Boundedness of global positive solutions of a porous medium equation with a moving localized source

In this paper a porous medium equation with a moving localized source ut=ur(Δu+af(u(x0(t),t))) is considered. It is shown that under certain conditions solutions of the above equation blow up in finite time for large a or large initial data while there exist global positive solutions to the above equation for small a or small initial data. Moreover, in one space dimension case, it is also shown that all global positive solutions of the above equation are uniformly bounded, and this differs from that of a porous medium equation with a local source.