Propagation of regularity and persistence of decay for fifth order dispersive models

TL;DR

This paper investigates how initial regularity and decay properties of solutions to a fifth order dispersive PDE influence the evolution of these properties over time, demonstrating persistence of regularity and decay in solutions.

Contribution

It establishes that regularity and decay in initial data on the positive half-line persist in solutions for positive times for a class of fifth order dispersive models.

Findings

Regularity on the positive half-line persists over time.

Polynomial decay of initial data is maintained in solutions.

Results apply to the specific fifth order KdV-type equation.

Abstract

This paper considers the initial value problem for a class of fifth order dispersive models containing the fifth order KdV equation $$\partial_tu - \partial_x^5u - 30u^2\partial_xu + 20\partial_xu\partial_x^2u + 10u\partial_x^3u = 0.$$ The main results show that regularity or polynomial decay of the data on the positive half-line yields regularity in the solution for positive times.