American Institute of Mathematical Sciences

Advanced
February  2017, 10(1): 75-85. doi: 10.3934/dcdss.2017004

Equipartition of energy for nonautonomous wave equations

Gisèle Ruiz Goldstein 1, , Jerome A. Goldstein 2, and  Fabiana Travessini De Cezaro 3,

1. 

Department of Mathematical Sciences, The University of Memphis, Dunn Hall, 337, Memphis, TN 38152, USA
2. 

Department of Mathematical Sciences, The University of Memphis, Dunn Hall, 343, Memphis, TN 38152, USA
3. 

Department of Mathematics, Statistics and Physics, Federal University of Rio Grande, Av. Italia, Km 08, Campus Carreiros, Rio Grande, RS 96203-900, Brazil

Received  March 2015 Revised  September 2015 Published  December 2016

Fund Project: The third author is supported by CAPES -Brazil grant 12220-13-2.

Consider wave equations of the form
$\begin{align*}u''(t)+ A^2u(t)=0\end{align*}$
with $A$ an injective selfadjoint operator on a complex Hilbert space
$\mathcal{H}$
. The kinetic, potential, and total energies of a solution $u$ are
$\begin{align*}K(t)= \| u'(t)\|^2, P(t)= \|Au(t)\|^2, E(t) = K(t)+P(t).\end{align*}$
Finite energy solutions are those mild solutions for which
$E(t)$
 is finite. For such solutions
$E(t)= E(0)$
, that is, energy is conserved, and asymptotic equipartition of energy
$\begin{align*}\lim_{t \longrightarrow ± ∞}K(t) = \lim_{t \longrightarrow ± ∞}P(t) = \frac{E(0)}{2}\end{align*}$
holds for all finite energy mild solutions iff
$e^{itA}\longrightarrow 0$
 in the weak operator topology. In this paper we present the first extension of this result to the case where
$A$
 is time dependent.
Keywords: Equipartition of energy, nonautonomous system, asymptotics, wave equations.
Mathematics Subject Classification: Primary: 34G10, 35L90; Secondary: 76D33.
Citation: Gisèle Ruiz Goldstein, Jerome A. Goldstein, Fabiana Travessini De Cezaro. Equipartition of energy for nonautonomous wave equations. Discrete and Continuous Dynamical Systems - S, 2017, 10 (1) : 75-85. doi: 10.3934/dcdss.2017004
References:
[1]

J. L. Doob, Stochastic Processes Reprint of the 1953 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc. , New York, 1990. doi: 10.1007/0-471-52369-0.

[2]

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.

[3]

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.

[4]

J. A. Goldstein, Semigroups of Linear Operators and Applications 1$^{st}$ edition, Oxford University Press, New York and Oxford, 1985. doi: 10.1007/0-19-503540-2.

[5]

J. A. Goldstein and G. Reyes, Equipartition of operator weighted energies in damped wave equations, Asymptotic Anal., 81 (2013), 171-187. 

show all references

References:
[1]

J. L. Doob, Stochastic Processes Reprint of the 1953 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc. , New York, 1990. doi: 10.1007/0-471-52369-0.

[2]

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.

[3]

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.

[4]

J. A. Goldstein, Semigroups of Linear Operators and Applications 1$^{st}$ edition, Oxford University Press, New York and Oxford, 1985. doi: 10.1007/0-19-503540-2.

[5]

J. A. Goldstein and G. Reyes, Equipartition of operator weighted energies in damped wave equations, Asymptotic Anal., 81 (2013), 171-187. 

[1]

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
[2]

Ioana Moise, Ricardo Rosa, Xiaoming Wang. Attractors for noncompact nonautonomous systems via energy equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 473-496. doi: 10.3934/dcds.2004.10.473
[3]

Simone Paleari, Tiziano Penati. Equipartition times in a Fermi-Pasta-Ulam system. Conference Publications, 2005, 2005 (Special) : 710-719. doi: 10.3934/proc.2005.2005.710
[4]

Salim A. Messaoudi, Ala A. Talahmeh. Blow up of negative initial-energy solutions of a system of nonlinear wave equations with variable-exponent nonlinearities. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1233-1245. doi: 10.3934/dcdss.2021107
[5]

Dandan Li. Asymptotics of singularly perturbed damped wave equations with super-cubic exponent. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 583-600. doi: 10.3934/dcdsb.2021056
[6]

Kazumasa Fujiwara, Shuji Machihara, Tohru Ozawa. On a system of semirelativistic equations in the energy space. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1343-1355. doi: 10.3934/cpaa.2015.14.1343
[7]

Carlos E. Kenig. The method of energy channels for nonlinear wave equations. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6979-6993. doi: 10.3934/dcds.2019240
[8]

Geng Chen, Ping Zhang, Yuxi Zheng. Energy conservative solutions to a nonlinear wave system of nematic liquid crystals. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1445-1468. doi: 10.3934/cpaa.2013.12.1445
[9]

Zhaoquan Xu, Jiying Ma. Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 5107-5131. doi: 10.3934/dcds.2015.35.5107
[10]

Ludwig Gauckler, Daniel Weiss. Metastable energy strata in numerical discretizations of weakly nonlinear wave equations. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3721-3747. doi: 10.3934/dcds.2017158
[11]

Ryo Ikehata, Shingo Kitazaki. Optimal energy decay rates for some wave equations with double damping terms. Evolution Equations and Control Theory, 2019, 8 (4) : 825-846. doi: 10.3934/eect.2019040
[12]

Petronela Radu, Grozdena Todorova, Borislav Yordanov. Higher order energy decay rates for damped wave equations with variable coefficients. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 609-629. doi: 10.3934/dcdss.2009.2.609
[13]

Carlos E. Kenig, Frank Merle. Radial solutions to energy supercritical wave equations in odd dimensions. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1365-1381. doi: 10.3934/dcds.2011.31.1365
[14]

Kazuhiro Kurata, Yuki Osada. Asymptotic expansion of the ground state energy for nonlinear Schrödinger system with three wave interaction. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4239-4251. doi: 10.3934/cpaa.2021157
[15]

Thomas Duyckaerts, Carlos E. Kenig, Frank Merle. Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1275-1326. doi: 10.3934/cpaa.2015.14.1275
[16]

Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631
[17]

Genni Fragnelli, Dimitri Mugnai. Nonlinear delay equations with nonautonomous past. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1159-1183. doi: 10.3934/dcds.2008.21.1159
[18]

Ge Zu, Bin Guo. Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy. Evolution Equations and Control Theory, 2021, 10 (2) : 259-270. doi: 10.3934/eect.2020065
[19]

Bilgesu A. Bilgin, Varga K. Kalantarov. Non-existence of global solutions to nonlinear wave equations with positive initial energy. Communications on Pure and Applied Analysis, 2018, 17 (3) : 987-999. doi: 10.3934/cpaa.2018048
[20]

Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control and Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45

2021 Impact Factor: 1.865

Tools

Metrics

  • PDF downloads (184)
  • HTML views (118)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy