# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [5] J. A. Goldstein and G. Reyes, Equipartition of operator weighted energies in damped wave equations, Asymptotic Anal., 81 (2013), 171-187.

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