Abstract
AbstractIt is shown that the Dirichlet problem for where Ω⊂ℝn 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.
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