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Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation. (English) Zbl 1009.74034
Summary: The linear Euler-Bernoulli viscoelastic equation \[ u_{tt}+ \Delta^2u- \int^t_0 g(t-\tau)\Delta^2 u(\tau)\, d\tau= 0\quad\text{in }\Omega\times (0,\infty) \] subject to nonlinear boundary conditions is considered. We prove existence and uniform decay rates of the energy by assuming a nonlinear and nonlocal feedback acting on the boundary, and provided that the kernel of the memory decays exponentially.

74H55 Stability of dynamical problems in solid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
45K05 Integro-partial differential equations
74D05 Linear constitutive equations for materials with memory
74M05 Control, switches and devices (“smart materials”) in solid mechanics
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
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