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.
The University of Wisconsin-Milwaukee, 2017 Under the Supervision of Professor Hans Volkmer The Fourier transform, F, on R N (N ≥ 1) transforms the Cauchy problem for the strongly damped wave equation u tt − ∆u t − ∆u = 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 ν(t, ξ) be the asymptotic profile of F(u)(t, ξ) =û(t, ξ) found by Ikehata in [4].In this thesis we study the asymptotic expansions of the squared L 2 -norms of u(t, x), u(t, ξ) − ν(t, ξ), and ν(t, ξ) 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 L 2 -norms of u(t, x) and ν(t, ξ) as t → ∞.By noting the cancellation of leading terms of their respective expansions, we conclude that the rate of convergence betweenû(t, ξ) and ν(t, ξ) in the L 2 -norm occurs quickly relative to their individual behaviors. Finally we consider three examples in order to illustrate the results.ii Conclusion 129Bibliography 132 Appendix: Computing B m,n ,B m,n , and C m,n with Mathematica 134Curriculum Vitae 136
<p style='text-indent:20px;'>The Fourier transform, <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{F} $\end{document}</tex-math></inline-formula>, on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M3">\begin{document}$ (N \geq 1) $\end{document}</tex-math></inline-formula> transforms the Cauchy problem for a strongly damped beam equation with structural damping <inline-formula><tex-math id="M4">\begin{document}$ u_{tt} - \Delta u_t + \alpha (\Delta^2)u - \Delta u = 0, \ \alpha \geq 0 $\end{document}</tex-math></inline-formula>, to an ordinary differential equation in time. With <inline-formula><tex-math id="M5">\begin{document}$ u(t, x) $\end{document}</tex-math></inline-formula> being the weak solution of the problem given by the Fourier transform, the goal of the paper is to determine the asymptotic expansion of the squared <inline-formula><tex-math id="M6">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm of <inline-formula><tex-math id="M7">\begin{document}$ u(t, x) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M8">\begin{document}$ t \to \infty $\end{document}</tex-math></inline-formula>. With suitable, additional assumptions on the initial data <inline-formula><tex-math id="M9">\begin{document}$ u(0, x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ u_t(0, x) $\end{document}</tex-math></inline-formula>, we establish the behavior of the squared <inline-formula><tex-math id="M11">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm of <inline-formula><tex-math id="M12">\begin{document}$ u(t, x) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M13">\begin{document}$ t \to \infty $\end{document}</tex-math></inline-formula>.</p>
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