656 research outputs found
Decay of semilinear damped wave equations:cases without geometric control condition
We consider the semilinear damped wave equation . In
this article, we obtain the first results concerning the stabilization of this
semilinear equation in cases where does not satisfy the geometric
control condition. When some of the geodesic rays are trapped, the
stabilization of the linear semigroup is semi-uniform in the sense that
for some function with when
. We provide general tools to deal with the semilinear
stabilization problem in the case where has a sufficiently fast decay
Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions
In this paper we consider a multi-dimensional wave equation with dynamic
boundary conditions, related to the Kelvin-Voigt damping. Global existence and
asymptotic stability of solutions starting in a stable set are proved. Blow up
for solutions of the problem with linear dynamic boundary conditions with
initial data in the unstable set is also obtained
Global Stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers
In this paper we introduce a finite-parameters feedback control algorithm for
stabilizing solutions of the Navier-Stokes-Voigt equations, the strongly damped
nonlinear wave equations and the nonlinear wave equation with nonlinear damping
term, the Benjamin-Bona-Mahony-Burgers equation and the KdV-Burgers equation.
This algorithm capitalizes on the fact that such infinite-dimensional
dissipative dynamical systems posses finite-dimensional long-time behavior
which is represented by, for instance, the finitely many determining parameters
of their long-time dynamics, such as determining Fourier modes, determining
volume elements, determining nodes , etc..The algorithm utilizes these finite
parameters in the form of feedback control to stabilize the relevant solutions.
For the sake of clarity, and in order to fix ideas, we focus in this work on
the case of low Fourier modes feedback controller, however, our results and
tools are equally valid for using other feedback controllers employing other
spatial coarse mesh interpolants
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