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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 |
$\begin{align*}u''(t)+ A^2u(t)=0\end{align*}$ |
$\mathcal{H}$ |
$\begin{align*}K(t)= \| u'(t)\|^2, P(t)= \|Au(t)\|^2, E(t) = K(t)+P(t).\end{align*}$ |
$E(t)$ |
$E(t)= E(0)$ |
$\begin{align*}\lim_{t \longrightarrow ± ∞}K(t) = \lim_{t \longrightarrow ± ∞}P(t) = \frac{E(0)}{2}\end{align*}$ |
$e^{itA}\longrightarrow 0$ |
$A$ |
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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