Energy decay for the modified Kawahara equation posed in a bounded domain

Araruna, F. D.; Capistrano–Filho, Roberto de A.; Doronin, Gleb G.

doi:10.1016/j.jmaa.2011.07.003

Cited by 33 publications

(25 citation statements)

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“…Exponential decay of weak solutions for small initial data has been established. In [2], decay of weak solutions for l = 2, k = 2 was established. Our goal in this work is to prove the existence and uniqueness of local regular solutions for all k ∈ N and for all finite positive L. Our paper has the following structure: Chapter 1 is Introduction.…”

Section: Introductionmentioning

confidence: 99%

Initial-Boundary Value Problems for Generalized Dispersive Equations of Higher Orders Posed on Bounded Intervals

Larkin

Luchesi

2019

Appl Math Optim

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Initial-boundary value problems for nonlinear dispersive equations of evolution of order 2l + 1, l ∈ N with a convective term of the form u k u x , k ∈ N have been considered on intervals (0, L), L ∈ (0, +∞). The existence and uniqueness of local regular solutions have been established.where x ∈ (0, L), Q T = (0, T ) × (0, L); l, k ∈ N; T, L are real positive numbers. This equation includes as special cases classical dispersive equations: when l = k = 1, we have the well-known Kortewegde Vries (KdV) equation, see [15,20,31], and when k = 1, l = 2, we have the Kawahara equation [5,16,23]. For k = 1, the Cauchy problem for dispersive equations of higher orders has been studied in [3,6,11,12,17,28,32] and initial boundary value problems have been studied in [4,5,7,23,30]. Although dispersive equations were deduced for the whole real line, necessity to calculate numerically the Cauchy problem, approximating the real line by finite intervals [4], implies to study initial-boundary value problems posed on bounded and unbounded intervals, [5,7,19,21,23]. What concerns (1.1) with k > 1, l = 1, called generalized Korteweg-de Vries equations, the Cauchy problem for (1.1) has been studied in [8,9,10,18,25,26], where was proved that for k = 4, called the critical case, the initial

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Section: Introductionmentioning

confidence: 99%

Initial-Boundary Value Problems for Generalized Dispersive Equations of Higher Orders Posed on Bounded Intervals

Larkin

Luchesi

2019

Appl Math Optim

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“…In [12,29,30,31] it was proved that a supercritical equation does not have global solutions and a critical one has a global solution for "small" initial data and the right-hand side. For l = 2, k = 2 the generalized Kawahara equation has been studied in [2]. Initial value problems for the Kawahara equation, l = 2, which had been derived in [19] as a perturbation of the KdV equation, have been considered in [3,8,12,14,16,18,20,21,34,35] and attracted attention due to various applications of those results in mechanics and physics such as dynamics of long small-amplitude waves in various media [13,15,17].…”

Section: Introductionmentioning

confidence: 99%

“…Initial value problems for the Kawahara equation, l = 2, which had been derived in [19] as a perturbation of the KdV equation, have been considered in [3,8,12,14,16,18,20,21,34,35] and attracted attention due to various applications of those results in mechanics and physics such as dynamics of long small-amplitude waves in various media [13,15,17]. On the other hand, last years appeared publications on solvability of initial-boundary value problems for various dispersive equations (which included the KdV and Kawahara equations) in bounded and unbounded domains [2,4,5,7,11,22,23,26,27,28]. In spite of the fact that there is not some clear physical interpretation for the problems on bounded intervals, their study is motivated by numerics [6].…”

Section: Introductionmentioning

confidence: 99%

On connection between the order of a stationary one-dimensional dispersive equation and the growth of its convective term

Larkin

Luchesi

2021

bspm

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“…Initial value problems for the Kawahara equation, which had been derived in [2] as a perturbation of the Korteweg-de Vries (KdV) equation, have been considered in [3][4][5][6][7][8][9][10][11][12] and attracted attention due to various applications of those results in mechanics and physics such as dynamics of long smallamplitude waves in various media [13][14][15]. On the other hand, last years appeared publications on solvability of initialboundary value problems for dispersive equations (which included the KdV and Kawahara equations) in bounded and unbounded domains [16][17][18][19][20][21][22][23]. In spite of the fact that there is not some clear physical interpretation for the problems on bounded intervals, their study is motivated by numerics [24].…”

Section: Introductionmentioning

confidence: 99%