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Addendum to: Qualitative aspects of solutions in resonators

Quintanilla R., Racke R.

Archives of Mechanics

2011

Vol. 63, nr 4

429-435

In correcting a small mistake in [10], we can prove a new result on nonexponential stability for a coupled system arising in resonators. This also gives another (surprising and simple) example for a thermoelastic system changing from exponential stability to non-exponential stability, when changing from Fourier’s law to Cattaneo’s law in modeling of the heat conduction.

Qualitative aspects of solutions in resonators

Quintanilla R., Racke R.

Archives of Mechanics

2008

Vol. 60, nr 4

345-360

We consider the system of micro-beam resonators in the thermoelastic theory of Lord and Shulmann. First, we prove the uniqueness and instability of solutions when the sign of a parameter is not prescribed. Existence of solutions and uniform bounds for the real part of the spectrum have been found. We finish the paper by proving the impossibility of the time localization of solutions.

Mathematical analysis of thermoplasticity with linear kinematic hardening

Chełmiński K., Racke R.

Journal of Applied Analysis

2006

Vol. 12, nr 1

37-57

We study thermoplasticity with the Prandtl-Reuss flow rule and with a linear evolution equation for the kinematic hardening. The yield function associated with the system under consideration depends explicitly on the temperature. To have a control on the temperature, we slightly modify the heat equation and prove that an approximation process, based on the Yosida approximation, converges to a global in time solution of the (modified) system of thermoplasticity.