A third-order conservation law for the Kirchhoff-Pokhozhaev equation

TL;DR

This paper establishes a third-order conservation law for the Kirchhoff-Pokhozhaev equation and demonstrates boundedness of derivatives' norms for small energy solutions over time.

Contribution

It introduces a novel third-order conservation law for the Kirchhoff-Pokhozhaev equation and analyzes the boundedness of derivatives for small energy solutions.

Findings

Existence of a third-order conservation law.

Boundedness of derivatives' norms for small energy solutions.

Implication for long-term behavior of solutions.

Abstract

We prove that the special Kirchhoff equation studied by Pokhozhaev admits a third-order conservation law. We further show that if the energy of the solution is sufficiently small, then the $L^2$-norms of the derivatives up to third order of the solution remain uniformly bounded with respect to time.