168 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
Averaging of equations of viscoelasticity with singularly oscillating external forces
Given , we consider for the nonautonomous
viscoelastic equation with a singularly oscillating external force together with the
{\it averaged} equation Under suitable assumptions on
the nonlinearity and on the external force, the related solution processes
acting on the natural weak energy space
are shown to possess uniform attractors . Within the
further assumption , the family turns out to
be bounded in , uniformly with respect to .
The convergence of the attractors to the attractor
of the averaged equation as is also
established
Time-Dependent Attractor for the Oscillon Equation
We investigate the asymptotic behavior of the nonautonomous evolution problem
generated by the Klein-Gordon equation in an expanding background, in one space
dimension with periodic boundary conditions, with a nonlinear potential of
arbitrary polynomial growth. After constructing a suitable dynamical framework
to deal with the explicit time dependence of the energy of the solution, we
establish the existence of a regular, time-dependent global attractor. The
sections of the attractor at given times have finite fractal dimension.Comment: to appear in Discrete and Continuous Dynamical System
Existence Results for Some Damped Second-Order Volterra Integro-Differential Equations
In this paper we make a subtle use of operator theory techniques and the
well-known Schauder fixed-point principle to establish the existence of
pseudo-almost automorphic solutions to some second-order damped
integro-differential equations with pseudo-almost automorphic coefficients. In
order to illustrate our main results, we will study the existence of
pseudo-almost automorphic solutions to a structurally damped plate-like
boundary value problem.Comment: 20 pages. arXiv admin note: substantial text overlap with
arXiv:1402.563
Convergence of non-autonomous attractors for subquintic weakly damped wave equation
We study the non-autonomous weakly damped wave equation with subquintic
growth condition on the nonlinearity. Our main focus is the class of
Shatah--Struwe solutions, which satisfy the Strichartz estimates and are
coincide with the class of solutions obtained by the Galerkin method. For this
class we show the existence and smoothness of pullback, uniform, and cocycle
attractors and the relations between them. We also prove that these
non-autonomous attractors converge upper-semicontinuously to the global
attractor for the limit autonomous problem if the time-dependent nonlinearity
tends to time independent function in an appropriate way
Рівномірний атрактор хвильового рівняння з нелінійним демпфуванням, що явно залежить від часу
The paper deals with long-time behavior of the solutions to the initial-boundary value problem for a non-autonomous non-linear wave equation. The peculiarity of the equation is the non-linear damping term depending explicitly on time. The problem is studied in the framework of the theory of processes and their attractors. The family of processes generated by the initial-boundary value problem is introduced. It is proved that this family is uniformly (with respect to the time-dependent damping coefficient) dissipative and asymptotically compact, thus possesses a uniqueuniform attractor. The attractor is a compact set in the common phase space of the processes.Вивчається асимптотична поведінка розв’язків початково-крайової задачі для неавтономного нелінійного хвильового рівняння. Особливістю рівняння є те, що доданок рівняння, який відповідає за демпфування, є нелінійним і залежить явно від часу. Дослідження проведено у рамках теорії процесів та їх атракторів. Побудовано сім’юпроцесів, що відповідає початково-крайовій задачі. Доведено, що ця сім’яє рівномірно (відносно коефіцієнта демпфування, який залежить від часу) дисипативною та асимптотично компактною, отже має єдиний рівномірний атрактор. Атрактор є компактною множиною у спільному фазовому просторі процесів
Attractors for processes on time-dependent spaces. Applications to wave equations
For a process U(t,s) acting on a one-parameter family of normed spaces, we
present a notion of time-dependent attractor based only on the minimality with
respect to the pullback attraction property. Such an attractor is shown to be
invariant whenever the process is T-closed for some T>0, a much weaker property
than continuity (defined in the text). As a byproduct, we generalize the recent
theory of attractors in time-dependent spaces developed in [10]. Finally, we
exploit the new framework to study the longterm behavior of wave equations with
time-dependent speed of propagation
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