Recovering time-dependent inclusion in heat-conductive bodies using a dynamical probe method

We consider an inverse boundary value problem for the heat equation: ∂tv=divx(γ∇xv) in (0,T)×Ω(0,T)×Ω, where Ω is a bounded domain of R3R3, and the heat conductivity γ(t,x)γ(t,x) admits a surface of discontinuity, which depends on time without any spatial smoothness. The reconstruction and, implicitly, the uniqueness of the moving inclusion based on the knowledge of the Dirichlet-to-Neumann operator is achieved using a dynamical probe method according to the construction of fundamental solutions of the elliptic operator −Δ+τ2⋅−Δ+τ2⋅, where τ is a large real parameter, and a pair of inequalities relate the data and integrals on the inclusion, in a similar manner to the elliptic case. These solutions depend on the pole of the fundamental solution but also on the large parameter τ, which allows the method to be applied in a very general situation.