Persistence properties for the dispersion generalized BO-ZK equation in weighted anisotropic Sobolev spaces

TL;DR

This paper investigates the persistence and decay properties of solutions to a generalized dispersive PDE in weighted anisotropic Sobolev spaces, establishing sharp results and unique continuation properties.

Contribution

It introduces new results on the persistence and decay of solutions to the dispersion generalized BO-ZK equation in weighted anisotropic Sobolev spaces, including sharpness and unique continuation.

Findings

Solutions persist in weighted anisotropic Sobolev spaces under certain conditions.

Decay properties in the x-direction are shown to be sharp.

Unique continuation properties are established for the solutions.

Abstract

In this paper we study the initial-value problem associated with the dispersion generalized-Benjamin-Ono-Zakharov-Kuznetsov equation, $$ u_{t}+D^{a+1}_x \partial_{x}u+u_{xyy}+uu_{x}=0, \qquad a\in(0,1). $$ More specifically, we study the persistence property of the solution in the weighted anisotropic Sobolev spaces $$ H^{(1+a)s,2s}(\R^{2})\cap L^{2}((x^{2r_1} +y^{2r_2})dxdy), $$ for appropriate $s$, $r_1$ and $r_2$. By establishing unique continuation properties we also show that our results are sharp with respect to the decay in the $x$-direction.