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Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity. (English) Zbl 0804.35132
This paper is concerned with the Dirichlet initial-boundary value problem in one-dimensional nonlinear thermoelasticity. Using Munõz Rivera’s idea on the treatment of some boundary terms for linear thermoelasticity [J. E. Munõz Rivera, Funkc. Ekvacioj, Ser. Int. 35, No. 1, 19-30 (1992)] and the energy method, the authors prove the global existence and exponential stability of smooth solutions for small initial data. Moreover, the existence of periodic solutions for small, periodic right- hand sides is established. This paper improves Racke and Shibata’s results [R. Racke and Y. Shibata, Arch. Ration. Mech. Anal. 116, No. 1, 1-34 (1991; Zbl 0756.73012)]. The improvement is threefold: the exponential decay is obtained, less regularity of the data is required, and the proof is much simpler.
Reviewer: S.Jiang (Bonn)

MSC:
35Q72 Other PDE from mechanics (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
74B20 Nonlinear elasticity
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