The generalized fractional KdV equation in weighted Sobolev spaces

Abstract

This work concerns the study of persistence property in polynomial weighted spaces for solutions of the generalized fractional KdV equation in any spatial dimension $d\geq 1$. By establishing well-posedness results in conjunction with some asymptotic at infinity unique continuation principles, it is verified that dispersive effects and dimensionality mainly determine the maximum spatial decay allowed by solutions of this model. In particular, we recover and extend some known results on weighted spaces for different models such as the Benjamin-Ono equation, and the dispersion generalized Benjamin-Ono equation. The estimates obtained for the linear equation seem to be of independent interest, and they are useful to obtain persistence properties in weighted spaces for models with different nonlinearities as the fractional KdV equation with combined nonlinearities.