Boundedness of global solutions for a porous medium system with moving localized sources

This paper deals with a class of porous medium systems with moving localized sources ut=ur1(Δu+af(v(x0(t),t))),vt=vr2(Δv+bg(u(x0(t),t)))ut=ur1(Δu+af(v(x0(t),t))),vt=vr2(Δv+bg(u(x0(t),t))) with homogeneous Dirichlet boundary conditions. It is shown that under certain conditions, solutions of the above system blow up in finite time for large aa and bb or large initial data while there exist global positive solutions to the above system for small aa and bb or small initial data. Moreover, in the one dimensional space case, it is also shown that all global positive solutions of the above problem are uniformly bounded.