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The exponential stabilization of the higher-dimensional linear system of thermoviscoelasticity. (English) Zbl 0922.35018
The author studies the exponential decay of solutions to the linear system of thermoviscoelasticity: $$u'' - \mu \Delta u -(\lambda+\mu)\nabla \text{div }u +\mu g*\Delta u +(\lambda+\mu)g*\nabla\text{div }u +\alpha\nabla\theta=0$$, $$\theta' - \Delta \theta +\beta \text{div } u' =0$$ subject to initial and Dirichlet boundary conditions. The assumptions made on the relaxation function $$g$$ are standard and they imply that the static modulus is positive. The main result of this paper is to prove that the energy functional $$E(u,\theta,t)$$ decays exponentially as $$t\to\infty$$. It is known that in the higher-dimensional case the energy functional defined as $$E(u,\theta,t)={1\over 2} (\| u'(t)\|_2^2 +{\alpha\over \beta}\| \theta(t)\|_2^2 +\mu \| \nabla u(t)\|_2^2 +(\lambda+\mu) \| \text{div }u(t)\|_2^2)$$, in general, does not tend to zero. In order to increase the loss of energy and subsequently to ensure the exponential decay of solutions the author introduces a boundary velocity feedback on a part of the boundary of a thermoviscoelastic body, which is clamped along the rest of its boundary.

##### MSC:
 35B35 Stability in context of PDEs 74Hxx Dynamical problems in solid mechanics 74A15 Thermodynamics in solid mechanics 45K05 Integro-partial differential equations
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