Global, Unbounded Solutions to a Parabolic Equation

Nonlinear parabolic equations of the form ut = Δu + ƒ(u) and which are capable of exhibiting blow-up are considered. We find that if there is a unique steady state then it is possible to choose initial conditions so that the solution is unbouded but exists, at least in a weak sense, for all time. For more restricted problems, e.g., ƒ(u) = eu in three dimensions, we find some particular solutions to the equation which blow up at a finite time but continue to exist afterwards.