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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.

MSC:
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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