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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

 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.
Citation: Yuxi Hu, Na Wang. On global solutions in one-dimensional thermoelasticity with second sound in the half line. Communications on Pure and 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, Arch. Ration. Mech. Anal., 199 (2011), 177-227. doi: 10.1007/s00205-010-0321-y. [2] C. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations-First-order Systems and Application, Clarendon Press, Oxford, 2007. [3] S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math., 60 (2007), 1559-1622. doi: 10.1002/cpa.20195. [4] C. M. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity, Quart. Appl. Math., 44 (1986), 463-474. [5] H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems-Cattaneo versus Fourier law, Arch. Ration. Mech. Anal., 194 (2009), 221-251. doi: 10.1007/s00205-009-0220-2. [6] I. Hansen, Lebensdauer von klassischen Lösungen nichtlinearer Thermoelastizitätsgleichungen, Diploma thesis, University of Bonn, 1994. [7] Y. Hu, Global solvability in thermoelasticity with second sound on the semi-axis, J. Part. Diff. Eq., 25 (2012), 37-68. [8] Y. Hu and R. Racke, Formation of singularities in one-dimensional thermoelasticity with second sound, Quart. Appl. Math., 72 (2014) 311-321. doi: 10.1090/S0033-569X-2014-01336-2. [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, Arch. Rational Mech. Anal., 63 (1977), 273-294. [10] W. J. Hrusa and S. A. Messaoudi, On formation of singularities in one-dimensional nonlinear thermoelasticity, Arch. Ration. Mech. Anal., 111 (1990), 135-151. doi: 10.1007/BF00375405. [11] W. J. Hrusa and M. A. Tarabek, On smooth solutions of the Cauchy problem in one-dimensional nonlinear thermoelasticity, Quart. Appl. Math., 47 (1989), 631-644. [12] S. Jiang, Global existence and asymptotic behavior of smooth solutions in one-dimensional nonlinear thermoelasticity, Proc. Roy. Soc. Edinburgh, 115A (1990), 257-274. doi: 10.1017/S0308210500020631. [13] S. Jiang, Global solutions of the Dirichlet problem in one-dimensional nonlinear thermoelasticity, SFB, Univ. Bonn, 138, 1990. [14] S. Jiang, On global smooth solutions to the one-dimensional equations of nonlnear inhomogeneous thermoelasticity, Nonlinear Anal., TMA, 20 (1993), 1245-1256. doi: 10.1016/0362-546X(93)90154-K. [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. [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, Applicable Analysis, 93 (2014), 911-935. doi: 10.1080/00036811.2013.801457. [17] T.-T, Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Masson, Paris, 1994. [18] S. A. Messaoudi and B. Said-Houari, Exponetial stability in one-dimensional non-linear thermoelasticity with second sound, Math. Methods Appl. Sci., 28 (2005) 205-232. doi: 10.1002/mma.556. [19] R. Racke and Y. Shibata, Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal., 116 (1991), 1-34. doi: 10.1007/BF00375601. [20] R. Racke and Y. Shibada, Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity, Quart. Appl. Math., 51 (1993), 751-763. [21] R. Racke, Thermoelasticity with second sound--Exponential stability in linear and non-linear 1-d, Math. Meth. Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298. [22] R. Racke, Thermoelasticity, Handbook of Differential Equations, Chapter 4, (C. M. Dafermos and M. Pokornýeds.), Elsevier (2009), 315-386. doi: 10.1016/S1874-5717(08)00211-9. [23] R. Racke and Y. G. Wang, Nonlinear well-posedness and rates of decay in thermoelasticity with second sound, J. Hyperbolic Differential Equations, 5 (2008), 25-43. doi: 10.1142/S021989160800143X. [24] J. E. M. Rivera, Energy decay rates in linear thermoelasticity, Funkcial. Ekvac., 35 (1992), 19-30. [25] Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1995), 249-275. doi: 10.14492/hokmj/1381757663. [26] M. A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound, Quart. Appl. Math., 50 (1992), 727-742.

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##### References:
 [1] K. Beauchard and E. Zuazua, Large time asymptotics for partially dissipative hyperbolic systems, Arch. Ration. Mech. Anal., 199 (2011), 177-227. doi: 10.1007/s00205-010-0321-y. [2] C. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations-First-order Systems and Application, Clarendon Press, Oxford, 2007. [3] S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math., 60 (2007), 1559-1622. doi: 10.1002/cpa.20195. [4] C. M. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity, Quart. Appl. Math., 44 (1986), 463-474. [5] H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems-Cattaneo versus Fourier law, Arch. Ration. Mech. Anal., 194 (2009), 221-251. doi: 10.1007/s00205-009-0220-2. [6] I. Hansen, Lebensdauer von klassischen Lösungen nichtlinearer Thermoelastizitätsgleichungen, Diploma thesis, University of Bonn, 1994. [7] Y. Hu, Global solvability in thermoelasticity with second sound on the semi-axis, J. Part. Diff. Eq., 25 (2012), 37-68. [8] Y. Hu and R. Racke, Formation of singularities in one-dimensional thermoelasticity with second sound, Quart. Appl. Math., 72 (2014) 311-321. doi: 10.1090/S0033-569X-2014-01336-2. [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, Arch. Rational Mech. Anal., 63 (1977), 273-294. [10] W. J. Hrusa and S. A. Messaoudi, On formation of singularities in one-dimensional nonlinear thermoelasticity, Arch. Ration. Mech. Anal., 111 (1990), 135-151. doi: 10.1007/BF00375405. [11] W. J. Hrusa and M. A. Tarabek, On smooth solutions of the Cauchy problem in one-dimensional nonlinear thermoelasticity, Quart. Appl. Math., 47 (1989), 631-644. [12] S. Jiang, Global existence and asymptotic behavior of smooth solutions in one-dimensional nonlinear thermoelasticity, Proc. Roy. Soc. Edinburgh, 115A (1990), 257-274. doi: 10.1017/S0308210500020631. [13] S. Jiang, Global solutions of the Dirichlet problem in one-dimensional nonlinear thermoelasticity, SFB, Univ. Bonn, 138, 1990. [14] S. Jiang, On global smooth solutions to the one-dimensional equations of nonlnear inhomogeneous thermoelasticity, Nonlinear Anal., TMA, 20 (1993), 1245-1256. doi: 10.1016/0362-546X(93)90154-K. [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. [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, Applicable Analysis, 93 (2014), 911-935. doi: 10.1080/00036811.2013.801457. [17] T.-T, Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Masson, Paris, 1994. [18] S. A. Messaoudi and B. Said-Houari, Exponetial stability in one-dimensional non-linear thermoelasticity with second sound, Math. Methods Appl. Sci., 28 (2005) 205-232. doi: 10.1002/mma.556. [19] R. Racke and Y. Shibata, Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal., 116 (1991), 1-34. doi: 10.1007/BF00375601. [20] R. Racke and Y. Shibada, Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity, Quart. Appl. Math., 51 (1993), 751-763. [21] R. Racke, Thermoelasticity with second sound--Exponential stability in linear and non-linear 1-d, Math. Meth. Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298. [22] R. Racke, Thermoelasticity, Handbook of Differential Equations, Chapter 4, (C. M. Dafermos and M. Pokornýeds.), Elsevier (2009), 315-386. doi: 10.1016/S1874-5717(08)00211-9. [23] R. Racke and Y. G. Wang, Nonlinear well-posedness and rates of decay in thermoelasticity with second sound, J. Hyperbolic Differential Equations, 5 (2008), 25-43. doi: 10.1142/S021989160800143X. [24] J. E. M. Rivera, Energy decay rates in linear thermoelasticity, Funkcial. Ekvac., 35 (1992), 19-30. [25] Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1995), 249-275. doi: 10.14492/hokmj/1381757663. [26] M. A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound, Quart. Appl. Math., 50 (1992), 727-742.
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