Equipartition of energy for nonautonomous wave equations

Gisèle Ruiz Goldstein; Jerome A. Goldstein; Fabiana Travessini De Cezaro

doi:10.3934/dcdss.2017004

Article Contents

2017, Volume 10, Issue 1: 75-85. Doi: 10.3934/dcdss.2017004

This issue Previous Article A note on $3$d-$1$d dimension reduction with differential constraints Next Article Boundary control of the number of interfaces for the one-dimensional Allen-Cahn equation

Equipartition of energy for nonautonomous wave equations

1.
Department of Mathematical Sciences, The University of Memphis, Dunn Hall, 337, Memphis, TN 38152, USA
2.
Department of Mathematical Sciences, The University of Memphis, Dunn Hall, 343, Memphis, TN 38152, USA
3.
Department of Mathematics, Statistics and Physics, Federal University of Rio Grande, Av. Italia, Km 08, Campus Carreiros, Rio Grande, RS 96203-900, Brazil

Received: March 2015

Revised: September 2015

Published: February 2018

The third author is supported by CAPES -Brazil grant 12220-13-2.

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Abstract

Consider wave equations of the form

$\begin{align*}u''(t)+ A^2u(t)=0\end{align*}$

with $A$ an injective selfadjoint operator on a complex Hilbert space $\mathcal{H}$. The kinetic, potential, and total energies of a solution $u$ are

$\begin{align*}K(t)= \| u'(t)\|^2, P(t)= \|Au(t)\|^2, E(t) = K(t)+P(t).\end{align*}$

Finite energy solutions are those mild solutions for which $E(t)$ is finite. For such solutions $E(t)= E(0)$, that is, energy is conserved, and asymptotic equipartition of energy

$\begin{align*}\lim_{t \longrightarrow ± ∞}K(t) = \lim_{t \longrightarrow ± ∞}P(t) = \frac{E(0)}{2}\end{align*}$

holds for all finite energy mild solutions iff $e^{itA}\longrightarrow 0$ in the weak operator topology. In this paper we present the first extension of this result to the case where $A$ is time dependent.

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Mathematics Subject Classification: Primary: 34G10, 35L90; Secondary: 76D33.

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References

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