Uniform attractors for measure-driven quintic wave equations

Savostianov, Anton; Zelik, Sergey

doi:10.1070/rm9932

Cited by 12 publications

(33 citation statements)

References 44 publications

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“…However, the energy-to-Strichartz estimate (3.10) is problematic in the critical case p = 4. To the best of our knowledge it is proved for the periodic boundary conditions only and its validity for other boundary conditions is an open problem, see [59]. In the critical case, the global existence of SSsolutions is verified using the so-called Pohozhaev-Morawetz inequality and related non-concentration estimates, see [29,61,65].…”

Section: Statement Of the Problem And Dissipativitymentioning

confidence: 99%

“…Note that the prove given there used in a crucial way the Lyapunov functional and cannot be extended to non-autonomous case. This drawback has been overcome in [59] for the case of periodic boundary conditions using the so-called energy-to-Strichartz estimates, but obtaining such estimates in a critical case for general domains is still an open problem. This is the reason why we mainly consider the subcritical case p < 4 in this paper.…”

Section: Introductionmentioning

confidence: 99%

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Deterministic and random attractors for a wave equation with sign changing damping

Chang,

Li,

Sun

et al. 2019

Preprint

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The paper gives a detailed study of long-time dynamics generated by weakly damped wave equations in bounded 3D domains where the damping exponent depends explicitly on time and may change sign. It is shown that in the case when the non-linearity is superlinear, the considered equation remains dissipative if the weighted mean value of the dissipation rate remains positive and that the conditions of this type are not sufficient in the linear case. Two principally different cases are considered. In the case when this mean is uniform (which corresponds to deterministic dissipation rates), it is shown that the considered system possesses smooth uniform attractors as well as non-autonomous exponential attractors. In the case where the mean is not uniform (which corresponds to the random dissipation rate, for instance, when this dissipation rate is generated by the Bernoulli process), the tempered random attractor is constructed. In contrast to the usual situation, this random attractor is expected to have infinite Hausdorff and fractal dimension. The simplified model example which demonstrates infinite-dimensionality of the random attractor is also presented.

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Section: Statement Of the Problem And Dissipativitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Deterministic and random attractors for a wave equation with sign changing damping

Chang,

Li,

Sun

et al. 2019

Preprint

View full text Add to dashboard Cite

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“…by space or time regular ones, see [33,19,20] and references therein for more details. Note also that it looks natural to consider external forces which are Borel measures in time (g ∈ M b (R, H)) for the case of damped hyperbolic equations, see [25] and Section 2 for more details.…”

Section: Introductionmentioning

confidence: 99%

Entropy estimates for uniform attractors of dissipative PDEs with non translation-compact external forces

Xiong¹,

Kostianko²,

Sun³

et al. 2022

Preprint

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We study the Kolmogorov's entropy of uniform attractors for non-autonomous dissipative PDEs. The main attention is payed to the case where the external forces are not translation-compact. We present a new general scheme which allows us to give the upper bounds of this entropy for various classes of external forces through the entropy of proper projections of their hulls to the space of translation-compact functions. This result generalizes well known estimates of Vishik and Chepyzhov for the translation-compact case. The obtained results are applied to three model problems: sub-quintic 3D damped wave equation with Dirichlet boundary conditions, quintic 3D wave equation with periodic boundary conditions and 2D Navier-Stokes system in a bounded domain. The examples of finite-dimensional uniform attractors for some special external forces which are not translation-compact are also given. Contents 1. Introduction 1 2. Preliminaries. Classes of external forces 4 3. Uniform attractors and entropy: an abstract scheme 10 4. Application 1. Subcritical damped wave equation with Dirichlet BC 15 5. Application 2. Critical damped wave equation with periodic BC 20 6. Application 3. 2D Navier-Stokes system 24 7. Examples and concluding remarks 28 References 31

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“…There are several recent results on the non-autonomous version of the weakly damped wave equation with quintic, or at least supercubic, growth condition which use the concept of Shatah-Struwe solutions. Savostianov and Zelik in the article [37] obtained the existence of the uniform attractor for the problem governed by…”

Section: Introductionmentioning

confidence: 99%

Convergence of Non-autonomous Attractors for Subquintic Weakly Damped Wave Equation

Banaśkiewicz

Kalita

2021

Appl Math Optim

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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 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 a time independent function in an appropriate way.

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Uniform attractors for measure-driven quintic wave equations

Cited by 12 publications

References 44 publications

Deterministic and random attractors for a wave equation with sign changing damping

Deterministic and random attractors for a wave equation with sign changing damping

Entropy estimates for uniform attractors of dissipative PDEs with non translation-compact external forces

Convergence of Non-autonomous Attractors for Subquintic Weakly Damped Wave Equation

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