Global Existence and Asymptotic Behavior of Solutions to the Generalized Damped Boussinesq Equation

Wang, Yinxia; Zhao, Hongbin

doi:10.1155/2013/364165

Cited by 9 publications

(10 citation statements)

References 16 publications

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“…Our equation (1.1) or (1.5) is a fourth-order wave equation. For more higher-order wave equations, we refer to [23][24][25][26]21] and the references therein. Also we refer to [7][8][9]17] for various aspects of dissipation of the plate equation.…”

Section: Generalized Cubic Double Dispersion Equationmentioning

confidence: 99%

Global existence and asymptotic behavior of solutions to the generalized cubic double dispersion equation

Kawashima

Wang

2015

Anal. Appl.

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Section: Generalized Cubic Double Dispersion Equationmentioning

confidence: 99%

Global existence and asymptotic behavior of solutions to the generalized cubic double dispersion equation

Kawashima

Wang

2015

Anal. Appl.

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“…Proof We shall prove () by applying to () the energy method in the Fourier space. Such an energy method was first developed in and then used in many papers (e.g., ). Multiplying () by

{\overset{û}{true}}_{t}

and taking the real part, we arrive at

\frac{1}{2} \frac{d}{dt} ({}, ((), 1 + | ξ |^{2} + | ξ |^{4}) | {\overset{u}{false}}_{t} |^{2} + ((), | ξ |^{2} + | ξ |^{4}) | \overset{u}{false} |^{2}) + μ | ξ |^{2} | {\overset{u}{false}}_{t} |^{2} = 0 .

We multiply () by

\overset{û}{true}

and take the real part.…”

Section: Decay Rate Of Solutions To Linear Problemmentioning

confidence: 99%

“…We shall prove (3.1) by applying to (2.1) the energy method in the Fourier space. Such an energy method was first developed in [15] and then used in many papers (e.g., [16][17][18][19]). Multiplying (2.1) by N O u t and taking the real part, we arrive at…”

Section: Decay Rate Of Solutions To Linear Problemmentioning

confidence: 99%

Asymptotic behavior of solutions to a class of nonlinear wave equations of sixth order with damping

Wang

2016

Math Methods in App Sciences

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We investigate the initial value problem for a class of nonlinear wave equations of sixth order with damping. The decay structure of this equation is of the regularity‐loss type, which causes difficulty in high‐frequency region. By using the Fourier splitting frequency technique and energy method in Fourier space, we establish asymptotic profiles of solutions to the linear equation that is given by the convolution of the fundamental solutions of heat and free wave equation. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear equation obtained in this paper that is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to this equation are also established. Copyright © 2016 John Wiley & Sons, Ltd.

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“…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%

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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Global Existence and Asymptotic Behavior of Solutions to the Generalized Damped Boussinesq Equation

Cited by 9 publications

References 16 publications

Global existence and asymptotic behavior of solutions to the generalized cubic double dispersion equation

Global existence and asymptotic behavior of solutions to the generalized cubic double dispersion equation

Asymptotic behavior of solutions to a class of nonlinear wave equations of sixth order with damping

On properties of solutions to the improved modified Boussinesq equation

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