Recovering memory kernels in parabolic transmission problems

In this paper we recover two unknown kernels related to a thermal body W with memory consisting of two different sub-bodies W1 and W2, when the boundaries of W1 and W2 have a common (closed) surface G intersecting the boundary @W of W. The additional measurements are performed on two (accessible) subsets of @W1 and @W2. For this problem we prove existence, uniqueness and continuous dependence on the data in the framework of Sobolev spaces of L2-type in space.