Energy bounds and virial theorems for abstract wave equations

Larry Eugene Bobisud and James Calvert

Vol. 47 (1973), No. 1, 27–37

DOI: 10.2140/pjm.1973.47.27

Abstract

The abstract wave equation u′′ = A²u + f(t,u) is considered on a Banach or Hilbert space, where A generates a (C₀) group. Under suitable conditions on f, a representation of the solution of the initial-value problem is used to establish bounds on the growth of the energy 1∕2∥Au(t)∥² + 1∕2∥u′(t)∥². For f ≡ 0 it is shown that neither the potential energy 1∕2∥Au(t)∥² nor the kinetic energy 1∕2∥u′(t)∥² tends to zero as t →∞, and necessary and sufficient conditions for the kinetic and potential energies to be equal for large time are given.

Mathematical Subject Classification 2000

Primary: 35R20

Secondary: 47D05, 34G05

Milestones

Received: 12 May 1972

Published: 1 July 1973