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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  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 & 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.  Google Scholar

[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.  Google Scholar

[3]

J. L. Doob, Stochastic Processes, John Wiley and Sons, Inc., New York, Chapman and Hall, Ltd., 1953.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. A. Goldstein, Semigroups of Linear Operators and Applications, 2nd edition, Dover Publications, Inc., Mineola, New York, 2017.  Google Scholar

[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.  Google Scholar

[9]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

J. L. Doob, Stochastic Processes, John Wiley and Sons, Inc., New York, Chapman and Hall, Ltd., 1953.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. A. Goldstein, Semigroups of Linear Operators and Applications, 2nd edition, Dover Publications, Inc., Mineola, New York, 2017.  Google Scholar

[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.  Google Scholar

[9]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.  Google Scholar

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