Equipartition of energy for nonautonomous damped wave equations

Marcello D'Abbicco; Giovanni Girardi; Giséle Ruiz Goldstein; Jerome A. Goldstein; Silvia Romanelli

doi:10.3934/dcdss.2020364

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February 2021, 14(2): 597-613. doi: 10.3934/dcdss.2020364

Equipartition of energy for nonautonomous damped wave equations

Marcello D'Abbicco ^1, , Giovanni Girardi ^1, , Giséle Ruiz Goldstein ^2, , Jerome A. Goldstein ^2,, and Silvia Romanelli ^1,

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

^* Corresponding author: Jerome A. Goldstein

Dedicated to Michel Pierre on his seventieth birthday

Received December 2019 Published February 2021 Early access May 2020

The kinetic and potential energies for the damped wave equation

$ \begin{equation} u''+2Bu'+A^2u = 0 \;\;\;\;\;\;({\rm DWE})\end{equation} $

are defined by

$ K(t) = \Vert u'(t)\Vert^2,\, P(t) = \Vert Au(t)\Vert^2, $

where

$ A,B $

are suitable commuting selfadjoint operators. Asymptotic equipartition of energy means

$\begin{equation} \lim\limits_{t\to\infty} \frac{K(t)}{P(t)} = 1 \;\;\;\;\;\;({\rm AEE})\end{equation}$

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).

Keywords: Equipartition of energy, nonautonomous systems, asymptotics, damped wave equations.

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

Citation: Marcello D'Abbicco, Giovanni Girardi, Giséle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Equipartition of energy for nonautonomous damped wave equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 597-613. doi: 10.3934/dcdss.2020364

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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