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Equipartition of energy for nonautonomous damped wave equations. (English) Zbl 1454.34087

Summary: The kinetic and potential energies for the damped wave equation \[ \tag{DWE}u''+2Bu'+A^2u = 0 \] are defined by \[ \begin{aligned} K(t)=\Vert u'(t)\Vert^2,\,P(t)=\Vert Au(t)\Vert^2, \end{aligned} \] where \(A,B\) are suitable commuting selfadjoint operators. Asymptotic equipartition of energy means \[ \tag{AEE}\lim\limits_{t\to\infty} \frac{K(t)}{P(t)}=1 \] for all (finite energy) non-zero solutions of (DWE). The main result of this paper is the proof of a result analogous to (AEE) for a nonautonomous version of (DWE).

MSC:

34G10 Linear differential equations in abstract spaces
35L90 Abstract hyperbolic equations
76D33 Waves for incompressible viscous fluids
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