Analysis on the blow-up of solutions to a class of integrable peakon equations

We investigate the blow-up mechanism of solutions to a class of quasilinear integrable equations which could possess peakons. The dynamics of the blow-up quantity along the characteristics is established by the Riccati-type differential inequality which involves the interaction among three parts: a local nonlinearity, a nonlocal term, and a term stemming from the weak linear dispersion. To analyse the interplay among these quantities, we provide two different approaches. The first one is designed for the case when the equations do not exhibit a weak linear dispersion and hence focuses on the interplay between the first two parts. The method is based on a refined analysis on either evolution of the solution u and its gradient ux, that is, Cu±ux or the growth rate of the relative ratio ux/u. The second one handles the general situation when all of three parts are present. The idea is to extract the “truly” blow-up component from the Riccati-type differential inequality and utilizes the Morawetz-type identity or higher order conservation laws to show that such a component blows up in finite time before the other component degenerates.