Asymptotic expansion of the L2-norm of a solution of the strongly damped wave equation

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:267

Article

Barrera, Joseph^1,2  Volkmer, Hans² 

[1] Converse Coll, Dept Math & Comp Sci, Spartanburg, SC 29302 USA

[2] Univ Wisconsin, Dept Math Sci, Milwaukee, WI 53201 USA

关键词: Asymptotic analysis;    Asymptotic expansion of L-2-norm;    Wave equation;    Strong damping;    Fourier analysis;    Weighted L-1-initial data;   

DOI  :  10.1016/j.jde.2019.01.028

来源: Elsevier

【 摘 要 】

The Fourier transform, F, on R-N (N >= 3) transforms the Cauchy problem for the strongly damped wave equation u(tt)- Delta(ut)- Delta u = 0 to an ordinary differential equation in time. We let u(t, x) be the solution of the problem given by the Fourier transform, and v(t, xi) be the asymptotic profile of F(u)(t, xi) = (u) over cap (t, xi) found by Ikehata in the paper Asymptotic profiles for wave equations with strong damping (2014). In this paper we study the asymptotic expansions of the squared L-2-norms of u(t, x), (u) over cap (t, xi) - v(t, xi), and v(t, xi) as t -> infinity. With suitable initial data u(0, x) and u(t)(0, x), we establish the rate of decay of the squared L-2-norms of u(t, x) and v(t, xi) as t oo. By noting the cancellation of leading terms of their respective expansions, we conclude that the rate of convergence between u(t, xi) and v(t, xi) in the L-2-norm occurs quickly relative to their individual behaviors. This observation is similar to the diffusion phenomenon, which has been well studied. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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