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

We consider an inverse boundary value problem for the heat equation ${\mathrm{\partial }}_{t}v={\mathrm{d}\mathrm{i}\mathrm{v}}_x(\gamma {\mathrm{\nabla }}_{x}v)$ in $(0,T)\times \mathrm{\Omega }$, where $\mathrm{\Omega }$ is a bounded domain of $R^3$, the heat conductivity $\gamma (t,x)$ admits a surface of discontinuity which depends on time and without any spatial smoothness. The reconstruction and, implicitly, uniqueness of the moving inclusion, from the knowledge of the Dirichlet-to-Neumann operator, is realised by a dynamical probe method based on the construction of fundamental solutions of the elliptic operator $-\mathrm{\Delta }+{\tau }^2\cdot$, where $\tau$ is a large real parameter, and a couple of inequalities relating data and integrals on the inclusion, which are similar to the elliptic case. That these solutions depend not only on the pole of the fundamental solution, but on the large parameter $\tau$ also, allows the method to work in the very general situation.