Pullback attractors for nonclassical diffusion equations with delays

Zhu, Kaixuan; Sun, Chunyou

doi:10.1063/1.4931480

Cited by 23 publications

(10 citation statements)

References 21 publications

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“…In recent years there is an increasing interest on this topic for both retarded ODEs and PDEs; see e.g. [5,6,8,9,26,36,44,53,60]. However, we find that the existing works mainly focus on the case where the terms involving time lags have at most sublinear nonlinearities.…”

Section: Introductionmentioning

confidence: 93%

Uniform Decay Estimates for Solutions of a Class of Retarded Integral Inequalities

Liu

2019

Preprint

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Some uniform decay estimates are established for solutions of the following type of retarded integral inequalities:As a simple example of application, the retarded scalar functional differential equation ẋ = −a(t)x+B(t, x t ) is considered, and the global asymptotic stability of the equation is proved under weaker conditions. Another example is the ODE system ẋ = F 0 (t, x) + m i=1 F i (t, x(t − r i (t))) on R n with superlinear nonlinearities F i (0 ≤ i ≤ m). The existence of a global pullback attractor of the system is established under appropriate dissipation conditions.The third example for application concerns the study of the dynamics of the functional cocycle system du dt + Au = F (θ t p, u t ) in a Banach space X with sublinear nonlinearity. In particular, the existence and uniqueness of a nonautonomous stationary solution Γ is obtained under the hyperbolicity assumption on operator A and some additional hypotheses, and the global asymptotic stability of Γ is also addressed.

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Section: Introductionmentioning

confidence: 93%

Uniform Decay Estimates for Solutions of a Class of Retarded Integral Inequalities

Liu

2019

Preprint

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“…We now recall some definitions and results concerning dynamical systems and pullback attractors. These definitions and results can be found in [7,8,9,10,12,13,19,23,34,36].…”

Section: Preliminariesmentioning

confidence: 99%

Dynamics of a non-autonomous incompressible non-Newtonian fluid with delay

Liu

Caraballo

2017

Dynamics of Partial Differential Equations

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We first study the well-posedness of a non-autonomous incompressible non-Newtonian fluid with delay. The existence of global solution is obtained by classical Galerkin approximation and the energy method. Actually, we also prove the uniqueness of solution as well as the continuous dependence on the initial value. Then we analyze the long time behavior of the dynamical system associated to the incompressible non-Newtonian fluid. Finally, we establish the existence of pullback attractors for the non-autonomous dynamical system associated to the problem.

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“…This type of problems have been studied by many authors in the past decades; see e.g. [1,2,3,6,8,11,13,14,17,18,19,20]. However, due to technical difficulties induced by time lags in the equations, we find that most of the existing works mainly focus on the case where the delay terms have at most sublinear nonlinearities.…”

Section: Introductionmentioning

confidence: 99%

Global Estimates and Regularity of Retarded Parabolic Equations with Fast-growing Nonlinearities

Li¹

2019

Preprint

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This paper is concerned with global estimates and regularity of solutions for the initial value problem of the retarded parabolic equationin a bounded domain Ω ⊂ R n with fast-growing nonlinearities and a dissipative structure, which is associated with the homogeneous Dirichlet boundary condition. Our results reveal some deeper inherent connections between dissipative structures and the regularity of solutions for such problems.

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Pullback attractors for nonclassical diffusion equations with delays

Cited by 23 publications

References 21 publications

Uniform Decay Estimates for Solutions of a Class of Retarded Integral Inequalities

Uniform Decay Estimates for Solutions of a Class of Retarded Integral Inequalities

Dynamics of a non-autonomous incompressible non-Newtonian fluid with delay

Global Estimates and Regularity of Retarded Parabolic Equations with Fast-growing Nonlinearities

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