hoc method to solve a nonlinear stationary transport equation that is itself of interest.

Mots clés: Inverse problems; Parabolic equation; Final data; Lipschitz stability estimate

Domaines: Equations aux dérivées partielles [math.AP]

Faouzi Triki. Coefficient identification in parabolic equations with final data. Journal de Mathématiques Pures et Appliquées, 2021, 148, pp.342-359. ⟨10.1016/j.matpur.2021.02.004⟩. ⟨hal-03606729⟩