# American Institute of Mathematical Sciences

## 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, 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
 [1] Gisèle Ruiz Goldstein, Jerome A. Goldstein, Fabiana Travessini De Cezaro. Equipartition of energy for nonautonomous wave equations. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 75-85. doi: 10.3934/dcdss.2017004 [2] Ioana Moise, Ricardo Rosa, Xiaoming Wang. Attractors for noncompact nonautonomous systems via energy equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 473-496. doi: 10.3934/dcds.2004.10.473 [3] Petronela Radu, Grozdena Todorova, Borislav Yordanov. Higher order energy decay rates for damped wave equations with variable coefficients. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 609-629. doi: 10.3934/dcdss.2009.2.609 [4] Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations & Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017 [5] Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure & Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921 [6] Mohammed Aassila. On energy decay rate for linear damped systems. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 851-864. doi: 10.3934/dcds.2002.8.851 [7] John M. Ball. Global attractors for damped semilinear wave equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31 [8] Lili Fan, Hongxia Liu, Huijiang Zhao, Qingyang Zou. Global stability of stationary waves for damped wave equations. Kinetic & Related Models, 2013, 6 (4) : 729-760. doi: 10.3934/krm.2013.6.729 [9] P. Fabrie, C. Galusinski, A. Miranville. Uniform inertial sets for damped wave equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 393-418. doi: 10.3934/dcds.2000.6.393 [10] Alexandre Nolasco de Carvalho, Jan W. Cholewa, Tomasz Dlotko. Damped wave equations with fast growing dissipative nonlinearities. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1147-1165. doi: 10.3934/dcds.2009.24.1147 [11] Carlos E. Kenig. The method of energy channels for nonlinear wave equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6979-6993. doi: 10.3934/dcds.2019240 [12] Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065 [13] Bouthaina Abdelhedi. Existence of periodic solutions of a system of damped wave equations in thin domains. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 767-800. doi: 10.3934/dcds.2008.20.767 [14] Feng Zhou, Chunyou Sun, Xin Li. Dynamics for the damped wave equations on time-dependent domains. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1645-1674. doi: 10.3934/dcdsb.2018068 [15] Filippo Dell'Oro. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1015-1027. doi: 10.3934/cpaa.2013.12.1015 [16] Kyouhei Wakasa. The lifespan of solutions to semilinear damped wave equations in one space dimension. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1265-1283. doi: 10.3934/cpaa.2016.15.1265 [17] Sergey Zelik. Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 351-392. doi: 10.3934/dcds.2004.11.351 [18] Shengfan Zhou, Linshan Wang. Kernel sections for damped non-autonomous wave equations with critical exponent. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 399-412. doi: 10.3934/dcds.2003.9.399 [19] Veronica Belleri, Vittorino Pata. Attractors for semilinear strongly damped wave equations on $\mathbb R^3$. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 719-735. doi: 10.3934/dcds.2001.7.719 [20] Hiroshi Takeda. Global existence of solutions for higher order nonlinear damped wave equations. Conference Publications, 2011, 2011 (Special) : 1358-1367. doi: 10.3934/proc.2011.2011.1358

2018 Impact Factor: 0.545

Article outline