A Liouville theorem for the Degasperis-Procesi equation

TL;DR

This paper proves a Liouville theorem for the Degasperis-Procesi equation, showing that under certain conditions, the only global periodic solution that vanishes at a point is the trivial solution, including cases with an added dispersive term.

Contribution

It establishes a Liouville-type theorem for the Degasperis-Procesi equation, extending to cases with an additional dispersive term, which was not previously known.

Findings

Only the zero solution exists under the given conditions.

The theorem applies to both standard and dispersive variants.

Provides a uniqueness result for solutions vanishing at a point.

Abstract

We prove that the only global, strong, spatially periodic solution to the Degasperis-Procesi equation, vanishing at some point (t0, x0), is the identically zero solution. We also establish the analogue of such Liouville-type theorem for the Degasperis-Procesi equation with an additional dispersive term.