Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term

Mourad Choulli (1) , Yavar Kian (2, 3)

1 IECL - Institut Élie Cartan de Lorraine
2 CPT - Centre de Physique Théorique - UMR 7332
3 CPT - E8 Dynamique quantique et analyse spectrale

We give a new stability estimate for the problem of determining the time-dependent zero order coefficient in a parabolic equation from a partial parabolic Dirichlet-to-Neumann map. The novelty of our result is that, contrary to the previous works, we do not need any measurement on the final time. We also show how this result can be used to establish a stability estimate for the problem of determining the nonlinear term in a semilinear parabolic equation from the corresponding " linearized " Dirichlet-to-Neumann map. The key ingredient in our analysis is a parabolic version of an elliptic Carleman inequality due to Bukhgeim and Uhlmann. This parabolic Carleman inequality enters in an essential way in the construction of Complex geometric optics solutions that vanish at a part of the lateral boundary.

Mots clés: Parabolic equation; Carleman inequality; partial Dirichlet-to-Neumann map; semilinear parabolic equation; logarithmic stability

Domaines: Equations aux dérivées partielles [math.AP]

Mourad Choulli, Yavar Kian. Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term. Journal de Mathématiques Pures et Appliquées, 2018, 114, pp.235-261. ⟨10.1016/j.matpur.2017.12.003⟩. ⟨hal-01322796⟩