Global existence of solutions for the thermoelastic Bresse system

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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 and 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.
[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.
[3]	C. M. Dafermos, Asymptotic stability in viscoelascity, Arch. Rational Mech. Anal., 37 (1970), 297-308.
[4]	C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelascity, J. Differential Equations, 7 (1990), 554-569.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, Berlin, Heidelbery, Tokyo, 1983.
[19]	J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
[20]	Y. Qin and Z. Ma, Global existence of the higher-dimensional linear system of thermoviscoelasticity, Preprint. doi: 10.1142/S0129167X12500279.
[21]	Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Operator Theory, Advances and Applications, Vol. 184, Birkh\"auser, Basel-Boston-Berlin.
[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.
[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.
[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.
[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.
[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.
[27]	Y. Shibata, Neumann problem for one-dimensional nonlinear thermoelascity, Banach Center Publication, 27 (1992), 457-480.
[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.
[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.
[30]	X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III, Comm. Contemp. Math., 5 (2003), 25-83.
[31]	S. Zheng, Nonlinear Evolution Equations, Pitman Monogr. Survey. Pure Appl. Math., Vol. 133, CRC Press, USA, 2004.
[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.
[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.

show all references

References:

[1]	J. A. C. Bresse, Cours de Méchanique Appliquée, Mallet Bachelier, 1859. doi: 10.3934/krm.2011.4.901.
[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.
[3]	C. M. Dafermos, Asymptotic stability in viscoelascity, Arch. Rational Mech. Anal., 37 (1970), 297-308.
[4]	C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelascity, J. Differential Equations, 7 (1990), 554-569.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, Berlin, Heidelbery, Tokyo, 1983.
[19]	J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
[20]	Y. Qin and Z. Ma, Global existence of the higher-dimensional linear system of thermoviscoelasticity, Preprint. doi: 10.1142/S0129167X12500279.
[21]	Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Operator Theory, Advances and Applications, Vol. 184, Birkh\"auser, Basel-Boston-Berlin.
[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.
[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.
[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.
[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.
[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.
[27]	Y. Shibata, Neumann problem for one-dimensional nonlinear thermoelascity, Banach Center Publication, 27 (1992), 457-480.
[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.
[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.
[30]	X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III, Comm. Contemp. Math., 5 (2003), 25-83.
[31]	S. Zheng, Nonlinear Evolution Equations, Pitman Monogr. Survey. Pure Appl. Math., Vol. 133, CRC Press, USA, 2004.
[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.
[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.

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