Inverse problem for an equation of parabolic-hyperbolic type with a nonlocal boundary condition (Q425234)

The paper deals with equation of parabolic-hyperbolic type \(Lu=f\), \[ Lu= u_t-u_{xx}+b^2 u ,\;t>0, \] \[ Lu= u_{tt}-u{xx} + b^2 u , \;t<0 \] in the rectangular domain \(D= \{ (x, t): \;0<x<1; \;-\alpha <t<\beta \}\), where \(\alpha> 0\) and \(\beta > 0\) are given real numbers. An inverse problem for the above equation with a nonlocal boundary condition relating solution derivatives is considered. A uniqueness criterion is justified and the existence of a solution is proved by the spectral analysis method. The stability of the solution with respect to the nonlocal boundary conditions is proved.