Enhanced existence time of solutions to evolution equations of Whitham type

TL;DR

This paper demonstrates that solutions to a class of Whitham type evolution equations with Fourier multiplier operators can be extended beyond their usual existence time, especially for small initial data, within a specific dispersive range.

Contribution

It establishes extended existence times for small solutions to a broad class of dispersive equations of Whitham type with general Fourier multipliers.

Findings

Small solutions can be extended beyond expected lifespan.

Results apply to equations with Fourier multipliers of order a7 a0 [-1,1], a7 eq 0.

Applicable to a range of quadratic dispersive equations.

Abstract

We show that Whitham type equations u_t + u u_x -L u_x = 0, where L is a general Fourier multiplier operator of order \alpha \in [-1,1], \alpha\neq 0, allow for small solutions to be extended beyond their expected existence time. The result is valid for a range of quadratic dispersive equations with inhomogeneous symbols in the dispersive range given by \alpha, and should be extendable to other equations of the same relative dispersive strength.