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

Wang, Shubin; Xu, Huiyang

doi:10.1007/s00033-012-0257-1

Cited by 12 publications

(9 citation statements)

References 17 publications

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“…25,LemmaA4 Our proofs of Theorem 3 to 6 mainly use the contraction principle and the linear estimates and follow the ideas of the previous. 26,27 To avoid repetition, we only give the proofs of Theorems 3 and 6.…”

Section: Global Solutions and Decay Estimates For The Problem (1) (2)mentioning

confidence: 99%

Optimal decay rates and small global solutions to the dissipative Boussinesq equation

Wang

2019

Math Methods in App Sciences

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This work is devoted to the small amplitude solutions for the initial value problem of the multidimensional dissipative Boussinesq equation. We firstly derive the pointwise estimates of the fundamental solutions by the energy method in the Fourier space. We give the asymptotic profiles of solutions to the corresponding linear problem to get the optimal decay rate for the . H s -norm of solutions in all space dimensions. Under smallness assumptions on the initial data, we study the global existence and uniqueness of solutions by the contractive mapping principle in the solution spaces with time weighted norm. KEYWORDS dissipative Boussinesq equation, optimal decay rates, small global solutions MSC CLASSIFICATION 35Q35; 35L30; 35B40He was the first to give a scientific explanation to the existence of solitary waves, long, shallow, and water waves of permanent form, found by Scott Russell. 2 A generalization of Equation (4) is considered, which arises in the modeling of nonlinear strings, namely, Math Meth Appl Sci. 2020;43:174-198. wileyonlinelibrary.com/journal/mma

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Section: Global Solutions and Decay Estimates For The Problem (1) (2)mentioning

confidence: 99%

Optimal decay rates and small global solutions to the dissipative Boussinesq equation

Wang

2019

Math Methods in App Sciences

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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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“…A generalization of the equation (1.1) to the higher dimensional case is the plate equation. For example, in Luz-Charão [12], Charão-Luz-Ikehata [3] and Luz-Ikehata-Charão [13] [22] and Wang-Xue [24] investigated decay estimates of some norms of solutions and its application to nonlinear problems. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Remarks on the Decay Rate of the Energy for Damped Modified Boussinesq-Beam Equations on the 1-D Half Line

Charão

Ikehata

2017

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Abstract. We consider the mixed problem for weakly damped modified BoussinesqBeam equations on the one dimensional half line ð0; þyÞ. We shall derive fast decay results of the total energy and L 2 -norm of solutions based on the idea due to [7], which is an essential modification of that developed by Morawetz [15]. In order to apply that idea due to [7] to the one dimensional exterior mixed problem, one also constructs an important Hardy-Sobolev type inequality, which holds only in the 1-D half line case.

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

Cited by 12 publications

References 17 publications

Optimal decay rates and small global solutions to the dissipative Boussinesq equation

Optimal decay rates and small global solutions to the dissipative Boussinesq equation

Asymptotic behavior for abstract evolution differential equations of second order

Remarks on the Decay Rate of the Energy for Damped Modified Boussinesq-Beam Equations on the 1-D Half Line

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