ASYMPTOTIC EXPANSION OF THE L-NORM OF A SOLUTION OF THE STRONGLY DAMPED WAVE EQUATION by Joseph Barrera The University of Wisconsin-Milwaukee, 2017 Under the Supervision of Professor Hans Volkmer The Fourier transform, F , on R (N ≥ 1) transforms the Cauchy problem for the strongly damped wave equation utt − ∆ut − ∆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-norms of u(t, x), û(t, ξ) − ν(t, ξ), and ν(t, ξ) as t → ∞. With suitable initial data u(0, x) and ut(0, x), we establish the rate of growth or decay of the squared L-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 between û(t, ξ) and ν(t, ξ) in the L-norm occurs quickly relative to their individual behaviors. Finally we consider three examples in order to illustrate the results.

中文翻译：

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空间维度 1 和 2 中强阻尼波动方程解的 L 2 -范数的渐近展开

强阻尼波方程解的 L 范数的渐近扩展 Joseph Barrera 威斯康星大学密尔沃基分校，2017 在 Hans Volkmer 教授的指导下 傅里叶变换 F 对 R (N ≥ 1) 进行了柯西变换强阻尼波动方程 utt − ∆ut − ∆u = 0 到时间 t 的常微分方程的问题。我们让 u(t, x) 是傅立叶变换给出的问题的解，而 ν(t, ξ) 是 F(u)(t, ξ) = û(t, ξ) 的渐近轮廓，通过池畑[4]。在本论文中，我们研究了 u(t, x)、û(t, ξ) − ν(t, ξ) 和 ν(t, ξ) 的平方 L 范数在 t → ∞ 时的渐近展开。使用合适的初始数据 u(0, x) 和 ut(0, x)，我们建立 u(t, x) 和 ν(t, ξ) 的平方 L 范数的增长或衰减速率为 t→∞ . 通过注意到它们各自展开的前导项的取消，我们得出结论，L 范数中 û(t, ξ) 和 ν(t, ξ) 之间的收敛速度相对于它们的个体行为发生得很快。最后，我们考虑三个例子来说明结果。