The energy space and norm growth for abstract wave equations

For the wave equation α2u αt2 = Σ i=i N α2u αx(i)α2 (for t ε R and ξ ε R None could think that the natural associated energy space might be K:= H 1(R N) × L 2(R N). This is misleading and only partially correct. The purpose of this paper is to explain the role of the energy spaces associated with a wave equation. This is done in an abstract context, when the negative Laplacian is replaced by an arbitrary nonnegative self-adjoint operator on a Hilbert space. For the wave equation on κ, the norm of the governing group of operators is shown to grow linearly in time (as t → ±∞). © 2003 Elsevier Science Ltd. All rights reserved