Identification problems for semilinear integro-differential hyperbolic equations with transformed arguments

The aim of the present paper is to generalize the results in [Denisov, J. Inverse Ill-Posed Probl. 16: 837–848, 2008.] devoted to a one-dimensional semilinear wave equation and consisting of recovering a time-dependent function α representing the transformed argument. More exactly: (i) we will deal with a general semilinear integro-differential hyperbolic d-dimensional equation in divergence form (d = 1, 2, 3); (ii) the space-time set considered here is a smooth cylinder where surface boundary conditions are prescribed; (iii) the term with transformed arguments is allowed to contain integral operators; (iv) the additional information is of integral type. The existence and uniqueness results for our specific problem are deduced as consequences of similar results for an operator integro-differential identification problem in a Hilbert space.