1,203 research outputs found
Global attractors for the one dimensional wave equation with displacement dependent damping
We study the long-time behavior of solutions of the one dimensional wave
equation with nonlinear damping coefficient. We prove that if the damping
coefficient function is strictly positive near the origin then this equation
possesses a global attractor
A note on a strongly damped wave equation with fast growing nonlinearities
A strongly damped wave equation including the displacement depending
nonlinear damping term and nonlinear interaction function is considered. The
main aim of the note is to show that under the standard dissipativity
restrictions on the nonlinearities involved the initial boundary value problem
for the considered equation is globally well-posed in the class of sufficiently
regular solutions and the semigroup generated by the problem possesses a global
attractor in the corresponding phase space. These results are obtained for the
nonlinearities of an arbitrary polynomial growth and without the assumption
that the considered problem has a global Lyapunov function
Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent
In this paper the long time behaviour of the solutions of 3-D strongly damped
wave equation is studied. It is shown that the semigroup generated by this
equation possesses a global attractor in H_{0}^{1}(\Omega)\times L_{2}(\Omega)
and then it is proved that this global attractor is a bounded subset of
H^{2}(\Omega)\times H^{2}(\Omega) and also a global attractor in
H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\times H_{0}^{1}(\Omega)
Finite dimensional attractor for a composite system of wave/plate equations with localised damping
The long-term behaviour of solutions to a model for acoustic-structure
interactions is addressed; the system is comprised of coupled semilinear wave
(3D) and plate equations with nonlinear damping and critical sources. The
questions of interest are: existence of a global attractor for the dynamics
generated by this composite system, as well as dimensionality and regularity of
the attractor. A distinct and challenging feature of the problem is the
geometrically restricted dissipation on the wave component of the system. It is
shown that the existence of a global attractor of finite fractal dimension --
established in a previous work by Bucci, Chueshov and Lasiecka (Comm. Pure
Appl. Anal., 2007) only in the presence of full interior acoustic damping --
holds even in the case of localised dissipation. This nontrivial generalization
is inspired by and consistent with the recent advances in the study of wave
equations with nonlinear localised damping.Comment: 40 pages, 1 figure; v2: added references for Section 1, submitte
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