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December  2012, 1(2): 251-270. doi: 10.3934/eect.2012.1.251

## Memory relaxation of type III thermoelastic extensible beams and Berger plates

 1 Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, Via Bonardi 9, Milano 20133 2 Dipartimento di Matematica “Francesco Brioschi”, Politecnico di Milano, Via Bonardi 9, Milano 20133

Received  February 2012 Revised  July 2012 Published  October 2012

We analyze an abstract version of the evolution system ruling the dynamics of a memory relaxation of a type III thermoelastic extensible beam or Berger plate occupying a volume $\Omega$ $$\begin{cases} u_{tt}-ωΔ u_{tt}+Δ^2 u-[b +||\nabla u\|^2_{L^2(\Omega)}]\Delta u+Δ α_t=g\\ α_{tt}-Δ α-∫_0^\infty u(s)Δ[α(t)-α(t-s)]d s-Δ u_t=0 \end{cases}$$ subject to hinged boundary conditions for $u$ and to the Dirichlet boundary condition for $\alpha$, where the dissipation is entirely contributed by the convolution term in the second equation. The study of the asymptotic properties of the related solution semigroup is addressed.
Citation: Filippo Dell'Oro, Vittorino Pata. Memory relaxation of type III thermoelastic extensible beams and Berger plates. Evolution Equations & Control Theory, 2012, 1 (2) : 251-270. doi: 10.3934/eect.2012.1.251
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