Global existence and long-time behavior of the initial–boundary value problem for the dissipative Boussinesq equation

Wang, Shubin; Su, Xiao

doi:10.1016/j.nonrwa.2016.03.002

Cited by 5 publications

(4 citation statements)

References 24 publications

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“…Equation (8) possesses two types of damping terms (−Δ)u t and (−Δ) 2 u t . From (14) and (15), we find that the decay rates of solutions for the Cauchy problem of Equation (8) are mainly dominated by (−Δ)u t (as t → +∞).…”

Section: Introductionmentioning

confidence: 99%

“…Obviously, Equation (1) is the general multidimensional form of Equation (7). Wang and Su studied the existence and uniqueness of global solutions, the finite time blow-up of solutions, and the long-time behavior of solutions for the initial-boundary value problem of Equation (1) in bounded domains, 13,14 and the authors also considered the existence and nonexistence of the global solutions for the Cauchy problem of Equation (1) without the decay estimates of the solutions. 15 The main purpose of the paper is to study the asymptotic behavior and the optimal decay rate of the linearized equation of Equation (1) and the global existence and the optimal decay estimates of small solutions for the nonlinear problem (1), (2).…”

Section: Introductionmentioning

confidence: 99%

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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: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

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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“…The authors showed that finite time blow-up happens for a natural class of initial data. More background introductions about Boussinesq equation can be found in references [4,5]. An and Peirce [6] used a weakly nonlinear analysis and explored the immediate postcritical behavior of the solution of elastoplastic-microstructure models for a longitudinal movement of elastic-plastic rod, just like the following equation:…”

Section: Introductionmentioning

confidence: 99%

“…where a is a constant. Wang and Chen [7] studied the existence and uniqueness of the global solution for the Cauchy problem of the generalized double dispersion equation 5) and they proved the blow-up result of the solution by using the concavity method and under some suitable conditions. Schneider and Wayne [8] considered the following Boussinesq equation that models the water wave problem with surface tension:…”

Section: Introductionmentioning

confidence: 99%