Recovering memory kernels in parabolic transmission problems in infinite time intervals: the non-accessible case

In this paper we study two problems concerned with recovering memory kernels related to two sub-bodies $\Omega_1$ and $\Omega_2$ of an open thermal body $\Omega={\overline \Omega_1}\cup\Omega_2$ under the assumptions that $\overline\Omega_1\subset \Omega$ and $\overline\Omega_1$ is not accessible for the measurements. Additional measurements of temperature gradient or flux type are provided on $\partial \Omega$. In the first problem the memory kernel related to $\Omega_1$ is unknown and a single measurement is given. In the second problem both kernels are to be determined from two measurements on $\partial \Omega$. Making use of Laplace transforms, we prove the uniqueness for these identification problems in the infinite time interval $(0,\infty)$.