An Inverse Problem for a Semilinear Wave Equation
V. G. Romanov
Abstract
Abstract For the equation $${{u}_{{tt}}} - \Delta u - f(x,u) = 0, (x,t) \in {{\mathbb{R}}^{4}},$$ where $$f(x,u)$$ is a smooth function of its variables and is compact in x , the inverse problem of recovering this function from given information on solutions of Cauchy problems for the differential equation is studied. Plane waves with a strong front that propagate in a homogeneous medium in the direction of the unit vector ν and then impinge on an inhomogeneity localized inside some ball B ( R ) are considered. It is supposed that the solutions of the Cauchy problems can be measured on the boundary of this ball for all ν at times close to the arriving time of the front. The forward Cauchy problem is studied, and the existence of a unique bounded solution in a neighborhood of a characteristic wedge is stated. An amplitude formula for the derivative of the solution with respect to t on the front of the wave is derived. It is demonstrated that the solution of the inverse problem reduces to a series of X-ray tomography problems.