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Equipartition of energy for nonautonomous wave equations

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

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  • Consider wave equations of the form

    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.


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