467 research outputs found
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
Nonlinear Nonoverlapping Schwarz Waveform Relaxation for Semilinear Wave Propagation
We introduce a non-overlapping variant of the Schwarz waveform relaxation
algorithm for semilinear wave propagation in one dimension. Using the theory of
absorbing boundary conditions, we derive a new nonlinear algorithm. We show
that the algorithm is well-posed and we prove its convergence by energy
estimates and a Galerkin method. We then introduce an explicit scheme. We prove
the convergence of the discrete algorithm with suitable assumptions on the
nonlinearity. We finally illustrate our analysis with numerical experiments.Comment: 20 page
Hyperboloidal layers for hyperbolic equations on unbounded domains
We show how to solve hyperbolic equations numerically on unbounded domains by
compactification, thereby avoiding the introduction of an artificial outer
boundary. The essential ingredient is a suitable transformation of the time
coordinate in combination with spatial compactification. We construct a new
layer method based on this idea, called the hyperboloidal layer. The method is
demonstrated on numerical tests including the one dimensional Maxwell equations
using finite differences and the three dimensional wave equation with and
without nonlinear source terms using spectral techniques.Comment: 23 pages, 23 figure
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
Open Boundaries for the Nonlinear Schrodinger Equation
We present a new algorithm, the Time Dependent Phase Space Filter (TDPSF)
which is used to solve time dependent Nonlinear Schrodinger Equations (NLS).
The algorithm consists of solving the NLS on a box with periodic boundary
conditions (by any algorithm). Periodically in time we decompose the solution
into a family of coherent states. Coherent states which are outgoing are
deleted, while those which are not are kept, reducing the problem of reflected
(wrapped) waves. Numerical results are given, and rigorous error estimates are
described.
The TDPSF is compatible with spectral methods for solving the interior
problem. The TDPSF also fails gracefully, in the sense that the algorithm
notifies the user when the result is incorrect. We are aware of no other method
with this capability.Comment: 21 pages, 4 figure
Dynamics of wave equations with moving boundary
This paper is concerned with long-time dynamics of weakly damped semilinear wave equations defined on domains with moving boundary. Since the boundary is a function of the time variable the problem is intrinsically non-autonomous. Under the hypothesis that the lateral boundary is time-like, the solution operator of the problem generates an evolution process U(t, τ ) : Xτ → Xt, where Xt are timedependent Sobolev spaces. Then, by assuming the domains are expanding, we establish the existence of minimal pullback attractors with respect to a universe of tempered sets defined by the forcing terms. Our assumptions allow nonlinear perturbations with critical growth and unbounded time-dependent external forces.Conselho Nacional de Desenvolvimento Científico e TecnológicoMinisterio de EducaciónMinisterio de Ciencia e Innovació
Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term
We consider a differential model describing nonisothermal fast phase
separation processes taking place in a three-dimensional bounded domain. This
model consists of a viscous Cahn-Hilliard equation characterized by the
presence of an inertial term , being the order parameter,
which is linearly coupled with an evolution equation for the (relative)
temperature \teta. The latter can be of hyperbolic type if the
Cattaneo-Maxwell heat conduction law is assumed. The state variables and the
chemical potential are subject to the homogeneous Neumann boundary conditions.
We first provide conditions which ensure the well-posedness of the initial and
boundary value problem. Then, we prove that the corresponding dynamical system
is dissipative and possesses a global attractor. Moreover, assuming that the
nonlinear potential is real analytic, we establish that each trajectory
converges to a single steady state by using a suitable version of the
Lojasiewicz-Simon inequality. We also obtain an estimate of the decay rate to
equilibrium
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