Abstract
The forward and inverse problems are investigated for the quasilinear wave equation \(\square u -q(x)u^{2}-K\ast u=0\) where the kernel \(K(x,t)\) is represented in the form \(K(x,t)=p(x) K_0(t)\) with \(p(x)\) being a continuous function. The inverse problem is devoted to thedetermination of the compact functions \(q(x)\) and \(p(x)\) . Traces of the derivative with respect to \(x\) of two solutions to the forward initial–boundary value problem related to twoarbitrary boundary data are given for \(x=0\) on the finite segment \([0,T]\) as an additional information for the solution to the inverse problem. Theconditions for the unique solvability of the forward problem are found. A local existence anduniqueness theorem is proved for the inverse problem.
