The inverse problem of the operator \(L_q=\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} + q(t)\) in semi-unbounded domain. (Chinese. English summary) Zbl 0928.35205
Summary: The study of the inverse problem of identifying unknown functions \((u,q)\) is studied which satisfy the following problem:
\[
\begin{cases} u_{tt}-u_{xx} +q(t)u= F(x,t),\;x>0,\;t>0,\\ u(x,0)= \varphi(x),\;u_t(x,0)= \psi (x),\;x\geq 0,\\ u(0,t)= f(t),\;t\geq 0,\\ u_x(0,t)=g(t),\;t\geq 0.\end{cases}
\]
By using a fixed point theorem and a priori estimates, existence, uniqueness and stability of the solution are demonstrated.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
