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Some remarks on the behaviour of the solution in dynamic processes for rate-type models. (English) Zbl 0743.73012

The dynamic problem for the semilinear rate-type consitutive equations involving a parameter \(k\) is considered. The continuous dependence of the solution on \(k\) is obtained and the problem of finite time stability is discussed. When \(k\) is the absolute temperature, the dynamic problem is studied in the framework of Cattaneo-type heat law and classical Fourier law. In the case when \(k\) is interpreted to be an internal state variable an existence and uniqueness result is obtained using a fixed point method: the problem of finite time stability is also investigated.

MSC:

74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B65 Smoothness and regularity of solutions to PDEs
74C99 Plastic materials, materials of stress-rate and internal-variable type
74A20 Theory of constitutive functions in solid mechanics
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