On the asymptotic behavior of solution for the generalized IBq equation with hydrodynamical damped term

Wang, Shubin; Xu, Huiyang

doi:10.1016/j.jde.2011.12.016

Cited by 42 publications

(23 citation statements)

References 23 publications

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“…(1.6) Under smallness condition on the initial data, Wang and Xu [23] obtained asymptotic behavior of global solutions to (1.5) and (1.6) by the contraction mapping principle. Later, global existence and asymptotic behavior of solutions were refined in [24].…”

Section: Equation (14) Has the Following Generalized Formmentioning

confidence: 99%

“…We may refer to [6,8,[16][17][18][21][22][23][24]. For quantum stochastic evolution inclusions and variational inclusions, some related results have been established in [12].…”

Section: Equation (14) Has the Following Generalized Formmentioning

confidence: 99%

“…Next, we state the decay estimates of the solution operators G(t) and H(t) appearing in the solution formula (2.8), which was established in [23] and [24].…”

Section: Decay Property Of Solution Operatormentioning

confidence: 99%

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On properties of solutions to the improved modified Boussinesq equation

Wang¹,

Wang²

2016

J. Nonlinear Sci. Appl.

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In this paper, we investigate the Cauchy problem for the generalized IBq equation with damping in one dimensional space. When σ = 1, the nonlinear approximation of the global solutions is established under small condition on the initial value. Moreover, we show that as time tends to infinity, the solution is asymptotic to the superposition of nonlinear diffusion waves which are given explicitly in terms of the selfsimilar solution of the viscous Burgers equation. When σ ≥ 2, we prove that our global solution converges to the superposition of diffusion waves which are given explicitly in terms of the solution of linear parabolic equation. c 2016 all rights reserved.

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Section: Equation (14) Has the Following Generalized Formmentioning

confidence: 99%

“…We may refer to [6,8,[16][17][18][21][22][23][24]. For quantum stochastic evolution inclusions and variational inclusions, some related results have been established in [12].…”

Section: Equation (14) Has the Following Generalized Formmentioning

confidence: 99%

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On properties of solutions to the improved modified Boussinesq equation

Wang¹,

Wang²

2016

J. Nonlinear Sci. Appl.

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“…Estimates for the case θ = 0 or θ = 1 appear, for example, in the articles [21] and [22]. Note that the case θ = 0 represents a frictional damping and the case θ = 1 implies a hydrodynamics damping.…”

Section: The Linearized Improved Boussinesq Equationmentioning

confidence: 99%

“…Moreover, the results obtained in Section 3 are new and, in particular, the decay rates obtained in Subsection 3.3 for the total energy of the IBq equation with fractional damping (− ) θ u t (0 ≤ θ ≤ 1) have a novelty for the case 0 < θ < 1. The cases θ = 0 and θ = 1 are treated by Wang-Xu in [22] and [21], respectively. We also observe that in the applications given in Section 3 one can deal with more general operators than the Laplace operator, and due to this fact the decay rates depend on the order of such operators.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behavior for abstract evolution differential equations of second order

Luz

Ikehata

Charão

2015

Journal of Differential Equations

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evolution differential equations of second order in time are studied in order to get (almost) optimal decay estimates to the corresponding energy functional of the equations. The framework is supported by a special energy method in the associated Fourier space. The constructed abstract theory can be applied to several concrete evolutionary partial differential equations as is illustrated in the last section of the paper.

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Asymptotic behavior for the singularly perturbed damped Boussinesq equation

Yang

2014

Math. Meth. Appl. Sci.

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Communicated by T. HishidaThis work is focused on the long-time behavior of solutions to the singularly perturbed damped Boussinesq equation in a 3D case u tt C 2 u u t f.u/ D g.x/, where > 0 is small enough. Without any growth restrictions on the nonlinearity f.u/, we establish in an appropriate bounded phase space a finite dimensional global attractor as well as an exponential attractor of optimal regularity. The key step is the estimate of the difference between the solutions of the damped Boussinesq equation and the corresponding pseudo-parabolic equation. (1.5) and its generalization form (see [3])which is used to describe the small nonlinear oscillations of elastic membranes in the presence of viscosity.

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On the asymptotic behavior of solution for the generalized IBq equation with hydrodynamical damped term

Cited by 42 publications

References 23 publications

On properties of solutions to the improved modified Boussinesq equation

On properties of solutions to the improved modified Boussinesq equation

Asymptotic behavior for abstract evolution differential equations of second order

Asymptotic behavior for the singularly perturbed damped Boussinesq equation

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