Stabilization of Regular Solutions for the Zakharov-Kuznetsov Equation Posed on Bounded Rectangles and on a Strip

Abstract

Initial-boundary value problems for the 2D Zakharov-Kuznetsov equation posed on bounded rectangles and on a strip are considered. Spectral properties of a linearized operator and critical sizes of domains are studied. Exponential decay of regular solutions for the original nonlinear problems is proved. 1991 Mathematics Subject Classification. 35M20, 35Q72.

Initial-boundary value problems in a rectangle for two-dimensional Zakharov–Kuznetsov equation

“…(1.11) Remark 1.8. In the case a) if b = 1 , ν 1 ≡ 0 a similar result for regular solutions in a slightly different form was previously established in [4].…”

“…In [36] either homogeneous Dirichlet or periodic boundary conditions with respect to y were considered and results on global existence and uniqueness of weak solutions were established. In [4] similar results in more regular classes for homogeneous Dirichlet boundary conditions were obtained. In both papers boundary conditions with respect to x were homogeneous.…”

“…Initial-boundary value problems in a bounded rectangle were studied in [36,4]. In [36] either homogeneous Dirichlet or periodic boundary conditions with respect to y were considered and results on global existence and uniqueness of weak solutions were established.…”

Regular solutions to initial-boundary value problems in a half-strip for two-dimensional Zakharov–Kuznetsov equation

“…Initial-boundary value problems, where y varies in a bounded interval, are less studied, however, from the physical point of view they seem at least the same important ( [19,22,3,18,16,21,5,9,10,11]). For example, there are no results on existence of global solutions in Sobolev spaces with any prescribed regularity.…”

Stabilization for the linear Zakharov–Kuznetsov equation without critical size restrictions

“…It has been showed in [7] that the above restrictions are stipulated by the spectral properties of the corresponding stationary operator. More precisely, considering (1.1) posed on a rectangle…”