Inverse problem for a quasilinear wave equation

Abstract

The present paper deals with the proof of the existence and uniqueness of a solution of an inverse problem for a quasilinear wave equation with an unknown coefficient q(x) multiplying a lower-order derivative. The values of the solution of the Cauchy problem for this equation on some curve serve as additional information for the solution of the inverse problem. The proof of the existence and uniqueness of the solution of the inverse problem is based on the reduction of the problem to an integro-functional equation for the unknown function q(x). A similar approach was used in [1, 2] for the analysis of the inverse problem of finding an unknown function occurring in the nonlinear term specifying the source in the wave equation. Inverse problems for the wave equation were considered in a number of papers (e.g., see [3–7]). 1. STATEMENT OF THE PROBLEM AND THE MAIN RESULT Consider the Cauchy problem for the quasilinear wave equation utt(x, t )= a 2 uxx(x, t )+ f (x, ux(x, t)) q(x), (x, t) ∈ ∆d, (1.1) u(x, 0) = ϕ(x) ,u t(x, 0) = ψ(x), 0 ≤ x ≤ d,