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

TL;DR

This paper studies initial-boundary value problems for a generalized Kawahara-Zakharov-Kuznetsov equation on a half-strip, establishing global existence, uniqueness, and decay of solutions using weighted Sobolev spaces and new inequalities.

Contribution

It introduces new interpolating inequalities in weighted anisotropic Sobolev spaces and analyzes solutions with higher-order nonlinearities for the first time.

Findings

Proved global existence and uniqueness of solutions.

Established large-time decay of small solutions.

Developed new inequalities in weighted Sobolev spaces.

Abstract

Initial-boundary value problems on a half-strip with different types of boundary conditions for the generalized Kawahara-Zakharov-Kuznetsov equation with nonlinearity of higher order are considered. In particular, nonlinearity can be quadratic and cubic. Results on global existence and uniqueness in classes of weak and strong solutions and large-time decay of small solutions are established. The solutions are considered in weighted at infinity Sobolev spaces. The use of weighted spaces is crucial for the study. To this end new interpolating inequalities in weighted anisotropic Sobolev spaces are established. Both exponential and power weights are admissible.