Global existence and blowup of solutions for the multidimensional sixth-order “good” Boussinesq equation

Xu, Runzhang; Yue, Yang; Liu, Bowei; Shen, Jihong; Huang, Shaobin

doi:10.1007/s00033-014-0459-9

Cited by 21 publications

(8 citation statements)

References 21 publications

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“…Most of them are results for blow up of solutions, see [29,25,16]. Similar analysis, for E 0 ≥ d, have been performed to prove the same qualitative properties for other equations, see [14,28] to cite just some of the most influential papers on the subject, and see [4,19,20,26,30,31,35,36,37] and references therein for some recent works. In [6] a numerical study for the Cauchy problem of the focusing cubic nonlinear Klein-Gordon equation in three dimensions for radial initial data shows that for high energies, the qualitative behavior seems to be much more complicated than for E < d, and more research is required to find a threshold between globality and blow up of solutions.…”

Section: The Abstract Formulation Related Results and Preliminariesmentioning

confidence: 84%

Blow-up in damped abstract nonlinear equations

Esquivel-Avila

2020

Electronic Research Archive

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As a typical example of our analysis we consider a generalized Boussinesq equation, linearly damped and with a nonlinear source term. For any positive value of the initial energy, in particular for high energies, we give sufficient conditions on the initial data to conclude nonexistence of global solutions. We do our analysis in an abstract framework. We compare our results with those in the literature and we give more examples to illustrate the applicability of the abstract formulation.

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Section: The Abstract Formulation Related Results and Preliminariesmentioning

confidence: 84%

Blow-up in damped abstract nonlinear equations

Esquivel-Avila

2020

Electronic Research Archive

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“…For the SHBVM method, we used a tolerance tol∼10 −11 in (46), providing s = 12 (and, then, k = 18). 6 As in the previous cases, this latter method turns out to be the most effective one, conserving all the invariant and with a negligible solution error, and a small execution time.…”

Section: Collision Of Two Solitary Wavesmentioning

confidence: 55%

“…Theorem 1 Assume that the solution of (4) and (5) is C 3 [a, b] as a periodic function. Then, with reference to (6), for all t ≥ 0 one has:…”

Section: Hamiltonian Formulationmentioning

confidence: 99%

“…It is for this reason that the equation is often considered in several physical contexts, such as ocean and coastal engineering (as stressed, e.g., in [1,2]). Moreover, the equation provides a balance between dispersion and nonlinearity that may lead to either the existence of solitons, or blowup solutions [3][4][5][6][7][8][9]. It is customary to simplify Equation (1), by defining the shifted variable u(x, t) = w(x, t)…”

Section: Introductionmentioning

confidence: 99%

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Spectrally accurate energy‐preserving methods for the numerical solution of the “good” Boussinesq equation

Brugnano

Gurioli

Zhang

2019

Numerical Methods Partial

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In this paper we study the geometric numerical solution of the so called "good" Boussinesq equation. This goal is achieved by using a convenient space semi-discretization, able to preserve the corresponding Hamiltonian structure, then using energy-conserving Runge-Kutta methods in the Hamiltonian boundary value method class for the time integration. Numerical tests are reported, confirming the effectiveness of the proposed method.

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“…The global existence condition of problem (1) is not studied; 4. The vacuum isolating phenomenon is important for evolution equation (see [4,13,21,15,16] and references therein), whether the vacuum isolating phenomenon happens for the solutions of problem (1) is not studied.…”

Section: A Parabolic Equation Associated With the Fraction P-laplaciamentioning

confidence: 99%

Blow-up and global existence of solutions to a parabolic equation associated with the fraction <i>p</i>-Laplacian

Jiang¹,

Zhou²

2019

Communications on Pure &Amp; Applied Analysis

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We consider a nonlocal parabolic equation associated with the fractional p-laplace operator, which was studied by Gal and Warm in [On some degenerate non-local parabolic equation associated with the fractional p-Laplacian. Dyn. Partial Differ. Equ., 14(1): 47-77, 2017]. By exploiting the boundary condition and the variational structure of the equation, according to the size of the initial dada, we prove the finite time blow-up, global existence, vacuum isolating phenomenon of the solutions. Furthermore, the upper and lower bounds of the blow-up time for blow-up solutions are also studied. The results generalize the results got by Gal and Warm.

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Global existence and blowup of solutions for the multidimensional sixth-order “good” Boussinesq equation

Cited by 21 publications

References 21 publications

Blow-up in damped abstract nonlinear equations

Blow-up in damped abstract nonlinear equations

Spectrally accurate energy‐preserving methods for the numerical solution of the “good” Boussinesq equation

Blow-up and global existence of solutions to a parabolic equation associated with the fraction <i>p</i>-Laplacian

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