On the Non-local Problem for a Boussinesq Type Equations

Abstract

The problem of finding a solution, satisfying the non-local condition \(u(\xi_{0})=\alpha u(+0)+\varphi\) in time for the Boussinesq type equation of the form \(u_{tt}+Au_{tt}+Au=f\) is studied in the article. Here \(\alpha\) and \(\xi_{0}\), \(\xi_{0}\in(0,T],\) are the given numbers, \(A:H\rightarrow H\) is the self-adjoint, unbounded, positive operator defined in the Hilbert separable space \(H\). By using the Fourier method, it was shown that the solution to the problem exists and is unique. The effect of parameter \(\alpha\) on the existence and uniqueness of the solution is studied in the article. The inverse problem of determining the right-hand side of the equation is also considered.