Asymptotic stability and global existence in thermoelasticity with symmetry
Authors:
S. Jiang, J. E. Muñoz Rivera and R. Racke
Journal:
Quart. Appl. Math. 56 (1998), 259-275
MSC:
Primary 35Q72; Secondary 35B40, 73B30
DOI:
https://doi.org/10.1090/qam/1622566
MathSciNet review:
MR1622566
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Abstract: First we prove an exponential decay result for solutions of the equations of linear, homogeneous, isotropic thermoelasticity in bounded regions in two or three space dimensions if the rotation of the displacement vanishes. As a consequence, we describe the decay in radially symmetrical situations, and in a cylinder in ${\mathbb {R}^{3}}$. Then we establish the global existence of solutions to the corresponding nonlinear equations for small smooth initial data and a certain class of nonlinearities.
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R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975
J. A. Burns, Z. Liu, and S. Zheng, On the energy decay of a linear thermoelastic bar, J. Math. Anal. Appl. 179, 574–591 (1993)
D. E. Carlson, Linear thermoelasticity, Handbuch der Physik VIa/2, Springer-Verlag, Berlin, 1972, pp. 297–346
D. Carvalho Pereira and G. Perla Menzala, Exponential decay of solutions to a coupled system of equations of linear thermoelasticity, Mat. Applic. Comp. 8, 193–240 (1989)
D. Carvalho Pereira and G. Perla Menzala, Exponential stability in linear thermoelasticity: the inhomogeneous case, Applic. Anal. 44, 21–36 (1992)
A. Chrzȩszczyk, Some existence results in dynamical thermoelasticity, Part I. Nonlinear case, Arch. Mech. 39, 605–617 (1987)
C. M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rat. Mech. Anal. 29, 241–271 (1968)
S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl. 167, 429–442 (1992)
D. B. Henry, Topics in Analysis, Publ. Secc. Mat., Univ. Autòn. Barc. 31, 29–84 (1987)
D. B. Henry, A. Perissinitto Jr., and O. Lopes, On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Anal. 21, 65–75 (1993)
S. Jiang, Global solutions of the Neumann problem in one-dimensional nonlinear thermoelasticity, Nonlinear Anal. 19, 107–121 (1992)
S. Jiang, Exponential decay and global existence of spherically symmetric solutions in thermoelasticity, SFB 256 Preprint, No. 351, Univ. Bonn, 1994
J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal. 23, 889–899 (1992)
I. Lasiecka, J.-L. Lions, and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures et Appl. 65, 149–192 (1986)
R. Leis, Initial boundary value problems in mathematical physics, B. G. Teubner Verlag, Stuttgart, John Wiley and Sons, Chichester, 1986
Z. Liu and S. Zheng, Exponential stability of semigroup associated with thermoelastic system, Quart. Appl. Math. 51, 535–545 (1993)
J. E. Muñoz Rivera, Energy decay rates in linear thermoelasticity, Funkcial. Ekvac. 35, 19–30 (1992)
J. E. Muñoz Rivera, Asymptotic behavior of the energy in thermo-visco-elasticity, Mat. Applic. Comp. 11, 45–71 (1992)
J. Ponce and R. Racke, Global existence of small solutions to the initial value problem for nonlinear thermoelasticity, J. Diff. Equations 87, 70–83 (1990)
R. Racke, On the time-asymptotic behaviour of solutions in thermoelasticity, Proc. Roy. Soc. Edinburgh 107A, 289–298 (1987)
R. Racke, On the Cauchy problem in nonlinear 3-d-thermoelasticity, Math. Z. 203, 649–682 (1990)
R. Racke, Blow-up in nonlinear three-dimensional thermoelasticity, Math. Meth. Appl. Sci. 12, 267–273 (1990)
R. Racke, Lectures on Nonlinear Evolution Equations, Aspects of Math., Vol. 19, Vieweg-Verlag, Braunschweig, 1992.
R. Racke, Exponential decay for a class of initial boundary value problems in thermoelasticity, Mat. Applic. Comp. 12, 67–80 (1993)
M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch. Rat. Mech. Anal. 76, 97–133 (1981)
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© Copyright 1998
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