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

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 H1(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, m0(x) = (u - dx2u)(0, x), we show strict decay on a similar region. Moreover, we show decay in an exterior region given the same nonnegativity condition.