On the structure of the solutions of the first initial boundary value problem for the Sobolev's equation (Q1326730)

Let \(\Omega\) be a plane convex bounded domain with corner points. The Cauchy-Dirichlet problem is considered for the Sobolev equation \[ {\partial^2 \over \partial t^2} \left( {\partial^2u \over \partial x_1^2} + {\partial^2u \over \partial x^2_2} \right) + {\partial^2u \over \partial x^2_2} = 0, \quad x \in \Omega, \;t > 0. \] This problem is reduced to the abstract Cauchy problem for the equation \(p_{tt} = Ap\), where \(A\) is a bounded selfadjoint operator in \(W_0^{1,2}(\Omega)\) with spectrum \(\sigma (A) = [-1,0]\). Under a certain condition on \(\partial \Omega\) it is shown that \(A\) has no eigenfunctions in \(W_0^{1,2}(\Omega)\) but for every point of \(\sigma (A)\) there exists a class of generalized eigenfunctions. The completeness of these generalized eigenfunctions in \(W_0^{1,2}(\Omega)\) is proved. Integral representation for solutions of the above abstract Cauchy problem is obtained.