Extended Decay Properties for Generalized BBM Equation

Kwak, Chulkwang; Muñoz, Claudio

doi:10.1007/978-1-4939-9806-7_8

Cited by 4 publications

(6 citation statements)

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“…Moreover, the results proved in this note and in [22,23,28,32] apply to equations which have long range nonlinearities, as well as very low or null decay rates. See also [34,26] for other applications of this technique to the case of Boussinesq equations, and [25] for an application to the case of the BBM equation.…”

Section: )mentioning

confidence: 99%

Breathers and the Dynamics of Solutions in KdV Type Equations

Muñoz

Ponce

2018

Commun. Math. Phys.

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In this paper our first aim is to identify a large class of non-linear functions f (·) for which the IVP for the generalized Korteweg-de Vries equation does not have breathers or "small" breathers solutions. Also we prove that all small, uniformly in time L 1 ∩ H 1 bounded solutions to KdV and related perturbations must converge to zero, as time goes to infinity, locally in an increasing-in-time region of space of order t 1/2 around any compact set in space. This set is included in the linearly dominated dispersive region x ≪ t. Moreover, we prove this result independently of the well-known supercritical character of KdV scattering. In particular, no standing breather-like nor solitary wave structures exists in this particular regime.

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Section: )mentioning

confidence: 99%

Breathers and the Dynamics of Solutions in KdV Type Equations

Muñoz

Ponce

2018

Commun. Math. Phys.

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“…The proof Theorem 1.1 follows the introduction of a new virial identity, in the spirit of the previous results by Martel and Merle [20,21] in the gKdV case, and [12,1] in the BBM case. Note however that in those cases the functional involved is related to the mass (L 2 norm) of the solution.…”

mentioning

confidence: 88%

“…Note however that u seems not locally L 2 integrable in time. However, (1.14) shows that this norm indeed decays to zero in time (even if it is not integrable in time The techniques that we use to prove Theorem 1.2 are not new, and have been used to show decay for the Born-Infeld equation [2], the good Boussinesq system [24], the Benjamin-Bona-Mahony (BBM) equation [12], and more recently in the more complex abcd Boussinesq system [14,13]. In all these works, suitable virial functionals were constructed to show decay to zero in compact/not compact regions of space.…”

Section: 2mentioning

confidence: 99%

“…Now we need the following result about equivalence of norms in terms of f and u. 12,14]). Let f be as in (3.2).…”

Section: Decay In Exterior Light Cones Proof Of Theorem 11mentioning

confidence: 99%

“…Controlling this last term requires some work. Indeed, we shall need the following definition (see [15,14,12] and references therein for more details)…”

Section: Theorem 12 (Full Decay In Interior Regionsmentioning

confidence: 99%

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Decay in the one dimensional generalized Improved Boussinesq equation

Maulén

Muñoz

2020

SN Partial Differ. Equ. Appl.

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We consider the decay problem for the generalized improved (or regularized) Boussinesq model with power type nonlinearity, a modification of the originally ill-posed shallow water waves model derived by Boussinesq. This equation has been extensively studied in the literature, describing plenty of interesting behavior, such as global existence in the space H 1 × H 2 , existence of super luminal solitons, and lack of a standard stability method to describe perturbations of solitons. The associated decay problem has been studied by Liu, and more recently by Cho-Ozawa, showing scattering in weighted spaces provided the power of the nonlinearity p is sufficiently large. In this paper we remove that condition on the power p and prove decay to zero in terms of the energy space norm L 2 × H 1 , for any p > 1, in two almost complementary regimes: (i) outside the light cone for all small, bounded in time H 1 × H 2 solutions, and (ii) decay on compact sets of arbitrarily large bounded in time H 1 × H 2 solutions. The proof consists in finding two new virial type estimates, one for the exterior cone problem based in the energy of the solution, and a more subtle virial identity for the interior cone problem, based in a modification of the momentum.Here the plus sign denotes the good Boussinesq system, which is locally and globally wellposed in standard Sobolev spaces [8,9], and the minus sign represents the "bad" equation originally derived by Boussinesq [4], which is strongly linearly ill-posed. Precisely, motivated Key words and phrases. Improved Boussinesq, decay, virial. (Ch.M.) Partially funded by Chilean research grants FONDECYT 1150202 and CONICYT PFCHA/DOCTORADO NACIONAL/2016-21160593. (Cl.M.) Partially funded by Chilean research grants FONDECYT 1150202 and 1191412, project France-Chile ECOS-Sud C18E06 and CMM Conicyt PIA AFB170001. Part of this work was done while Cl.M. was visiting the CMLS at

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