Initial-boundary value problems on a half-strip for the modified Zakharov–Kuznetsov equation

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.…”

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

“…For ZK equation there is a lot of literature, devoted to the initial value and initial-boundary value problems, where the variable y is considered on the whole line (see, for example, bibliography in [10,11] and recent papers [13,23]). In particular, global well-posedness in Sobolev spaces H k of arbitrary large regularity to the initial value problem and the initial-boundary value problem, posed on R 2 + = {(x, y) : x > 0} and (0, 1) × R , was established (see, for example, [6,7]).…”

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

“…In the two-dimensional case the corresponding results in different functional spaces can be found in [34,5,6,2,27,28,32,16,3,19,31,17,18]. For initial-boundary value problems such a theory is most developed for domains, where the variable y is considered in the whole line, ( [7,8,11,10,35,12,4]).…”

“…Proof. Solutions to the considered problem (similarly to the corresponding problem in [10]) are constructed in the form u(t, x, y) = w(t, x, y) + v(t, x, y),…”