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On formation of singularities in one-dimensional nonlinear thermoelasticity. (English) Zbl 0712.73023
In this paper the authors studied the formation of singularities in one- dimensional nonlinear thermoelastic materials. Their constitutive relations are similar to those used by C. M. Dafermos and L. Hsiao [Q. Appl. Math. 44, 463-474 (1986; Zbl 0661.35009)]. Although their constitutive equations may not be appropriate for any real material, they satisfy the assumptions used to establish the global existence of smooth solutions when the data are close to the equilibrium. Moreover, they are fully compatible with the second law of thermodynamics; in particular there is a free energy. They considered the Cauchy problem in which the body occupies the entire real line and the initial values of the strain, velocity, and temperature are prescribed.
As their main result they show that there are smooth initial data for which the solution will develop singularities at finite time. Their proof is similar to that of Dafermos and Hsiao. The primary difference is that their equations of evolution contain an additional term that prevents them from using the maximum principle to obtain an important a priori bound. This difficulty is overcome by exploiting some relations associated with the second law of thermodynamics.
This work might be useful for those researchers working in the area of propagation of singularities.
Reviewer: H.Demiray

MSC:
74J99 Waves in solid mechanics
74B20 Nonlinear elasticity
74A15 Thermodynamics in solid mechanics
35L67 Shocks and singularities for hyperbolic equations
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