February  2017, 10(1): 75-85. doi: 10.3934/dcdss.2017004

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  December 2016

Fund Project: The third author is supported by CAPES -Brazil grant 12220-13-2

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.
Citation: Gisèle Ruiz Goldstein, Jerome A. Goldstein, Fabiana Travessini De Cezaro. Equipartition of energy for nonautonomous wave equations. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 75-85. doi: 10.3934/dcdss.2017004
References:
[1]

J. L. Doob, Stochastic Processes Reprint of the 1953 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc. , New York, 1990. doi: 10.1007/0-471-52369-0.  Google Scholar

[2]

J. A. Goldstein, An asymptotic property of solutions of wave equations, Proc. Amer. Math. Soc., 23 (1969), 359-363.  doi: 10.1090/S0002-9939-1969-0250125-1.  Google Scholar

[3]

J. A. Goldstein, An asymptotic property of solutions of wave equations, Ⅱ, J. Math. Anal. Appl., 32 (1970), 392-399.  doi: 10.1016/0022-247X(70)90305-7.  Google Scholar

[4]

J. A. Goldstein, Semigroups of Linear Operators and Applications 1$^{st}$ edition, Oxford University Press, New York and Oxford, 1985. doi: 10.1007/0-19-503540-2.  Google Scholar

[5]

J. A. Goldstein and G. Reyes, Equipartition of operator weighted energies in damped wave equations, Asymptotic Anal., 81 (2013), 171-187.   Google Scholar

show all references

References:
[1]

J. L. Doob, Stochastic Processes Reprint of the 1953 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc. , New York, 1990. doi: 10.1007/0-471-52369-0.  Google Scholar

[2]

J. A. Goldstein, An asymptotic property of solutions of wave equations, Proc. Amer. Math. Soc., 23 (1969), 359-363.  doi: 10.1090/S0002-9939-1969-0250125-1.  Google Scholar

[3]

J. A. Goldstein, An asymptotic property of solutions of wave equations, Ⅱ, J. Math. Anal. Appl., 32 (1970), 392-399.  doi: 10.1016/0022-247X(70)90305-7.  Google Scholar

[4]

J. A. Goldstein, Semigroups of Linear Operators and Applications 1$^{st}$ edition, Oxford University Press, New York and Oxford, 1985. doi: 10.1007/0-19-503540-2.  Google Scholar

[5]

J. A. Goldstein and G. Reyes, Equipartition of operator weighted energies in damped wave equations, Asymptotic Anal., 81 (2013), 171-187.   Google Scholar

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