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

Abstract

Initial-boundary value problems in a half-strip with different types of boundary conditions for two-dimensional Zakharov-Kuznetsov equation are considered. Results on global existence, uniqueness and long-time decay of weak and regular solutions are established.

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

“…The main results of the paper are related to well-posedness in classes of smooth solutions and to internal regularity of solutions. Weak and less regular solutions to the considered problem were previously studied in [10].…”

“…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.…”

“…Notation, used in the present paper, in many respects repeats the one from [10]. In what follows (unless stated otherwise) i , j , k , l , m , n mean non-negative integers, p ∈ [1, +∞] , s ∈ R .…”

“…(here D α denotes the Riesz potential of the order −α ). In [10] the following result on global well-posedness was established.…”

“…In our study we use the following interpolating inequality from [10]. If ρ 1 (x) , ρ 2 (x) are two admissible weight functions, such that ρ 1 (x) ≤ c 0 ρ 2 (x) ∀x ≥ 0 for some constant c 0 > 0 , then there exists a constant c > 0 , such that for every function ϕ(x, y) , satisfying |Dϕ|ρ…”