Abstract linear hyperbolic equations with variable domain (Q795982)

This paper provides new results concerning the existence, uniqueness and regularity of the solution to the abstract Cauchy problem \(u''(s)+A(s)u(s)=0,\) \(s\in [0,T]\), \(u(0)=u_ 0\), \(u'(0)=u_ 1\). A(s) denotes for a.a. \(s\in [0,T]\) a self-adjoint positive operator on \(D\{\) A(s)\(\} \subset H\) onto H with everywhere defined inverse \(A^{-1}(s).\) The author treats the case in which the domain D[A(s)] varies with the time parameter s and gives an example: the vibrating membrane with moving Dirichlet-Neumann boundary conditions. On the other hand, he proves new results also for the case when D[A(s)]\(\equiv const\). As an application, he proves some regularity properties of the solution to the mixed problem for a linear second order hyperbolic partial differential equation. This paper provides a very large bibliography.