Boundedness of global solutions of a supercritical parabolic equation

We study radially symmetric classical solutions of the Dirichlet problem for a heat equation with a supercritical nonlinear source. We give a sufficient condition under which blow-up in infinite time cannot occur. This condition involves only the growth rate of the source term at infinity. We do not need the homogeneity property which played a key role in previous proofs of similar results. We also establish the blow-up rate for a class of solutions which blow up in finite time.