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

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 $H^1\left(R\right)$ 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, $m_0\left(x\right)=(u-{\partial }_x^{2}u)(0,x)$, we show strict decay on a similar region. Moreover, we show decay in an exterior region given the same nonnegativity condition.