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Equipartition of energy for nonautonomous wave equations

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.

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

 Citation:

•  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. 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. 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. 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. 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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