Multi-dimensional Kuramoto-Sivashinsky-Zakharov-Kuznetsov equation posed on admissible multi-dimensional domains

TL;DR

This paper investigates the existence, uniqueness, and decay of solutions for a multi-dimensional Kuramoto-Sivashinsky-Zakharov-Kuznetsov equation on bounded domains, revealing a link between domain dimension and solution properties.

Contribution

It establishes global regular solutions and their exponential decay for the n-dimensional equation, connecting domain dimension with solution admissibility.

Findings

Proved existence and uniqueness of global solutions.

Established exponential decay of solutions.

Identified relationship between domain dimension and stationary part of the equation.

Abstract

An initial-boundary value problem for the n-dimensional ($n$ is a natural number from the interval [2,7]) Kuramoto-Sivashinsky-Zakharov-Kuznetsov equation posed on smooth bounded domains in $\mathbb{R}^n$ was considered. The existence and uniqueness of global regular solutions as well as their exponential decay have been established. A connection between an order of the stationary part of the equation and admissible dimensions of a domain has been revealed. Keywords: Kuramoto-Sivashinsky equation, Zakharov-Kuznetsov equation, Global solutions