Regular solutions to 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 well-posedness in classes of regular solutions in the cases of periodic and Neumann boundary conditions, as well as on internal regularity of solutions for all types of boundary conditions are established. Also in the case of Dirichlet boundary conditions one result on long-time decay of regular solutions is obtained.

Initial-Boundary Value Problems on a Half-Strip for the Generalized Kawahara-Zakharov-Kuznetsov Equation

“…The initial value problem and initial-boundary value problems on semi-axes are considered, for example, in [2,3,4,6,7,14,15,16,17,21,22,23,26,27,32,34]. Also during last 30 years there were published several papers devoted to initial-boundary value problems (especially to the pure initial value problem) for the ZK equation and its extensions with more general nonlinearity (see, for example, bibliography in [11,12,13] and recent papers [5,20,25,29]). Wide classes of quasilinear dispersive equations were considered, for example, in [9,10,28].…”

“…Note that both ρ 0 and ρ ′ 0 are admissible weight functions. Note that, for example, in [11,12,13] condition (1.8) was not used for the definition of the admissible weight function. Here we have to introduce it in order to establish interpolating inequalities in anisotropic Sobolev spaces in Lemma 2.1.…”

On Inner Regularity of Solutions of Two-Dimensional Zakharov–Kuznetsov Equation

“…Use the same smooth solutions u h as in the proofs of Theorems 2.1-2.2. Use the following result from [20] on the additional regularity of such solutions for x > 0: ∂ n x u h ∈ X 3,β,exp (Π 0,x 0 T,L ) for each n, x 0 > 0, and β > 0. In particular, ∂ n x u ht , ∂ n x u hty , and ∂ n x ∂ j y u h ∈ L 2 (0, T ; L β,exp 2 (Σ L,x 0 )) (for j ≤ 4).…”

“…), and u 0 (0, y) ≡ μ(0, y) and in spaces X 3,ψ(x) (Π + T,L ) for u 0 ∈ H 3,ψ(x) (Σ L,+ ), μ ∈ H 4/3,4 (B T,L ), and u 0 (0, y) ≡ μ(0, y). In [20], the additional regularity is investigated for solutions from the space X 3,ψ(x) (Π + T,L ), constructed in [19]. If the boundary conditions are (b) or (d),…”

Generalized Variational Principle for the Fractal (2 + 1)-Dimensional Zakharov–Kuznetsov Equation in Quantum Magneto-Plasmas

“…where m, n, and k are non-zero constants. In recent years, the exact solutions of Equation ( 1) have been studied by many researchers, and many effective solutions have been obtained, such as the generalized exponential rational function method [8], Lie group analysis [9], group analysis approach [10], Exp-function method [11], Coupled Burgers' equations method [12], extended tanh method [13], and so on [14][15][16][17]. These results are important and can help us study the (2 + 1)-dimensional Z-K equation.…”