A doubly critical degenerate parabolic problem

It is shown that the Dirichlet problem for$$u_t=u^p(\Delta u + u)\quad {\rm in}\enspace \Omega \times (0,T),\quad p>0$$ where Ω⊂ℝ n is critical in that it has first eigenvalue one, is globally solvable for any continuous positive initial datum vanishing at ∂ Ω. Moreover, for p <3 all solutions are bounded and tend to some nonnegative eigenfunction of the Laplacian as t →∞, while if p ⩾3 then there are both bounded and unbounded solutions. Finally, it is shown that unlike the case p ∈[0,1), all steady states are unstable if p ⩾1. Copyright © 2004 John Wiley & Sons, Ltd.