Vol. 47, No. 1, 1973

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Energy bounds and virial theorems for abstract wave equations

Larry Eugene Bobisud and James Calvert

Vol. 47 (1973), No. 1, 27–37

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 12Au(t)2 + 12u(t)2. For f 0 it is shown that neither the potential energy 12Au(t)2 nor the kinetic energy 12u(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
Received: 12 May 1972
Published: 1 July 1973
Larry Eugene Bobisud
James Calvert