Date of Award
May 2017
Degree Type
Dissertation
Degree Name
Doctor of Philosophy
Department
Mathematics
First Advisor
Hans Volkmer
Committee Members
Dashan Fan, Peter Hinow, Lijing Sun, Bruce Wade
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.
Recommended Citation
Barrera, Joseph Silvio, "Asymptotic Expansion of the L^2-norm of a Solution of the Strongly Damped Wave Equation" (2017). Theses and Dissertations. 1445.
https://dc.uwm.edu/etd/1445