580 research outputs found
Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications
In this work we study certain invariant measures that can be associated to
the time averaged observation of a broad class of dissipative semigroups via
the notion of a generalized Banach limit. Consider an arbitrary complete
separable metric space which is acted on by any continuous semigroup
. Suppose that possesses a global
attractor . We show that, for any generalized Banach limit
and any distribution of initial
conditions , that there exists an invariant probability measure
, whose support is contained in , such that for all
observables living in a suitable function space of continuous mappings
on .
This work is based on a functional analytic framework simplifying and
generalizing previous works in this direction. In particular our results rely
on the novel use of a general but elementary topological observation, valid in
any metric space, which concerns the growth of continuous functions in the
neighborhood of compact sets. In the case when does not
possess a compact absorbing set, this lemma allows us to sidestep the use of
weak compactness arguments which require the imposition of cumbersome weak
continuity conditions and limits the phase space to the case of a reflexive
Banach space. Two examples of concrete dynamical systems where the semigroup is
known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic
Fast spatial behavior in higher order in time equations and systems
In this work, we consider the spatial decay for high-order parabolic (and combined with a hyperbolic) equation in a semi-infinite cylinder. We prove a Phragmén-Lindelöf alternative function and, by means of some appropriate inequalities, we show that the decay is of the type of the square of the distance to the bounded end face of the cylinder. The thermoelastic case is also considered when the heat conduction is modeled using a high-order parabolic equation. Though the arguments are similar to others usually applied, we obtain new relevant results by selecting appropriate functions never considered beforePeer ReviewedPostprint (published version
Twenty-eight years with “Hyperbolic Conservation Laws with Relaxation”
This paper is a review on the results inspired by the publication “Hyperbolic conservation laws with relaxation” by Tai-Ping Liu [1], with emphasis on the topic of nonlinear waves (specifically, rarefaction and shock waves). The aim is twofold: firstly, to report in details the impact of the article on the subsequent research in the area; secondly, to detect research trends which merit attention in the (near) future
Fast spatial behavior in higher order in time equations and systems
Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGIn this work, we consider the spatial decay for high-order parabolic (and combined with a hyperbolic) equation in a semi-infinite cylinder. We prove a Phragmén-Lindelöf alternative function and, by means of some appropriate inequalities, we show that the decay is of the type of the square of the distance to the bounded end face of the cylinder. The thermoelastic case is also considered when the heat conduction is modeled using a high-order parabolic equation. Though the arguments are similar to others usually applied, we obtain new relevant results by selecting appropriate functions never considered before.Agencia Estatal de Investigación | Ref. PGC2018-096696-B-I00Agencia Estatal de Investigación | Ref. PID2019-105118GB-I0
Exponential stability of the wave equation with memory and time delay
We study the asymptotic behaviour of the wave equation with viscoelastic
damping in presence of a time-delayed damping. We prove exponential stability
if the amplitude of the time delay term is small enough
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