Memory relaxation of type III thermoelastic extensible beams and Berger plates
Filippo Dell’Oro, Vittorino Pata · Evolution equations and control theory · 2012
We analyzean abstract version of the evolution systemruling the dynamics of a memory relaxation of a type III thermoelasticextensible beam or Berger plateoccupying a volume $\Omega$\begin{equation}\begin{cases}u_{tt}-ωΔ u_{tt}+Δ^2 u-[b+|| abla u\|^2_{L^2(\Omega)}]\Delta u+Δ α_t=g\\α_{tt}-Δ α-∫_0^\inftyu(s)Δ[α(t)-α(t-s)]d s-Δ u_t=0\end{cases}\end{equation}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 semigroupis addressed.