On local energy decay for solutions to the b-family of peakon equations

TL;DR

This paper investigates the long-term behavior of solutions to the b-family of peakon equations, demonstrating local energy decay under certain conditions and providing insights into the decay properties of solutions over time.

Contribution

It establishes local energy decay for solutions to the b-family peakon equations, including decay in exterior regions, under specific hypotheses and initial conditions.

Findings

Proves local energy decay for global solutions under hypotheses.

Shows decay along sequences of time in expanding regions.

Demonstrates decay in exterior regions with nonnegative initial momentum.

Abstract

This work is concerned with the long time behavior of solutions to the $b$-family of peakon equations. We prove local energy decay of global solutions under suitable hypotheses. Assuming the global bound of the $H^1(\mathbb R)$ norm, we show local energy decay along sequences of time in an expanding region around the origin. If we assume nonnegativity for an initial momentum, $m_0(x) = (u - \partial_x^2 u)(0,x)$, we show strict decay on a similar region. Moreover, we show decay in an exterior region given the same nonnegativity condition.