Asymptotic expansion of the L 2 -norm of a solution of the strongly damped wave equation in space dimension 1 and 2

Abstract

In previous work the authors found the asymptotic expansion of the L 2 -norm of the solution u ( t , x ) of the strongly damped wave equation u t t − Δ u t − Δ u = 0 and also of the L 2 -norm of the difference between u ( t , x ) and its asymptotic approximation ν ( t , x ). This was done in space dimension N ⩾ 3. In the present work results are extended to the exceptional cases N = 1 and N = 2. This extension is achieved by deriving new lemmas on the asymptotic expansion of some parameter dependent integrals.