American Institute of Mathematical Sciences

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

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

* 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
 $$$u''+2Bu'+A^2u = 0 \;\;\;\;\;\;({\rm DWE})$$$
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
 $$$\lim\limits_{t\to\infty} \frac{K(t)}{P(t)} = 1 \;\;\;\;\;\;({\rm AEE})$$$
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).
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:
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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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