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Equipartition of energy for nonautonomous damped wave equations
1. | Dipartimento di Matematica, University of Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy |
2. | Department of Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis, TN 38152-3240, USA |
$ \begin{equation} u''+2Bu'+A^2u = 0 \;\;\;\;\;\;({\rm DWE})\end{equation} $ |
$ K(t) = \Vert u'(t)\Vert^2,\, P(t) = \Vert Au(t)\Vert^2, $ |
$ A,B $ |
$\begin{equation} \lim\limits_{t\to\infty} \frac{K(t)}{P(t)} = 1 \;\;\;\;\;\;({\rm AEE})\end{equation}$ |
References:
[1] |
M. D'Abbicco, M. R. Ebert and S. Lucente,
Self-similar asymptotic profile of the solution to a nonlinear evolution equation with critical dissipation, Math. Methods Appl. Sci., 40 (2017), 6480-6494.
doi: 10.1002/mma.4469. |
[2] |
M. D'Abbicco, G. Girardi and M. Reissig,
A scale of critical exponents for semilinear waves with time-dependent damping and mass terms, Nonlinear Anal., 179 (2019), 15-40.
doi: 10.1016/j.na.2018.08.006. |
[3] |
J. L. Doob, Stochastic Processes, John Wiley and Sons, Inc., New York, Chapman and Hall, Ltd., 1953. |
[4] |
G. R. Goldstein, J. A. Goldstein and F. Travessini,
Equipartition of energy for nonautonomous wave equations, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 75-85.
doi: 10.3934/dcdss.2017004. |
[5] |
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. |
[6] |
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. |
[7] |
J. A. Goldstein, Semigroups of Linear Operators and Applications, 2nd edition, Dover Publications, Inc., Mineola, New York, 2017. |
[8] |
J. A. Goldstein and G. Reyes,
Equipartition of operator-weighted energies in damped wave equations, Asymptot. Anal., 81 (2013), 171-187.
doi: 10.3233/ASY-2012-1124. |
[9] |
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. |
show all references
References:
[1] |
M. D'Abbicco, M. R. Ebert and S. Lucente,
Self-similar asymptotic profile of the solution to a nonlinear evolution equation with critical dissipation, Math. Methods Appl. Sci., 40 (2017), 6480-6494.
doi: 10.1002/mma.4469. |
[2] |
M. D'Abbicco, G. Girardi and M. Reissig,
A scale of critical exponents for semilinear waves with time-dependent damping and mass terms, Nonlinear Anal., 179 (2019), 15-40.
doi: 10.1016/j.na.2018.08.006. |
[3] |
J. L. Doob, Stochastic Processes, John Wiley and Sons, Inc., New York, Chapman and Hall, Ltd., 1953. |
[4] |
G. R. Goldstein, J. A. Goldstein and F. Travessini,
Equipartition of energy for nonautonomous wave equations, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 75-85.
doi: 10.3934/dcdss.2017004. |
[5] |
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. |
[6] |
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. |
[7] |
J. A. Goldstein, Semigroups of Linear Operators and Applications, 2nd edition, Dover Publications, Inc., Mineola, New York, 2017. |
[8] |
J. A. Goldstein and G. Reyes,
Equipartition of operator-weighted energies in damped wave equations, Asymptot. Anal., 81 (2013), 171-187.
doi: 10.3233/ASY-2012-1124. |
[9] |
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. |
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