# American Institute of Mathematical Sciences

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

 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.
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.

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##### 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.
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