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Exponential stability of a flexible structure with history and thermal effect. (English) Zbl 07250669

The paper deals with the asymptotic behavior of a system modeling oscillations of a beam with a memory and a thermal effect given by the Green-Naghdi III model. The system has the form of initial-boundary value problem \begin{align*} & m(x)u_{tt}-g(0)u_{xx}-\sigma_x(u_x,u_{xt})-\int_0^\infty g'(s)u_{xx}(t-s)\mathrm{d}s -\xi\theta_{xt},\\ &\theta_{tt}-\kappa \theta_{xx} -\beta \theta_{xxt} = 0\quad \text{in}\quad (0,\ell)\times (0,\infty),\\ &u(0,t)=u(\ell,t)=\theta(0,t)=\theta(\ell,t)=0,\ t>0,\\ &u(x,0)=u_0(x),\ u_t(x,0)=u_1(x),\ \theta(x,0)=\theta_0(x),\ \theta_t(x,0)=\varphi_0(x)\, x\in (0,\ell) \end{align*} with the stress \(\sigma(u_x,u_{xt})=p(x)u_x+2\delta(x)u_{xt}\), positive constants \(\beta\), \(\kappa\), \(\xi\) and positive functions \(m,\ p,\ \delta\ \in W^{1,\infty}(0,\ell)\).
Using the arguments from spectral theory the authors prove the well-posedness and exponential stability of the problem.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
45N05 Abstract integral equations, integral equations in abstract spaces
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35R09 Integro-partial differential equations
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