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

In this paper a localized porous medium equation ut=ur(Δu+af(u(x0,t)))ut=ur(Δu+af(u(x0,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, 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.