580 research outputs found

    Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications

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    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 XX which is acted on by any continuous semigroup {S(t)}t0\{S(t)\}_{t \geq 0}. Suppose that §(t)}t0\S(t)\}_{t \geq 0} possesses a global attractor A\mathcal{A}. We show that, for any generalized Banach limit LIMT\underset{T \rightarrow \infty}{\rm{LIM}} and any distribution of initial conditions m0\mathfrak{m}_0, that there exists an invariant probability measure m\mathfrak{m}, whose support is contained in A\mathcal{A}, such that Xϕ(x)dm(x)=LIMT1T0TXϕ(S(t)x)dm0(x)dt, \int_{X} \phi(x) d\mathfrak{m} (x) = \underset{T\to \infty}{\rm{LIM}} \frac{1}{T}\int_0^T \int_X \phi(S(t) x) d \mathfrak{m}_0(x) d t, for all observables ϕ\phi living in a suitable function space of continuous mappings on XX. 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 {S(t)}t0\{S(t)\}_{t \geq 0} 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 XX 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

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    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”

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    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

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    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

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    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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