265 research outputs found
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
Attractors for damped quintic wave equations in bounded domains
The dissipative wave equation with a critical quintic nonlinearity in smooth
bounded three dimensional domain is considered. Based on the recent extension
of the Strichartz estimates to the case of bounded domains, the existence of a
compact global attractor for the solution semigroup of this equation is
established. Moreover, the smoothness of the obtained attractor is also shown
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
Preventing blow up by convective terms in dissipative PDEs
We study the impact of the convective terms on the global solvability or
finite time blow up of solutions of dissipative PDEs. We consider the model
examples of 1D Burger's type equations, convective Cahn-Hilliard equation,
generalized Kuramoto-Sivashinsky equation and KdV type equations, we establish
the following common scenario: adding sufficiently strong (in comparison with
the destabilizing nonlinearity) convective terms to equation prevents the
solutions from blowing up in finite time and makes the considered system
globally well-posed and dissipative and for weak enough convective terms the
finite time blow up may occur similarly to the case when the equation does not
involve convective term.
This kind of result has been previously known for the case of Burger's type
equations and has been strongly based on maximum principle. In contrast to
this, our results are based on the weighted energy estimates which do not
require the maximum principle for the considered problem
Global solvability and blow up for the convective Cahn-Hilliard equations with concave potentials
We study initial boundary value problems for the convective Cahn-Hilliard
equation \Dt u +\px^4u +u\px u+\px^2(|u|^pu)=0. It is well-known that without
the convective term, the solutions of this equation may blow up in finite time
for any . In contrast to that, we show that the presence of the convective
term u\px u in the Cahn-Hilliard equation prevents blow up at least for
. We also show that the blowing up solutions still exist if is
large enough (). The related equations like
Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard
equation, are also considered
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