Boundedness of global solutions for a nonlinear degenerate parabolic (porous medium) system with localized sources

This paper deals with nonlinear degenerate parabolic (porous medium) system with localized sources. It is shown that under certain conditions solutions of the equation blow up in finite time for large a and b or large initial data while there exist global positive solutions for small a and b or small initial data. Moreover, it is also shown that all global positive solutions of the above equation are uniformly bounded. These extend a recent work of Chen et al. in [Y. Chen, Q. Liu, H. Gao, Boundedness of global solutions of a porous medium equation with a localized source, Nonlinear Anal. 64 (2006) 2168-2182], which considered the single equation with localized source.