×

Exponential stability for the wave equations with local Kelvin-Voigt damping. (English) Zbl 1114.35023

Let \(\Omega \subset \mathbb R^N\) be a bounded open set with Lipschitz boundary. The following wave equation with local viscoelastic damping distributed around the boundary is considered \[ \rho (x)u_{tt}(x, t) = \text{div}(a(x)\nabla u + b(x)\nabla u_t)\quad (x\in\Omega,\;t>0). \] This equation involving a constructive viscoelastic damping \(\operatorname{div} b(x)\nabla u_t\), models the vibrations of an elastic body which has one part made of viscoelastic material. For the considered equation the exponential stability conditions are established.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
93D20 Asymptotic stability in control theory
35B37 PDE in connection with control problems (MSC2000)
35B35 Stability in context of PDEs
35L70 Second-order nonlinear hyperbolic equations
PDFBibTeX XMLCite
Full Text: DOI