On some class of the pseudohyperbolic equations with an unknown coefficient

We consider the inverse problems with an unknown coefficient $a$ for a pseudohyperbolic equation $u_{tt}-\Delta u-\Delta u_{tt}+au=f(x,t)$ and investigate the solvability of the problem. We prove the theorems of the existence of the problem's regular solutions. The distinctiveness of the problems is a presence of new overdetermination conditions for the considered class of equations.