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

Abstract

Abstract. 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 [6]. This parabolic Carleman inequality enters in an essential way in the construction of CGO solutions that vanish at a part of the lateral boundary.

Recovery of Non-Compactly Supported Coefficients of Elliptic Equations on an Infinite Waveguide

“…In dimension two, similar results with full and partial data have been stated in [5,22,23]. Moreover, without being exhaustive, we refer to the work of [8,9,15,36,43,44] dealing with the stability issue associated with this problem and some results inspired by this approach for other partial differential equations (PDEs) stated in [13,20,[31][32][33]. Let us remark that all the above-mentioned results have been proved in a bounded domain.…”

Determination of non-compactly supported electromagnetic potentials in an unbounded closed waveguide

“…Concerning results with partial data associated with this last problem, we mention the work of [17,18] and concerning the stability issue, without being exhaustive, we refer to [3,6,7,9,39,40,50]. We mention also the work of [12,22,29] related to problems for hyperbolic and parabolic equations treated with an approach similar to the one considered for elliptic equations. Note that all the above mentioned results have been stated in a bounded domain.…”

Determination of singular time-dependent coefficients for wave equations from full and partial data

“…Moreover, singular time-dependent coefficients can be associated to some unstable time-evolving phenomenon that can not be modeled by bounded time-dependent coefficients or time independent coefficients. Let us also observe that, according to [11,27], for parabolic equations the recovery of nonlinear terms, appearing in some suitable nonlinear equations, can be reduced to the determination of time-dependent coefficients. In this context, the information that allows to recover the nonlinear term is transferred, throw a linearization process, to a time-dependent coefficient depending explicitly on some solutions of the nonlinear problem.…”

“…In contrast to parabolic equations, due to the weak regularity of solutions, it is not clear that this process allows to transfer the recovery of nonlinear terms, appearing in a nonlinear wave equation, to a bounded time-dependent coefficient. Thus, in order to expect an application of the strategy set by [11,27] to the recovery of nonlinear terms for nonlinear wave equations, it seems important to consider recovery of singular time-dependent coefficients.…”