American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 197-206. doi: 10.3934/proc.2013.2013.197

Decay property of regularity-loss type for quasi-linear hyperbolic systems of viscoelasticity

Priyanjana M. N. Dharmawardane 1,

1. 

Graduate School of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

Received  August 2012 Revised  December 2012 Published  November 2013

In this paper, we consider a quasi-linear hyperbolic systems of viscoelasticity. This system has dissipative properties of the memory type and the friction type. The decay property of this system is of the regularity-loss type. To overcome the difficulty caused by the regularity-loss property, we employ a special time-weighted energy method. Moreover, we combine this time-weighted energy method with the semigroup argument to obtain the global existence and sharp decay estimate of solutions under the smallness conditions and enough regularity assumptions on the initial data.
Keywords: semigroup argument, time-weighted energy method, Viscoelasticity, decay property., dissipative terms.
Mathematics Subject Classification: Primary: 35D30, 35L52; Secondary:74D1.
Citation: Priyanjana M. N. Dharmawardane. Decay property of regularity-loss type for quasi-linear hyperbolic systems of viscoelasticity. Conference Publications, 2013, 2013 (special) : 197-206. doi: 10.3934/proc.2013.2013.197
References:
[1]

P. M. N. Dharmawardane, Global solutions and decay property of regularity-loss type for quasi-linear hyperbolic systems with dissipation, J. Hyperbolic Differ. Equ. 10 (2013), 37-76.

[2]

P. M. N. Dharmawardane, T. Nakamura and S. Kawashima, Decay estimates of solutions for quasi-linear hyperbolic systems of viscoelasticity, SIAM J. Math. Anal. 44 (2012), 1976-2001.

[3]

P. M. N. Dharmawardane, T. Nakamura and S. Kawashima, Global solutions to quasi-linear hyperbolic systems of viscoelasticity, Kyoto J. Math. 51 (2011), 467-483.

[4]

P. M. N. Dharmawardane, J. E. Muñoz Rivera and S. Kawashima, Decay property for second order hyperbolic systems of viscoelastic materials, J. Math. Anal. Appl. 366 (2010), 621-635.

[5]

T. Hosono and S. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Methods Appl. Sci. 16 (2006), 1839-1859.

[6]

T. J. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63 (1976), 273-294.

[7]

K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008), 647-667.

[8]

K. Ide and S. Kawashima, Decay property of regularity-loss type and nonlinear effects for dissipative Timoshenko system, Math. Models Meth. Appl. Sci. 18 (2008), 1001-1025.

[9]

Y. Liu and S. Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation, Discrete Continuous Dynamical Systems, A 29 (2011), 1113-1139.

[10]

Y. Liu and S. Kawashima, Decay property for a plate equation with memory-type dissipation, Kinetic and Related Models, 4 (2011), 531-547.

[11]

Y. Liu and S. Kawashima, Decay property for the Timoshenko system with memory-type dissipation, Math. Models Meth. Appl. Sci. 22 (2012), 1150012, 19 pp.

[12]

A. Matsumura, An energy method for the equations of motion of compressible viscous and heat-conductive fluids, MRC Technical Summary Report, Univ. of Wisconsin-Madison \#2194 (1981).

[13]

J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math. 52 (1994), 628-648.

[14]

J. E. Muñoz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl. 286 (2003), 692-704.

[15]

Y. Ueda and S. Kawashima, Decay property of regularity-loss type for the Euler-Maxwell system, Methods Appl. Anal. 18 (2011), 245-267.

show all references

References:
[1]

P. M. N. Dharmawardane, Global solutions and decay property of regularity-loss type for quasi-linear hyperbolic systems with dissipation, J. Hyperbolic Differ. Equ. 10 (2013), 37-76.

[2]

P. M. N. Dharmawardane, T. Nakamura and S. Kawashima, Decay estimates of solutions for quasi-linear hyperbolic systems of viscoelasticity, SIAM J. Math. Anal. 44 (2012), 1976-2001.

[3]

P. M. N. Dharmawardane, T. Nakamura and S. Kawashima, Global solutions to quasi-linear hyperbolic systems of viscoelasticity, Kyoto J. Math. 51 (2011), 467-483.

[4]

P. M. N. Dharmawardane, J. E. Muñoz Rivera and S. Kawashima, Decay property for second order hyperbolic systems of viscoelastic materials, J. Math. Anal. Appl. 366 (2010), 621-635.

[5]

T. Hosono and S. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Methods Appl. Sci. 16 (2006), 1839-1859.

[6]

T. J. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63 (1976), 273-294.

[7]

K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008), 647-667.

[8]

K. Ide and S. Kawashima, Decay property of regularity-loss type and nonlinear effects for dissipative Timoshenko system, Math. Models Meth. Appl. Sci. 18 (2008), 1001-1025.

[9]

Y. Liu and S. Kawashima, Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation, Discrete Continuous Dynamical Systems, A 29 (2011), 1113-1139.

[10]

Y. Liu and S. Kawashima, Decay property for a plate equation with memory-type dissipation, Kinetic and Related Models, 4 (2011), 531-547.

[11]

Y. Liu and S. Kawashima, Decay property for the Timoshenko system with memory-type dissipation, Math. Models Meth. Appl. Sci. 22 (2012), 1150012, 19 pp.

[12]

A. Matsumura, An energy method for the equations of motion of compressible viscous and heat-conductive fluids, MRC Technical Summary Report, Univ. of Wisconsin-Madison \#2194 (1981).

[13]

J. E. Muñoz Rivera, Asymptotic behaviour in linear viscoelasticity, Quart. Appl. Math. 52 (1994), 628-648.

[14]

J. E. Muñoz Rivera, M. G. Naso and F. M. Vegni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory, J. Math. Anal. Appl. 286 (2003), 692-704.

[15]

Y. Ueda and S. Kawashima, Decay property of regularity-loss type for the Euler-Maxwell system, Methods Appl. Anal. 18 (2011), 245-267.

[1]

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
[2]

Monica Conti, V. Pata. Weakly dissipative semilinear equations of viscoelasticity. Communications on Pure and Applied Analysis, 2005, 4 (4) : 705-720. doi: 10.3934/cpaa.2005.4.705
[3]

Nguyen Dinh Cong. Semigroup property of fractional differential operators and its applications. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022064
[4]

Nakao Hayashi, Chunhua Li, Pavel I. Naumkin. Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2089-2104. doi: 10.3934/cpaa.2017103
[5]

Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks and Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465
[6]

Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065
[7]

Marat Akhmet, Duygu Aruğaslan. Lyapunov-Razumikhin method for differential equations with piecewise constant argument. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 457-466. doi: 10.3934/dcds.2009.25.457
[8]

Ferhat Mohamed, Hakem Ali. Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 491-506. doi: 10.3934/dcdsb.2017024
[9]

Luciano Pandolfi. Riesz systems and moment method in the study of viscoelasticity in one space dimension. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1487-1510. doi: 10.3934/dcdsb.2010.14.1487
[10]

Vladislav Balashov, Alexander Zlotnik. An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations. Journal of Computational Dynamics, 2020, 7 (2) : 291-312. doi: 10.3934/jcd.2020012
[11]

Makoto Nakamura. Remarks on global solutions of dissipative wave equations with exponential nonlinear terms. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1533-1545. doi: 10.3934/cpaa.2015.14.1533
[12]

Yves Coudène. The Hopf argument. Journal of Modern Dynamics, 2007, 1 (1) : 147-153. doi: 10.3934/jmd.2007.1.147
[13]

Yi Yang, Robert J. Sacker. Periodic unimodal Allee maps, the semigroup property and the $\lambda$-Ricker map with Allee effect. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 589-606. doi: 10.3934/dcdsb.2014.19.589
[14]

Michinori Ishiwata, Makoto Nakamura, Hidemitsu Wadade. Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4889-4903. doi: 10.3934/dcds.2015.35.4889
[15]

Xin Yu, Guojie Zheng, Chao Xu. The $C$-regularized semigroup method for partial differential equations with delays. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5163-5181. doi: 10.3934/dcds.2016024
[16]

Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. A thermo piezoelectric model: Exponential decay of the total energy. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5273-5292. doi: 10.3934/dcds.2013.33.5273
[17]

Mohammed Aassila. On energy decay rate for linear damped systems. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 851-864. doi: 10.3934/dcds.2002.8.851
[18]

Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations and Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017
[19]

Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721
[20]

George J. Bautista, Ademir F. Pazoto. Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain. Communications on Pure and Applied Analysis, 2020, 19 (2) : 747-769. doi: 10.3934/cpaa.2020035

 Impact Factor: 

Tools

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy