598 research outputs found
Uniform attractors for non-autonomous wave equations with nonlinear damping
We consider dynamical behavior of non-autonomous wave-type evolutionary
equations with nonlinear damping, critical nonlinearity, and time-dependent
external forcing which is translation bounded but not translation compact
(i.e., external forcing is not necessarily time-periodic, quasi-periodic or
almost periodic). A sufficient and necessary condition for the existence of
uniform attractors is established using the concept of uniform asymptotic
compactness. The required compactness for the existence of uniform attractors
is then fulfilled by some new a priori estimates for concrete wave type
equations arising from applications. The structure of uniform attractors is
obtained by constructing a skew product flow on the extended phase space for
the norm-to-weak continuous process.Comment: 33 pages, no figur
Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit
We apply the dynamical approach to the study of the second order semi-linear
elliptic boundary value problem in a cylindrical domain with a small parameter
at the second derivative with respect to the "time" variable corresponding to
the axis of the cylinder.
We prove that, under natural assumptions on the nonlinear interaction
function and the external forces, this problem possesses the uniform attractors
and that these attractors tend to the attractor of the limit parabolic
equation. Moreover, in case where the limit attractor is regular, we give the
detailed description of the structure of these uniform attractors when the
perturbation parameter is small enough, and estimate the symmetric distance
between the perturbed and non-perturbed attractors
Violation of hyperbolicity in a diffusive medium with local hyperbolic attractor
Departing from a system of two non-autonomous amplitude equations,
demonstrating hyperbolic chaotic dynamics, we construct a 1D medium as ensemble
of such local elements introducing spatial coupling via diffusion. When the
length of the medium is small, all spatial cells oscillate synchronously,
reproducing the local hyperbolic dynamics. This regime is characterized by a
single positive Lyapunov exponent. The hyperbolicity survives when the system
gets larger in length so that the second Lyapunov exponent passes zero, and the
oscillations become inhomogeneous in space. However, at a point where the third
Lyapunov exponent becomes positive, some bifurcation occurs that results in
violation of the hyperbolicity due to the emergence of one-dimensional
intersections of contracting and expanding tangent subspaces along trajectories
on the attractor. Further growth of the length results in two-dimensional
intersections of expanding and contracting subspaces that we classify as a
stronger type of the violation. Beyond of the point of the hyperbolicity loss,
the system demonstrates an extensive spatiotemporal chaos typical for extended
chaotic systems: when the length of the system increases the Kaplan-Yorke
dimension, the number of positive Lyapunov exponents, and the upper estimate
for Kolmogorov-Sinai entropy grow linearly, while the Lyapunov spectrum tends
to a limiting curve.Comment: 11 pages, 11 figures, results reproduced with higher precision, new
figures added, text revise
Random Attractor for Stochastic Wave Equation with Arbitrary Exponent and Additive Noise on
Asymptotic random dynamics of weak solutions for a damped stochastic wave
equation with the nonlinearity of arbitrarily large exponent and the additive
noise on is investigated. The existence of a pullback random
attractor is proved in a parameter region with a breakthrough in proving the
pullback asymptotic compactness of the cocycle with the quasi-trajectories
defined on the integrable function space of arbitrary exponent and on the
unbounded domain of arbitrary dimension
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
Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction
We perform a thorough study of the blow up profiles associated to the
following second order reaction-diffusion equation with non-homogeneous
reaction: in the range
of exponents . We classify blow up solutions in
self-similar form, that are likely to represent typical blow up patterns for
general solutions. We thus show that the non-homogeneous coefficient
has a strong influence on the qualitative aspects related to the
finite time blow up. More precisely, for , blow up profiles have
similar behavior to the well-established profiles for the homogeneous case
, and typically \emph{global blow up} occurs, while for
sufficiently large, there exist blow up profiles for which blow up \emph{occurs
only at space infinity}, in strong contrast with the homogeneous case. This
work is a part of a larger program of understanding the influence of unbounded
weights on the blow up behavior for reaction-diffusion equations
A Non-Autonomous Strongly Damped Wave Equation: Existence and Continuity of the Pullback Attractor
In this paper we consider the strongly damped wave equation with time dependent terms
utt − u − γ(t) ut + β"(t)ut = f(u), in a bounded domain ⊂ Rn, under some restrictions on β"(t), γ(t) and growth restrictions on the non-linear term f. The function β"(t) depends on a parameter ε, β"(t)
"!0 −→ 0. We will prove, under suitable assumptions, local and global well posedness (using the uniform sectorial operators theory), the existence and regularity of pullback attractors {A"(t) :
t ∈ R}, uniform bounds for these pullback attractors, characterization of these pullback attractors and their upper and lower semicontinuity at ǫ = 0
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