A spectral approach to ill-posed problems for wave equations

Abstract

In this article, we consider the problem of finding a solution to ill-posed problems for abstract wave equations in a Hilbert space, of the form $$\frac{{\rm d}^{2}u}{{\rm d}t^{2}}\left( t\right) + Au \left(t\right) = 0,\quad t \in \left(0, T\right),\quad u \left(0\right) = 0,\quad u \left(T\right) = u_{0},$$ when A is a general linear selfadjoint operator. We study issues like existence, uniqueness and continuance dependance of data and stability for this problem. Under precise constraint conditions on T, we make such problems well posed and in effect, generalize known results about these equations.