Conditional stability and uniqueness for determining two coefficients in a hyperbolic–parabolic system

Abstract

We study the inverse problem of determining two spatially varying coefficients in a thermoelastic model with the following observation data: displacement in a subdomain ω satisfying ∂ω ⊃ ∂ along a sufficiently large time interval, both displacement and temperature at a suitable time over the whole spatial domain.Based on a Carleman estimate on the hyperbolic-parabolic system, we prove the Lipschitz stability and the uniqueness for this inverse problem under some a priori information.

Carleman estimate for Biot consolidation system in poro‐elasticity and application to inverse problems

“…The main subject of this paper is the inverse problem of determining not only a physical parameter arising in connection with secondary consolidation effects λ ∗ but also the two spatially varying densities ϱ 1 and ϱ 2 , in the Biot consolidation model in poro‐elasticity, uniquely from observed data of displacement vector u and the temperature θ on a suitable subdomain ω ⊂Ω and the observation data of u and θ at given a suitable time t 0 . Such kinds of observation data are similar to those considered in, for example, , which are typical for obtaining the corresponding stability results by the Carleman estimates.…”

“…and the observation data of u and  at given a suitable time t 0 . Such kinds of observation data are similar to those considered in, for example, [3,4], which are typical for obtaining the corresponding stability results by the Carleman estimates.…”

“…Then, in a thermoelastic model, B. Wu and J. Liu [4] study the inverse problem of determining two spatially varying coefficients with the following observation data: displacement in a subdomain ! satisfying @!…”

“…Then, in a thermoelastic model, B. Wu and J. Liu study the inverse problem of determining two spatially varying coefficients with the following observation data: displacement in a subdomain ω satisfying ∂ ω ⊃ ∂ Ω along a sufficiently large time interval and both displacement and temperature at a suitable time over the whole spatial domain. Based on a Carleman estimate on the hyperbolic–parabolic system, B. Wu and J. Liu prove the Lipschitz stability and the uniqueness for this inverse problem under some a priori information.…”

Uniqueness and stability of an inverse problem for a phase field model using data from one component

“…[5,15] or [16]. We think that it is acceptable, because less measurement data is more helpful to applications in practice.…”