On some differential equations arising in time domain scattering problems for a dissipative wave equation

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Abstract
The problem of identification of one spatially varying material property, defined within a slab, from boundary measurements is examined. This inverse problem is described by a Banach-valued Volterra integro-differential equation. Uniqueness and existence of the solution of this inverse problem is proven. The associated direct problem is examined and global existence, uniqueness and continuous dependence of the solution is proven. Of major importance in any inverse problem are the properties of the operator mapping the boundary measurements to the material property. It is shown that this operator is continuous and differentiable.

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