An Inverse Problem for the Semilinear Wave Equation with a Nonlinear Integral Operator

We consider an integrodifferential equation whose leading part coincides with the wave operator, while the lower part includes the nonlinear term $ q({\mathbf{x}})u^{m} $ with $ m>1 $ and a nonlinear integral operator. This operator models the memory of a medium and includes a variable coefficient $ p({\mathbf{x}}) $ . For the original equation we study the structure of the solution to the Cauchy problem with zero initial data and a pulsating source localized at some point $ ({\mathbf{y}},0) $ of the four-dimensional space of variables $ ({\mathbf{x}},t) $ . We assume that $ q({\mathbf{x}}) $ and $ p({\mathbf{x}}) $ are compactly supported functions with supports lying in the ball $ B_{0} $ centered at the origin and bounded by some sphere $ S_{0} $ , while the point $ {\mathbf{y}} $ lies on a sphere $ S $ concentric with $ S_{0} $ but of larger radius. The point $ {\mathbf{y}} $ is a parameter of the problem and can run over the whole sphere $ S $ . We study the inverse problem of determining the functions $ q({\mathbf{x}}) $ and $ p({\mathbf{x}}) $ on $ B_{0} $ . For that we use the following information. For every point $ {\mathbf{y}} $ on the sphere $ S $ and a point $ {\mathbf{x}} $ on a certain part of the same sphere we define the solution to the Cauchy problem for the original integrodifferential equation for the moments of time close to the arrival of the wave from the source at $ {\mathbf{y}} $ to $ {\mathbf{x}} $ . We show that this inverse problem reduces to two identical integral geometry problems on the family of straight lines with a prescribed weight function invariant under all rotations about the center of the ball $ B_{0} $ . We establish a uniqueness theorem and propose a method for solving these problems.