Boundedness of global solutions of nonlinear diffusion equations (Q1196635)

This paper deals with large time behaviour of solutions of Dirichlet- initial value problems for the equation \(u_ t=\Delta(| u|^{m- 1}u)+f(u)\) when \(f\) grows superlinearly. In particular the question of global existence without uniform a priori bounds is examined. The case when \(m=1\) and \(f\) is allowed to grow exponentially is of particular concern and it is shown that global existence without uniform bounds is impossible in one and two dimensions.