American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay
  • CPAA Home
  • This Issue
  • Next Article
    Sharp threshold for scattering of a generalized Davey-Stewartson system in three dimension
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

[1]

Virginie Bonnaillie-Noël. Harmonic oscillators with Neumann condition on the half-line. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2221-2237. doi: 10.3934/cpaa.2012.11.2221
[2]

Larissa V. Fardigola. Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition. Mathematical Control & Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31
[3]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[4]

Shitao Liu. Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement. Evolution Equations & Control Theory, 2013, 2 (2) : 355-364. doi: 10.3934/eect.2013.2.355
[5]

Nicola Abatangelo, Serena Dipierro, Mouhamed Moustapha Fall, Sven Jarohs, Alberto Saldaña. Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1205-1235. doi: 10.3934/dcds.2019052
[6]

Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295
[7]

Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070
[8]

Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095
[9]

Yongming Liu, Lei Yao. Global solution and decay rate for a reduced gravity two and a half layer model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2613-2638. doi: 10.3934/dcdsb.2018267
[10]

Lena-Susanne Hartmann, Ilya Pavlyukevich. Advection-diffusion equation on a half-line with boundary Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 637-655. doi: 10.3934/dcdsb.2018200
[11]

Türker Özsarı, Nermin Yolcu. The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3285-3316. doi: 10.3934/cpaa.2019148
[12]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[13]

Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061
[14]

Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic & Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601
[15]

Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191
[16]

Jian Zhai, Zhihui Cai. $\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1071-1085. doi: 10.3934/dcdsb.2009.11.1071
[17]

Lan Zeng, Guoxi Ni, Yingying Li. Low Mach number limit of strong solutions for 3-D full compressible MHD equations with Dirichlet boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5503-5522. doi: 10.3934/dcdsb.2019068
[18]

António J.G. Bento, Nicolae Lupa, Mihail Megan, César M. Silva. Integral conditions for nonuniform $μ$-dichotomy on the half-line. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3063-3077. doi: 10.3934/dcdsb.2017163
[19]

Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1391-1407. doi: 10.3934/dcds.2015.35.1391
[20]

Pedro M. Jordan. Second-sound phenomena in inviscid, thermally relaxing gases. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2189-2205. doi: 10.3934/dcdsb.2014.19.2189

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences