Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains. (English) Zbl 1174.74320

Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Ma, T. F.

Differ. Integral Equ. 17, No. 5-6, 495-510 (2004).

Summary: The viscoelastic Euler-Bernoulli equation with nonlinear and nonlocal damping \[ u_{tt}+\Delta ^2u-\int _0^t g(t-\tau )\Delta ^2u(\tau )\,\text d\tau +a(t)u_t=0 \text { in } \Omega \times \mathbb R^+, \] where \(a(t)=M(\int _\Omega | \nabla u(x,t)| ^2\,\text dx)\), is considered in bounded or unbounded domains \(\Omega \) of \(\mathbb R^n\). The existence of global solutions and decay rates of the energy are proved.

MSC:

74K20	Plates
74Dxx	Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
93D15	Stabilization of systems by feedback

Keywords:

viscoelastic Euler-Bernoulli equation; existence of global solutions