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Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity. (English) Zbl 0756.73012
It is established a global existence theorem of nonlinear classical dynamic coupled thermoelasticity for a one-dimensional displacement- temperature initial-boundary value problem with homogeneous boundary conditions. The theorem asserts that for sufficiently “small” initial data there is a smooth and unique solution to the problem, and suitably defined norms of the solution vanish or are bounded as the time goes to infinity. The reviewer notes a number of slips and typographical errors, e.g.: 1. The field equations (0.3)-(0.4) are not postulated in a dimensionless form. Therefore, introducing a unit interval \(0<x<1\) as a reference configuration makes a confusion. 2. For the same reason, the definition of the norm: \(| v|_{t,K,L}\) (p. 4) involving the coefficient \((1+s)^ K\), when \(0\leq s\leq t\), and \(t\) is the time, makes another confusion. 3. The hypothesis (4.5), p. 22, in which a constant \(\sigma\) is an upper bound for the three fields of different physical dimensions, can be hardly justified.

74A15 Thermodynamics in solid mechanics
74B20 Nonlinear elasticity
35Q72 Other PDE from mechanics (MSC2000)
Full Text: DOI
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