The Inverse Problem for a Quasilinear Wave Equation withMemory

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 the determination 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 two arbitrary boundary data are given for $$ x=0 $$ on the finite segment $$ [0,T] $$ as an additional information for the solution to the inverse problem. The conditions for the unique solvability of the forward problem are found. A local existence and uniqueness theorem is proved for the inverse problem.