Smoothing effect for higher order dispersive equations and applications to nonlinear initial value problems

TL;DR

This paper investigates the smoothing effects in higher order dispersive equations with variable coefficients and applies these results to establish existence and uniqueness of solutions for nonlinear initial value problems of physical interest.

Contribution

It introduces new smoothing estimates for p-evolution equations with variable coefficients and applies them to nonlinear problems, including KdV and Kawahara equations.

Findings

Established smoothing effects under decay assumptions on coefficients

Proved existence and uniqueness of solutions in Sobolev spaces

Applied results to physically relevant nonlinear dispersive equations

Abstract

In this paper we deal with the initial value problem related to a family of dispersive inhomogeneous evolution equations Pu=f with variable coefficients belonging to the class of p-evolution equations, $p\geq 2$. We study the smoothing effect produced by some spatial decay assumptions on the imaginary part of the subleading coefficient of the linear operator P. Then we apply this result to nonlinear problems with derivative nonlinearities obtaining existence and uniqueness of the solution in a suitable Sobolev class. The nonlinear equations considered include various equations of physical interest such as KdV-type and Kawahara-type equations.