Decay properties for solutions of fifth order nonlinear dispersive equations

Abstract

ABSTRACT. We consider the initial value problem associated to a large class of fifth order nonlinear dispersive equations. This class includes several models arising in the study of different physical phenomena. Our aim is to establish special (space) decay properties of solutions to these systems. These properties complement previous unique continuation results and in some case, show that they are optimal. These decay estimates reflect the "parabolic character" of these dispersive models in exponential weighted spaces. This principle was first obtained by T. Kato in solutions of the KdV equation.

Propagation of Regularity and Persistence of Decay for Fifth Order Dispersive Models

“…We show several lemmas which are needed to prove Theorems 1, 2 and 3. The first lemma is an analogue of (1.13) to implement Kato's energy estimate argument which is proved by Isaza-Linares-Ponce [6].…”

“…The use of asymmetric spaces leads to a result which is irreversible in time. Isaza, Linares and Ponce [6] extended the quasiparabolic smoothing effect to a large class of fifth order equations. corresponding to initial data u 0 ∈ H 6 (R) ∩ L 2 (e βx dx), β > 0, with…”

Higher-Order Stationary Dispersive Equations on Bounded Intervals

“…For = 1, we have the well-known generalized KdV equation and for = 2 the Kawahara equation. 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].…”

“…Last years, publications on dispersive equations of higher orders appeared [7,9,10,21,27]. Here, we propose (1) as a stationary analog of (2) because the last equation includes classical models such as the KdV and Kawahara equations.…”

Kawahara-Burgers Equation on a Strip

“…We are concerned with an initial-boundary value problem (IBVP) for the two-dimensional Kawahara-Burgers (KB) equation which includes dissipation and dispersion and has been studied intensively last years due to its applications in Mechanics and Physics [1,3,4,5,7,6,8,18,25]. Equations (1.1) and (1.2) are typical examples of so-called dispersive equations which attract considerable attention of both pure and applied mathematicians in the past decades.…”