Longtime dynamics of the Kirchhoff equations with fractional damping and supercritical nonlinearity

Yang, Zuodong; Ding, Peng-Xu; Li, Lei

doi:10.1016/j.jmaa.2016.04.079

Cited by 32 publications

(41 citation statements)

References 25 publications

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“…Therefore, the dynamical system (S(t), X) is gradient, A α = M + (N ), and every full trajectory γ = {ξ u (t)|t ∈ R} from A α is of property (i)-(ii) (cf. Theorem 2.28 in [12] and [33]). Now, we investigate the exponential attractors.…”

Section: Zhijian Yang Pengyan Ding and Xiaobin Liumentioning

confidence: 96%

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Attractors and their stability on Boussinesq type equations with gentle dissipation

Yang¹,

Ding²,

Liu³

2019

Communications on Pure &Amp; Applied Analysis

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The paper investigates longtime dynamics of Boussinesq type equations with gentle dissipation: utt + ∆ 2 u + (−∆) α ut − ∆f (u) = g(x), with α ∈ (0, 1). For general bounded domain Ω ⊂ R N (N ≥ 1), we show that there exists a critical exponent pα ≡ N +2(2α−1) (N −2) + depending on the dissipative index α such that when the growth p of the nonlinearity f (u) is up to the range: 1 ≤ p < pα, (i) the weak solutions of the equations are of additionally global smoothness when t > 0; (ii) the related dynamical system possesses a global attractor Aα and an exponential attractor A α exp in natural energy space for each α ∈ (0, 1), respectively; (iii) the family of global attractors {Aα} is upper semicontinuous at each point α 0 ∈ (0, 1], i.e., for any neighborhood U of Aα 0 , Aα ⊂ U when |α − α 0 | 1. These results extend those for structural damping case: α ∈ [1, 2) in [31, 32].

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Section: Zhijian Yang Pengyan Ding and Xiaobin Liumentioning

confidence: 96%

“…By the standard argument (cf. [33]) one easily knows that A α exp is an exponential attractor of the dynamical system (S(t), B) (equipped with Y α -topology). And by virtue of additional regularity of the weak solutions of Eq.…”

Section: Zhijian Yang Pengyan Ding and Xiaobin Liumentioning

confidence: 99%

Attractors and their stability on Boussinesq type equations with gentle dissipation

Yang¹,

Ding²,

Liu³

2019

Communications on Pure &Amp; Applied Analysis

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“…For the physical background of Eq. (4) and related researches, one can refer to [7,8,15,22,23,24] in detail. Recently, for the autonomous Kirchhoff wave model (4), i.e., g(x, t) ≡ g(x), Yang et al [24] have found an optimal supercritical…”

Section: Zhijian Yang and Yanan LImentioning

confidence: 99%

“…It is also shown in [24] that when the growth exponent p of the nonlinearity f (u) is up to the supercritical range 1 ≤ p < p α , (i) the well-posedness and longtime behavior of solutions of Eq. (4) are of characteristics of parabolic equations.…”

Section: Zhijian Yang and Yanan LImentioning

confidence: 99%

Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models

Yang¹,

Li²

2018

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we are concerned with the existence and stability of pullback exponential attractors for a non-autonomous dynamical system. (i) We propose two new criteria for the discrete dynamical system and continuous one, respectively. (ii) By applying the criteria to the non-autonomous Kirchhoff wave models with structural damping and supercritical nonlinearity we construct a family of pullback exponential attractors which are stable with respect to perturbations.

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“…We consider the following Higher-order Kirchhoff- where 1 m > is an integer constant, and Ω is a bounded domain of n This kind of wave models goes back to G. Kirchhoff [1] and has been studied by many authors under different types of hypotheses. There have been many researchers on the global attractors existence of Kirchhoff equation, we can refer [2] [3] [4] [5] [6].…”

Section: Introductionmentioning

confidence: 99%

The Global Attractors for the Higher-Order Kirchhoff-Type Equation with Nonlinear Strongly Damped Term

Sun¹,

Gao²,

Lin³

2016

IJMNTA

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We investigate the global well-posedness and the global attractors of the solutions for the Higher-order Kirchhoff-type wave equation with nonlinear strongly damping:proved by Galerkin method, and deal with the global attractors. KeywordsStrongly Nonlinear Damped, Higher-Order Kirchhoff Equation, The Existence and Uniqueness, The Global Attractors R , with a smooth dirichlet boundary ∂Ω and initial value. Moreover, ν is the unit outward normal on ∂Ω . σ and φ are scalar functions specified later, f is a given function.

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Longtime dynamics of the Kirchhoff equations with fractional damping and supercritical nonlinearity

Cited by 32 publications

References 25 publications

Attractors and their stability on Boussinesq type equations with gentle dissipation

Attractors and their stability on Boussinesq type equations with gentle dissipation

Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models

The Global Attractors for the Higher-Order Kirchhoff-Type Equation with Nonlinear Strongly Damped Term

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