Global existence of solutions for the thermoelastic
Bresse system

Yuming Qin; Xinguang Yang; Zhiyong Ma

doi:10.3934/cpaa.2014.13.1395

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July 2014, 13(4): 1395-1406. doi: 10.3934/cpaa.2014.13.1395

Global existence of solutions for the thermoelastic Bresse system

Yuming Qin ^1, , Xinguang Yang ^2, and Zhiyong Ma ^3,

1.	Department of Applied Mathematics, Donghua University, Shanghai 201620
2.	College of Information Science and Technology, Donghua University, Shanghai 201620
3.	College of Science, Shanghai Second Polytechnic University, Shanghai, 201209, China

Received June 2011 Revised December 2013 Published February 2014

Abstract
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In this paper, using the semigroup approach, we obtain the global existence of solutions for linear (nonlinear) homogeneous (nonhomogeneous) thermoelastic Bresse System.

Keywords: Semigroup approach., Global existence, Thermoelastic Bresse System.

Mathematics Subject Classification: Primary: 76B03; Secondary: 76W0.

Citation: Yuming Qin, Xinguang Yang, Zhiyong Ma. Global existence of solutions for the thermoelastic Bresse system. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1395-1406. doi: 10.3934/cpaa.2014.13.1395

References:

[1]	J. A. C. Bresse, Cours de Méchanique Appliquée, Mallet Bachelier, 1859. doi: 10.3934/krm.2011.4.901. Google Scholar
[2]	C. M. Dafermos, On the existence and the asymptotic stability of solution to the equations of linear thermoelasticity, Arch. Rational Mech. Anal., 29 (1968), 247-271. doi: 10.1016/j.anihpc.2013.04.006. Google Scholar
[3]	C. M. Dafermos, Asymptotic stability in viscoelascity, Arch. Rational Mech. Anal., 37 (1970), 297-308. Google Scholar
[4]	C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelascity, J. Differential Equations, 7 (1990), 554-569. Google Scholar
[5]	R. H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations arising in linear viscoelascity, SIAM J. Math. Anal., 21 (1990), 374-393. doi: 10.1002/cpa.3160360506. Google Scholar
[6]	L. H. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math., 75 (2010), 881-904. doi: 10.3934/dcds.2004.10.543. Google Scholar
[7]	S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl., 167 (1992), 429-442. doi: 10.1016/j.jfa.2005.06.009. Google Scholar
[8]	F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. doi: 10.1007/s10255-012-0175-1. Google Scholar
[9]	K. Liu and Z. Liu, On the type of $C_0$-semigroup associated with the abstract linear viscoelastic system, Z. angew. Math. Phys., 47 (1996), 1-15. doi: 10.3934/cpaa.2012.11.973. Google Scholar
[10]	W. J. Liu, Partial exact controllability and exponential stability in higher dimensional linear thermoelascity, ESAIM: Control Optim. Calc. Var., 3 (1998), 23-48. doi: 10.3934/dcds.2005.12.881. Google Scholar
[11]	W. J. Liu, The exponential stabilization of higher-dimensional linear system of thermoviscoelasticity, J. Math. Pures Appl., 77 (1998), 355-386. doi: 10.1016/j.anihpc.2006.03.014. Google Scholar
[12]	Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. angew. Math. Phys., 60 (2009), 54-69. doi: 10.1007/s00033-009-0023-1. Google Scholar
[13]	Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermoviscoelasticity, Quart. Appl. Math., LIV (1996), 21-31. doi: 10.1016/j.na.2009.12.045. Google Scholar
[14]	Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Research Notes in Mathematics, 389, Chapman & Hall/CRC, Boca Raton, FL, 1999. doi: 10.1515/form.2011.079. Google Scholar
[15]	Z. Ma, Exponential stability and global attractors for a thermoelastic Bresse system, Adv. Difference Equations, 1 (2010), 1-17. doi: 10.1016/j.nonrwa.2011.07.055. Google Scholar
[16]	S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type III, J. Math. Anal. Appl., 348 (2008), 298-307. doi: 10.1016/j.jde.2009.09.020. Google Scholar
[17]	J. E. Muñoz Rivera and H. D. Fernández Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502. doi: 10.3934/dcds.2009.25.575. Google Scholar
[18]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, Berlin, Heidelbery, Tokyo, 1983. Google Scholar
[19]	J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. Google Scholar
[20]	Y. Qin and Z. Ma, Global existence of the higher-dimensional linear system of thermoviscoelasticity,, Preprint., (). doi: 10.1142/S0129167X12500279. Google Scholar
[21]	Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors,, Operator Theory, 184 (). Google Scholar
[22]	Y. Qin, Universal attractor in $H^4$ for the nonlinear one-dimensional compressible Navier -Stokes equations, J. Differential Equations, 207 (2004), 21-72. doi: 10.1007/s002090100332. Google Scholar
[23]	Y. Qin and J. E. Muñoz Rivera, Universal attractors for a nonlinear one-dimensional heat-conductive viscous real gas, Proc. Roy. Soc. Edinburgh, 132(A) (2002), 685-709. doi: 10.1155/2011/128614. Google Scholar
[24]	Y. Qin, S. Deng and B. W. Schulze, Uniform compact attractors for a nonlinear non-autonomous equation of viscoelastisity, J. Partial Differential Equations, 22 (2009), 152-192. doi: 10.1186/1687-2770-2011-11. Google Scholar
[25]	R. Racke, Y. Shibata and S. Zheng, Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity, Quart. Appl. Math., 4 (1993), 751-763. doi: 10.1002/mma.1510. Google Scholar
[26]	R. Racke and J. E. Muñoz Rivera, Midly dissipative nonlinear Tymoshenko systems-Global existence and exponential stability, J. Math. Appl. Anal., 276 (2002), 248-278. Google Scholar
[27]	Y. Shibata, Neumann problem for one-dimensional nonlinear thermoelascity, Banach Center Publication, 27 (1992), 457-480. Google Scholar
[28]	M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelascity, Arch. Rational Mech. Anal., 76 (1981), 97-133. Google Scholar
[29]	S. P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, 6 (1921), 744-746. Google Scholar
[30]	X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III, Comm. Contemp. Math., 5 (2003), 25-83. Google Scholar
[31]	S. Zheng, Nonlinear Evolution Equations, Pitman Monogr. Survey. Pure Appl. Math., Vol. 133, CRC Press, USA, 2004. Google Scholar
[32]	S. Zheng and Y. Qin, Maximal attractor for the system of one-dimensional polytropic viscous ideal gas, Quart. Appl. Math., 3 (2001), 579-599. Google Scholar
[33]	S. Zheng and Y. Qin, Universal attractors for the Navier-Stokes equations of compressible and heat conductive fluids in bounded annular domains in $R^n$, Arch. Rational Mech. Anal., 160 (2001), 153-179. Google Scholar

show all references

References:

[1]	J. A. C. Bresse, Cours de Méchanique Appliquée, Mallet Bachelier, 1859. doi: 10.3934/krm.2011.4.901. Google Scholar
[2]	C. M. Dafermos, On the existence and the asymptotic stability of solution to the equations of linear thermoelasticity, Arch. Rational Mech. Anal., 29 (1968), 247-271. doi: 10.1016/j.anihpc.2013.04.006. Google Scholar
[3]	C. M. Dafermos, Asymptotic stability in viscoelascity, Arch. Rational Mech. Anal., 37 (1970), 297-308. Google Scholar
[4]	C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelascity, J. Differential Equations, 7 (1990), 554-569. Google Scholar
[5]	R. H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations arising in linear viscoelascity, SIAM J. Math. Anal., 21 (1990), 374-393. doi: 10.1002/cpa.3160360506. Google Scholar
[6]	L. H. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math., 75 (2010), 881-904. doi: 10.3934/dcds.2004.10.543. Google Scholar
[7]	S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl., 167 (1992), 429-442. doi: 10.1016/j.jfa.2005.06.009. Google Scholar
[8]	F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. doi: 10.1007/s10255-012-0175-1. Google Scholar
[9]	K. Liu and Z. Liu, On the type of $C_0$-semigroup associated with the abstract linear viscoelastic system, Z. angew. Math. Phys., 47 (1996), 1-15. doi: 10.3934/cpaa.2012.11.973. Google Scholar
[10]	W. J. Liu, Partial exact controllability and exponential stability in higher dimensional linear thermoelascity, ESAIM: Control Optim. Calc. Var., 3 (1998), 23-48. doi: 10.3934/dcds.2005.12.881. Google Scholar
[11]	W. J. Liu, The exponential stabilization of higher-dimensional linear system of thermoviscoelasticity, J. Math. Pures Appl., 77 (1998), 355-386. doi: 10.1016/j.anihpc.2006.03.014. Google Scholar
[12]	Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. angew. Math. Phys., 60 (2009), 54-69. doi: 10.1007/s00033-009-0023-1. Google Scholar
[13]	Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermoviscoelasticity, Quart. Appl. Math., LIV (1996), 21-31. doi: 10.1016/j.na.2009.12.045. Google Scholar
[14]	Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Research Notes in Mathematics, 389, Chapman & Hall/CRC, Boca Raton, FL, 1999. doi: 10.1515/form.2011.079. Google Scholar
[15]	Z. Ma, Exponential stability and global attractors for a thermoelastic Bresse system, Adv. Difference Equations, 1 (2010), 1-17. doi: 10.1016/j.nonrwa.2011.07.055. Google Scholar
[16]	S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type III, J. Math. Anal. Appl., 348 (2008), 298-307. doi: 10.1016/j.jde.2009.09.020. Google Scholar
[17]	J. E. Muñoz Rivera and H. D. Fernández Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502. doi: 10.3934/dcds.2009.25.575. Google Scholar
[18]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, Berlin, Heidelbery, Tokyo, 1983. Google Scholar
[19]	J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. Google Scholar
[20]	Y. Qin and Z. Ma, Global existence of the higher-dimensional linear system of thermoviscoelasticity,, Preprint., (). doi: 10.1142/S0129167X12500279. Google Scholar
[21]	Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors,, Operator Theory, 184 (). Google Scholar
[22]	Y. Qin, Universal attractor in $H^4$ for the nonlinear one-dimensional compressible Navier -Stokes equations, J. Differential Equations, 207 (2004), 21-72. doi: 10.1007/s002090100332. Google Scholar
[23]	Y. Qin and J. E. Muñoz Rivera, Universal attractors for a nonlinear one-dimensional heat-conductive viscous real gas, Proc. Roy. Soc. Edinburgh, 132(A) (2002), 685-709. doi: 10.1155/2011/128614. Google Scholar
[24]	Y. Qin, S. Deng and B. W. Schulze, Uniform compact attractors for a nonlinear non-autonomous equation of viscoelastisity, J. Partial Differential Equations, 22 (2009), 152-192. doi: 10.1186/1687-2770-2011-11. Google Scholar
[25]	R. Racke, Y. Shibata and S. Zheng, Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity, Quart. Appl. Math., 4 (1993), 751-763. doi: 10.1002/mma.1510. Google Scholar
[26]	R. Racke and J. E. Muñoz Rivera, Midly dissipative nonlinear Tymoshenko systems-Global existence and exponential stability, J. Math. Appl. Anal., 276 (2002), 248-278. Google Scholar
[27]	Y. Shibata, Neumann problem for one-dimensional nonlinear thermoelascity, Banach Center Publication, 27 (1992), 457-480. Google Scholar
[28]	M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelascity, Arch. Rational Mech. Anal., 76 (1981), 97-133. Google Scholar
[29]	S. P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, 6 (1921), 744-746. Google Scholar
[30]	X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III, Comm. Contemp. Math., 5 (2003), 25-83. Google Scholar
[31]	S. Zheng, Nonlinear Evolution Equations, Pitman Monogr. Survey. Pure Appl. Math., Vol. 133, CRC Press, USA, 2004. Google Scholar
[32]	S. Zheng and Y. Qin, Maximal attractor for the system of one-dimensional polytropic viscous ideal gas, Quart. Appl. Math., 3 (2001), 579-599. Google Scholar
[33]	S. Zheng and Y. Qin, Universal attractors for the Navier-Stokes equations of compressible and heat conductive fluids in bounded annular domains in $R^n$, Arch. Rational Mech. Anal., 160 (2001), 153-179. Google Scholar

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