Energy bounds and virial theorems for abstract wave equations
L. E. Bobisud, James Calvert · Pacific Journal of Mathematics · 1973
The abstract wave equation u" -A 2 u + f(t, u) is considered on a Banach or Hubert space, where A generates a (Co) group.Under suitable conditions on /, a representation of the solution of the initial-value problem is used to establish bounds on the growth of the energy 1/211 Au(t) 11 2 + 1/211 u'(t) | | 2 .For /Ξ= 0 it is shown that neither the potential energy 1/211 Au(t) 11 2 nor the kinetic energy 1/21| u f (t) || 2 tends to zero as £->oo, and necessary and sufficient conditions for the kinetic and potential energies to be equal for large time are given.