Almost-periodic solutions of a class of linear hyperbolic equations (Q1905273)

S. L. Sobolev proved in 1945 that all solutions of the problem \(u_{tt}(x, t)+ (- \Delta + q(x)) u(x, t)= 0\), \(u|_{\partial\Omega}= 0\), where \(\Omega\subset \mathbb{R}^n\) is a bounded region, \(q(x)\geq 0\), \(x\in \Omega\), \(u|_{t= 0}\in H_1(\Omega)\), \(u_t|_{t= 0}\in L_2(\Omega)\), belong to the class of almost periodic functions with values in the energy field. In this paper the operator \(-\Delta+ q(x)\) is replaced by an abstract selfadjoint operator \(A\) in the Hilbert space and the problem of almost periodic solutions of the corresponding equation is considered.