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

Global existence of solutions for the thermoelastic Bresse system

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

In this paper, using the semigroup approach, we obtain the global existence of solutions for linear (nonlinear) homogeneous (nonhomogeneous) thermoelastic Bresse System.
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,, \emph{Arch. Rational Mech. Anal., 29 (1968), 247.  doi: 10.1016/j.anihpc.2013.04.006.  Google Scholar

[3]

C. M. Dafermos, Asymptotic stability in viscoelascity,, \emph{Arch. Rational Mech. Anal., 37 (1970), 297.   Google Scholar

[4]

C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelascity,, \emph{J. Differential Equations, 7 (1990), 554.   Google Scholar

[5]

R. H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations arising in linear viscoelascity,, \emph{SIAM J. Math. Anal.}, 21 (1990), 374.  doi: 10.1002/cpa.3160360506.  Google Scholar

[6]

L. H. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system,, \emph{IMA J. Appl. Math.}, 75 (2010), 881.  doi: 10.3934/dcds.2004.10.543.  Google Scholar

[7]

S. W. Hansen, Exponential energy decay in a linear thermoelastic rod,, \emph{J. Math. Anal. Appl.}, 167 (1992), 429.  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,, \emph{Ann. Differential Equations}, 1 (1985), 43.  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,, \emph{Z. angew. Math. Phys.}, 47 (1996), 1.  doi: 10.3934/cpaa.2012.11.973.  Google Scholar

[10]

W. J. Liu, Partial exact controllability and exponential stability in higher dimensional linear thermoelascity,, \emph{ESAIM: Control Optim. Calc. Var.}, 3 (1998), 23.  doi: 10.3934/dcds.2005.12.881.  Google Scholar

[11]

W. J. Liu, The exponential stabilization of higher-dimensional linear system of thermoviscoelasticity,, \emph{J. Math. Pures Appl.}, 77 (1998), 355.  doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

[12]

Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system,, \emph{Z. angew. Math. Phys.}, 60 (2009), 54.  doi: 10.1007/s00033-009-0023-1.  Google Scholar

[13]

Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermoviscoelasticity,, \emph{Quart. Appl. Math.}, LIV (1996), 21.  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 (1999).  doi: 10.1515/form.2011.079.  Google Scholar

[15]

Z. Ma, Exponential stability and global attractors for a thermoelastic Bresse system,, \emph{Adv. Difference Equations}, 1 (2010), 1.  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,, \emph{J. Math. Anal. Appl.}, 348 (2008), 298.  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,, \emph{J. Math. Anal. Appl.}, 339 (2008), 482.  doi: 10.3934/dcds.2009.25.575.  Google Scholar

[18]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).   Google Scholar

[19]

J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.   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,, \emph{J. Differential Equations}, 207 (2004), 21.  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,, \emph{Proc. Roy. Soc. Edinburgh}, 132(A) (2002), 685.  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,, \emph{J. Partial Differential Equations}, 22 (2009), 152.  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,, \emph{Quart. Appl. Math.}, 4 (1993), 751.  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,, \emph{J. Math. Appl. Anal.}, 276 (2002), 248.   Google Scholar

[27]

Y. Shibata, Neumann problem for one-dimensional nonlinear thermoelascity,, \emph{Banach Center Publication}, 27 (1992), 457.   Google Scholar

[28]

M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelascity,, \emph{Arch. Rational Mech. Anal., 76 (1981), 97.   Google Scholar

[29]

S. P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars,, \emph{Philosophical Magazine}, 6 (1921), 744.   Google Scholar

[30]

X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III,, \emph{Comm. Contemp. Math., 5 (2003), 25.   Google Scholar

[31]

S. Zheng, Nonlinear Evolution Equations,, Pitman Monogr. Survey. Pure Appl. Math., 133 (2004).   Google Scholar

[32]

S. Zheng and Y. Qin, Maximal attractor for the system of one-dimensional polytropic viscous ideal gas,, \emph{Quart. Appl. Math., 3 (2001), 579.   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 $R^n$,, \emph{Arch. Rational Mech. Anal., 160 (2001), 153.   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,, \emph{Arch. Rational Mech. Anal., 29 (1968), 247.  doi: 10.1016/j.anihpc.2013.04.006.  Google Scholar

[3]

C. M. Dafermos, Asymptotic stability in viscoelascity,, \emph{Arch. Rational Mech. Anal., 37 (1970), 297.   Google Scholar

[4]

C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelascity,, \emph{J. Differential Equations, 7 (1990), 554.   Google Scholar

[5]

R. H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations arising in linear viscoelascity,, \emph{SIAM J. Math. Anal.}, 21 (1990), 374.  doi: 10.1002/cpa.3160360506.  Google Scholar

[6]

L. H. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system,, \emph{IMA J. Appl. Math.}, 75 (2010), 881.  doi: 10.3934/dcds.2004.10.543.  Google Scholar

[7]

S. W. Hansen, Exponential energy decay in a linear thermoelastic rod,, \emph{J. Math. Anal. Appl.}, 167 (1992), 429.  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,, \emph{Ann. Differential Equations}, 1 (1985), 43.  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,, \emph{Z. angew. Math. Phys.}, 47 (1996), 1.  doi: 10.3934/cpaa.2012.11.973.  Google Scholar

[10]

W. J. Liu, Partial exact controllability and exponential stability in higher dimensional linear thermoelascity,, \emph{ESAIM: Control Optim. Calc. Var.}, 3 (1998), 23.  doi: 10.3934/dcds.2005.12.881.  Google Scholar

[11]

W. J. Liu, The exponential stabilization of higher-dimensional linear system of thermoviscoelasticity,, \emph{J. Math. Pures Appl.}, 77 (1998), 355.  doi: 10.1016/j.anihpc.2006.03.014.  Google Scholar

[12]

Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system,, \emph{Z. angew. Math. Phys.}, 60 (2009), 54.  doi: 10.1007/s00033-009-0023-1.  Google Scholar

[13]

Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermoviscoelasticity,, \emph{Quart. Appl. Math.}, LIV (1996), 21.  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 (1999).  doi: 10.1515/form.2011.079.  Google Scholar

[15]

Z. Ma, Exponential stability and global attractors for a thermoelastic Bresse system,, \emph{Adv. Difference Equations}, 1 (2010), 1.  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,, \emph{J. Math. Anal. Appl.}, 348 (2008), 298.  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,, \emph{J. Math. Anal. Appl.}, 339 (2008), 482.  doi: 10.3934/dcds.2009.25.575.  Google Scholar

[18]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).   Google Scholar

[19]

J. Prüss, On the spectrum of $C_0$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.   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,, \emph{J. Differential Equations}, 207 (2004), 21.  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,, \emph{Proc. Roy. Soc. Edinburgh}, 132(A) (2002), 685.  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,, \emph{J. Partial Differential Equations}, 22 (2009), 152.  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,, \emph{Quart. Appl. Math.}, 4 (1993), 751.  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,, \emph{J. Math. Appl. Anal.}, 276 (2002), 248.   Google Scholar

[27]

Y. Shibata, Neumann problem for one-dimensional nonlinear thermoelascity,, \emph{Banach Center Publication}, 27 (1992), 457.   Google Scholar

[28]

M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelascity,, \emph{Arch. Rational Mech. Anal., 76 (1981), 97.   Google Scholar

[29]

S. P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars,, \emph{Philosophical Magazine}, 6 (1921), 744.   Google Scholar

[30]

X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III,, \emph{Comm. Contemp. Math., 5 (2003), 25.   Google Scholar

[31]

S. Zheng, Nonlinear Evolution Equations,, Pitman Monogr. Survey. Pure Appl. Math., 133 (2004).   Google Scholar

[32]

S. Zheng and Y. Qin, Maximal attractor for the system of one-dimensional polytropic viscous ideal gas,, \emph{Quart. Appl. Math., 3 (2001), 579.   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 $R^n$,, \emph{Arch. Rational Mech. Anal., 160 (2001), 153.   Google Scholar

[1]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[2]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055

[3]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[4]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[5]

Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020031

[6]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[7]

Xinpeng Wang, Bingo Wing-Kuen Ling, Wei-Chao Kuang, Zhijing Yang. Orthogonal intrinsic mode functions via optimization approach. Journal of Industrial & Management Optimization, 2021, 17 (1) : 51-66. doi: 10.3934/jimo.2019098

[8]

Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082

[9]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[10]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[11]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353

[12]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

[13]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[14]

Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325

[15]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[16]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

[17]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291

[18]

Yolanda Guerrero–Sánchez, Muhammad Umar, Zulqurnain Sabir, Juan L. G. Guirao, Muhammad Asif Zahoor Raja. Solving a class of biological HIV infection model of latently infected cells using heuristic approach. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020431

[19]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[20]

M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]