Stability estimates for the solution to the problem of determining the kernel of a viscoelastic equation

For an integrodifferential equation corresponding to a two-dimensional viscoelastic problem, we study the problem of defining the spatial part of the kernel involved in the integral term of the equation. The support of the sought function is assumed to belong to a compact domain Ω. As information for solving this inverse problem, the traces of the solution to the direct Cauchy problem and its normal derivative are given for some finite time interval on the boundary of Ω. An important feature in the statement of the problem is the fact that the solution of the direct problem corresponds to the zero initial data and a force impulse in time localized on a fixed straight line disjoint with Ω. The main result of the article consists in obtaining a Lipschitz estimate for the conditional stability of the solution to the inverse problem under consideration.

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