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September  2015, 14(5): 1671-1683. doi: 10.3934/cpaa.2015.14.1671

On global solutions in one-dimensional thermoelasticity with second sound in the half line

Yuxi Hu 1, and  Na Wang 2,

1. 

Department of Mathematics, China University of Mining and Technology, Beijing, 100083, China
2. 

Department of Mathematics, Tianjin University of Technology, Tianjin 300384, China

Received  December 2013 Revised  March 2014 Published  June 2015

In this paper, we investigate the initial boundary value problem for one-dimensional thermoelasticity with second sound in the half line. By using delicate energy estimates, together with a special form of Helmholtz free energy, we are able to show the global solutions exist under the Dirichlet boundary condition if the initial data are sufficient small.
Keywords: Second sound, half line, Dirichlet boundary condition, global solution..
Mathematics Subject Classification: 35L50, 74F05.
Citation: Yuxi Hu, Na Wang. On global solutions in one-dimensional thermoelasticity with second sound in the half line. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1671-1683. doi: 10.3934/cpaa.2015.14.1671
References:
[1]

K. Beauchard and E. Zuazua, Large time asymptotics for partially dissipative hyperbolic systems,, \emph{Arch. Ration. Mech. Anal.}, 199 (2011), 177.  doi: 10.1007/s00205-010-0321-y.  Google Scholar

[2]

C. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations-First-order Systems and Application,, Clarendon Press, (2007).   Google Scholar

[3]

S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 1559.  doi: 10.1002/cpa.20195.  Google Scholar

[4]

C. M. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity,, \emph{Quart. Appl. Math.}, 44 (1986), 463.   Google Scholar

[5]

H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems-Cattaneo versus Fourier law,, \emph{Arch. Ration. Mech. Anal.}, 194 (2009), 221.  doi: 10.1007/s00205-009-0220-2.  Google Scholar

[6]

I. Hansen, Lebensdauer von klassischen Lösungen nichtlinearer Thermoelastizitätsgleichungen,, Diploma thesis, (1994).   Google Scholar

[7]

Y. Hu, Global solvability in thermoelasticity with second sound on the semi-axis,, \emph{J. Part. Diff. Eq.}, 25 (2012), 37.   Google Scholar

[8]

Y. Hu and R. Racke, Formation of singularities in one-dimensional thermoelasticity with second sound,, \emph{Quart. Appl. Math.}, 72 (2014), 311.  doi: 10.1090/S0033-569X-2014-01336-2.  Google Scholar

[9]

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 relaticity,, \emph{Arch. Rational Mech. Anal.}, 63 (1977), 273.   Google Scholar

[10]

W. J. Hrusa and S. A. Messaoudi, On formation of singularities in one-dimensional nonlinear thermoelasticity,, \emph{Arch. Ration. Mech. Anal.}, 111 (1990), 135.  doi: 10.1007/BF00375405.  Google Scholar

[11]

W. J. Hrusa and M. A. Tarabek, On smooth solutions of the Cauchy problem in one-dimensional nonlinear thermoelasticity,, \emph{Quart. Appl. Math.}, 47 (1989), 631.   Google Scholar

[12]

S. Jiang, Global existence and asymptotic behavior of smooth solutions in one-dimensional nonlinear thermoelasticity,, \emph{Proc. Roy. Soc. Edinburgh}, 115A (1990), 257.  doi: 10.1017/S0308210500020631.  Google Scholar

[13]

S. Jiang, Global solutions of the Dirichlet problem in one-dimensional nonlinear thermoelasticity,, \emph{SFB, 138 (1990).   Google Scholar

[14]

S. Jiang, On global smooth solutions to the one-dimensional equations of nonlnear inhomogeneous thermoelasticity,, \emph{Nonlinear Anal., 20 (1993), 1245.  doi: 10.1016/0362-546X(93)90154-K.  Google Scholar

[15]

S. Jiang and R. Racke, Evolution Equations in Thermoelasticity,, Chapman and Hall/CRC Monographs and Surveys in Pure and Appl. Math. Vol. 112, (2000).   Google Scholar

[16]

A. Kasimov, R. Racke and B. Said-Houari, Global existence and decay properties for solutions of the Cauchy problem in one-dimensional thermoelasticity with second sound,, \emph{Applicable Analysis}, 93 (2014), 911.  doi: 10.1080/00036811.2013.801457.  Google Scholar

[17]

T.-T, Li, Global Classical Solutions for Quasilinear Hyperbolic Systems,, Masson, (1994).   Google Scholar

[18]

S. A. Messaoudi and B. Said-Houari, Exponetial stability in one-dimensional non-linear thermoelasticity with second sound,, \emph{Math. Methods Appl. Sci.}, 28 (2005), 205.  doi: 10.1002/mma.556.  Google Scholar

[19]

R. Racke and Y. Shibata, Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity,, \emph{Arch. Rational Mech. Anal.}, 116 (1991), 1.  doi: 10.1007/BF00375601.  Google Scholar

[20]

R. Racke and Y. Shibada, Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity,, \emph{Quart. Appl. Math.}, 51 (1993), 751.   Google Scholar

[21]

R. Racke, Thermoelasticity with second sound--Exponential stability in linear and non-linear 1-d,, \emph{Math. Meth. Appl. Sci.}, 25 (2002), 409.  doi: 10.1002/mma.298.  Google Scholar

[22]

R. Racke, Thermoelasticity, Handbook of Differential Equations,, Chapter 4, (2009), 315.  doi: 10.1016/S1874-5717(08)00211-9.  Google Scholar

[23]

R. Racke and Y. G. Wang, Nonlinear well-posedness and rates of decay in thermoelasticity with second sound,, \emph{J. Hyperbolic Differential Equations}, 5 (2008), 25.  doi: 10.1142/S021989160800143X.  Google Scholar

[24]

J. E. M. Rivera, Energy decay rates in linear thermoelasticity,, \emph{Funkcial. Ekvac.}, 35 (1992), 19.   Google Scholar

[25]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation,, \emph{Hokkaido Math. J.}, 14 (1995), 249.  doi: 10.14492/hokmj/1381757663.  Google Scholar

[26]

M. A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound,, \emph{Quart. Appl. Math.}, 50 (1992), 727.   Google Scholar

show all references

References:
[1]

K. Beauchard and E. Zuazua, Large time asymptotics for partially dissipative hyperbolic systems,, \emph{Arch. Ration. Mech. Anal.}, 199 (2011), 177.  doi: 10.1007/s00205-010-0321-y.  Google Scholar

[2]

C. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations-First-order Systems and Application,, Clarendon Press, (2007).   Google Scholar

[3]

S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 1559.  doi: 10.1002/cpa.20195.  Google Scholar

[4]

C. M. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity,, \emph{Quart. Appl. Math.}, 44 (1986), 463.   Google Scholar

[5]

H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems-Cattaneo versus Fourier law,, \emph{Arch. Ration. Mech. Anal.}, 194 (2009), 221.  doi: 10.1007/s00205-009-0220-2.  Google Scholar

[6]

I. Hansen, Lebensdauer von klassischen Lösungen nichtlinearer Thermoelastizitätsgleichungen,, Diploma thesis, (1994).   Google Scholar

[7]

Y. Hu, Global solvability in thermoelasticity with second sound on the semi-axis,, \emph{J. Part. Diff. Eq.}, 25 (2012), 37.   Google Scholar

[8]

Y. Hu and R. Racke, Formation of singularities in one-dimensional thermoelasticity with second sound,, \emph{Quart. Appl. Math.}, 72 (2014), 311.  doi: 10.1090/S0033-569X-2014-01336-2.  Google Scholar

[9]

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 relaticity,, \emph{Arch. Rational Mech. Anal.}, 63 (1977), 273.   Google Scholar

[10]

W. J. Hrusa and S. A. Messaoudi, On formation of singularities in one-dimensional nonlinear thermoelasticity,, \emph{Arch. Ration. Mech. Anal.}, 111 (1990), 135.  doi: 10.1007/BF00375405.  Google Scholar

[11]

W. J. Hrusa and M. A. Tarabek, On smooth solutions of the Cauchy problem in one-dimensional nonlinear thermoelasticity,, \emph{Quart. Appl. Math.}, 47 (1989), 631.   Google Scholar

[12]

S. Jiang, Global existence and asymptotic behavior of smooth solutions in one-dimensional nonlinear thermoelasticity,, \emph{Proc. Roy. Soc. Edinburgh}, 115A (1990), 257.  doi: 10.1017/S0308210500020631.  Google Scholar

[13]

S. Jiang, Global solutions of the Dirichlet problem in one-dimensional nonlinear thermoelasticity,, \emph{SFB, 138 (1990).   Google Scholar

[14]

S. Jiang, On global smooth solutions to the one-dimensional equations of nonlnear inhomogeneous thermoelasticity,, \emph{Nonlinear Anal., 20 (1993), 1245.  doi: 10.1016/0362-546X(93)90154-K.  Google Scholar

[15]

S. Jiang and R. Racke, Evolution Equations in Thermoelasticity,, Chapman and Hall/CRC Monographs and Surveys in Pure and Appl. Math. Vol. 112, (2000).   Google Scholar

[16]

A. Kasimov, R. Racke and B. Said-Houari, Global existence and decay properties for solutions of the Cauchy problem in one-dimensional thermoelasticity with second sound,, \emph{Applicable Analysis}, 93 (2014), 911.  doi: 10.1080/00036811.2013.801457.  Google Scholar

[17]

T.-T, Li, Global Classical Solutions for Quasilinear Hyperbolic Systems,, Masson, (1994).   Google Scholar

[18]

S. A. Messaoudi and B. Said-Houari, Exponetial stability in one-dimensional non-linear thermoelasticity with second sound,, \emph{Math. Methods Appl. Sci.}, 28 (2005), 205.  doi: 10.1002/mma.556.  Google Scholar

[19]

R. Racke and Y. Shibata, Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity,, \emph{Arch. Rational Mech. Anal.}, 116 (1991), 1.  doi: 10.1007/BF00375601.  Google Scholar

[20]

R. Racke and Y. Shibada, Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity,, \emph{Quart. Appl. Math.}, 51 (1993), 751.   Google Scholar

[21]

R. Racke, Thermoelasticity with second sound--Exponential stability in linear and non-linear 1-d,, \emph{Math. Meth. Appl. Sci.}, 25 (2002), 409.  doi: 10.1002/mma.298.  Google Scholar

[22]

R. Racke, Thermoelasticity, Handbook of Differential Equations,, Chapter 4, (2009), 315.  doi: 10.1016/S1874-5717(08)00211-9.  Google Scholar

[23]

R. Racke and Y. G. Wang, Nonlinear well-posedness and rates of decay in thermoelasticity with second sound,, \emph{J. Hyperbolic Differential Equations}, 5 (2008), 25.  doi: 10.1142/S021989160800143X.  Google Scholar

[24]

J. E. M. Rivera, Energy decay rates in linear thermoelasticity,, \emph{Funkcial. Ekvac.}, 35 (1992), 19.   Google Scholar

[25]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation,, \emph{Hokkaido Math. J.}, 14 (1995), 249.  doi: 10.14492/hokmj/1381757663.  Google Scholar

[26]

M. A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound,, \emph{Quart. Appl. Math.}, 50 (1992), 727.   Google Scholar

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