Asymptotic Expansion of the L^2-norm of a Solution of the Strongly Damped Wave Equation

Abstract

The Fourier transform, F, on R^N (N≥1) transforms the Cauchy problem for the strongly damped wave equation u_tt(t,x) - Δu_t(t,x) - Δu(t,x) = 0 to an ordinary differential equation in time t. We let u(t,x) be the solution of the problem given by the Fourier transform, and v(t,ƺ) be the asymptotic profile of F(u)(t,ƺ) = û(t,ƺ) found by Ikehata in [4]. In this thesis we study the asymptotic expansions of the squared L^2-norms of u(t,x), û(t,ƺ) - v(t,ƺ), and v(t,ƺ) as t → ∞. With suitable initial data u(0,x) and u_t(0,x), we establish the rate of growth or decay of the squared L2-norms of u(t,x) and v(t,ƺ) as t → ∞. By noting the cancellation of leading terms of their respective expansions, we conclude that the rate of convergence between û(t,ƺ) and v(t,ƺ) in the L^2-norm occurs quickly relative to their individual behaviors. Finally we consider three examples in order to illustrate the results.