Cauchy problem for a higher‐order Boussinesq equation

We study the initial-value problem for a general class of nonlinear nonlocal wave equations arising in one-dimensional nonlocal elasticity. The model involves a convolution integral operator with a general kernel function whose Fourier transform is nonnegative. We show that some well-known examples of nonlinear wave equations, such as Boussinesq-type equations, follow from the present model for suitable choices of the kernel function. We establish global existence of solutions of the model assuming enough smoothness on the initial data together with some positivity conditions on the nonlinear term. Furthermore, conditions for finite time blow-up are provided.

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“…The global existence of the Cauchy problem of the higher-order Boussinesq equation has been proved in [8,16,17]. (v) The Gaussian kernel [10]:…”

Section: Examples For the Kernelmentioning

confidence: 99%

Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity

2009

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“…However, in many cases, this kind of description is insufficient and it is necessary to apply the more general nonlocal elas ticity theory. The need to use this theory arises, for example, in describing the elastic properties of statis tically inhomogeneous media, geological media, com posite and structural materials, crystals with defects, and perturbations in perfect crystals for the case of the characteristic size of perturbation being comparable to the lattice constant [3,[9][10][11][12][13][14][15][16][17][18].…”

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confidence: 99%

Nonlinear solitary waves in nonlocal elastic solids

Pamyatnykh

Ursulov

2012

Acoust. Phys.

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The possibility of new weakly nonlinear solitary waves in nonlocal elastic media is demonstrated. The properties of these waves are determined by the characteristic features of wave dispersion in the linear approximation, and their velocity and amplitude cannot exceed certain limiting values. In the case of small amplitudes and velocities close to the velocity of sound, the profile of the waves under consideration coincides with the profile of the soliton described by the Korteweg-de Vries equation. When the amplitude and velocity of the aforementioned waves reach their limiting values, the wave profile sharpens. It is concluded that the propagation of such waves in rocks and soils is possible.

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“…In fact, the method presented in Wang & Chen (2006) for the generalized double dispersion equation was extended to the Cauchy problem (3.1-3.2) for the HBq equation in Duruk (2006). Summarizing the results in Duruk (2006), we prove in this section the global well-posedness when the non-linear term satisfies a positivity condition. Similar results also have been derived independently in Wang & Mu (2007).…”

Section: Cauchy Problemmentioning

confidence: 63%

A higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity

Duruk

Erkip

Erbay

2008

IMA Journal of Applied Mathematics

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Cauchy problem for a higher‐order Boussinesq equation

Abstract: In this study we establish global well-posedness of the Cauchy problem for a higher-order Boussinesq equation.

Cited by 4 publications

References 3 publications

Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity

Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity

Nonlinear solitary waves in nonlocal elastic solids

A higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity

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