| Sumario: | We study the blow-up phenomenon for non-negative solutions to the following parabolic problem: [equation presented] where 0 < p = min p p(x) max p = p+ is a smooth bounded function. After discussing existence and uniqueness, we characterize the critical exponents for this problem. We prove that there are solutions with blow-up in finite time if and only if p+ > 1. When ω = ℝ N we show that if p > 1 + 2/N, then there are global non-trivial solutions, while if 1 < p p+ 1 + 2/N, then all solutions to the problem blow up in finite time. Moreover, in the case when p < 1 + 2/N < p+, there are functions p(x) such that all solutions blow up in finite time and functions p(x) such that the problem possesses global non-trivial solutions. When ω is a bounded domain we prove that there are functions p(x) and domains ω such that all solutions to the problem blow up in finite time. On the other hand, if ω is small enough, then the problem possesses global non-trivial solutions regardless of the size of p(x). © 2012 Royal Society of Edinburgh.
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