1,323 research outputs found
Finite-dimensional attractors for the quasi-linear strongly-damped wave equation
We present a new method of investigating the so-called quasi-linear strongly
damped wave equations in bounded 3D domains. This method
allows us to establish the existence and uniqueness of energy solutions in the
case where the growth exponent of the non-linearity is less than 6 and
may have arbitrary polynomial growth rate. Moreover, the existence of a
finite-dimensional global and exponential attractors for the solution semigroup
associated with that equation and their additional regularity are also
established. In a particular case which corresponds to the
so-called semi-linear strongly damped wave equation, our result allows to
remove the long-standing growth restriction .Comment: 36 page
Attractors for Damped Semilinear Wave Equations with Singularly Perturbed Acoustic Boundary Conditions
Under consideration is the damped semilinear wave equation in a bounded domain in
subject to an acoustic boundary condition with a singular perturbation, which
we term "massless acoustic perturbation," \ep\delta_{tt}+\delta_t+\delta =
-u_t\quad\text{for}\quad \ep\in[0,1]. By adapting earlier work by S.
Frigeri, we prove the existence of a family of global attractors for each
\ep\in[0,1]. We also establish the optimal regularity for the global
attractors, as well as the existence of an exponential attractor, for each
\ep\in[0,1]. The later result insures the global attractors possess finite
(fractal) dimension, however, we cannot yet guarantee that this dimension is
independent of the perturbation parameter \ep. The family of global
attractors are upper-semicontinuous with respect to the perturbation parameter
\ep, a result which follows by an application of a new abstract result also
contained in this article. Finally, we show that it is possible to obtain the
global attractors using weaker assumptions on the nonlinear term , however,
in that case, the optimal regularity, the finite dimensionality, and the
upper-semicontinuity of the global attractors does not necessarily hold.Comment: To appear in EJDE. arXiv admin note: substantial text overlap with
arXiv:1503.01821 and text overlap with arXiv:1302.426
Hyperbolic Relaxation of Reaction Diffusion Equations with Dynamic Boundary Conditions
Under consideration is the hyperbolic relaxation of a semilinear
reaction-diffusion equation on a bounded domain, subject to a dynamic boundary
condition. We also consider the limit parabolic problem with the same dynamic
boundary condition. Each problem is well-posed in a suitable phase space where
the global weak solutions generate a Lipschitz continuous semiflow which admits
a bounded absorbing set. We prove the existence of a family of global
attractors of optimal regularity. After fitting both problems into a common
framework, a proof of the upper-semicontinuity of the family of global
attractors is given as the relaxation parameter goes to zero. Finally, we also
establish the existence of exponential attractors.Comment: to appear in Quarterly of Applied Mathematic
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 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
Strong Uniform Attractors for Non-Autonomous Dissipative PDEs with non translation-compact external forces
We give a comprehensive study of strong uniform attractors of non-autonomous
dissipative systems for the case where the external forces are not translation
compact. We introduce several new classes of external forces which are not
translation compact, but nevertheless allow to verify the attraction in a
strong topology of the phase space and discuss in a more detailed way the class
of so-called normal external forces introduced before. We also develop a
unified approach to verify the asymptotic compactness for such systems based on
the energy method and apply it to a number of equations of mathematical physics
including the Navier-Stokes equations, damped wave equations and
reaction-diffusing equations in unbounded domains
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