Blow-up of solutions to a degenerate parabolic equation not in divergence form

AbstractWe study nonglobal positive solutions to the Dirichlet problem for ut=up(Δu+u) in bounded domains, where 0<p<2. It is proved that the set of points at which u blows up has positive measure and the blow-up rate is exactly (T−t)−1p. If either the space dimension is one or p<1, the ω-limit set of (T−t)1pu(t) consists of continuous functions solving Δw+w=1pw1−p. In one space dimension it is shown that actually (T−t)1pu(t)→w as t→T, where w coincides with an element of a one-parameter family of functions inside each component of its positivity set; furthermore, we study the size of the components of {w>0} with the result that this size is uniquely determined by Ω in the case p<1, while for p>1, the positivity set can have the maximum possible size 2πp for certain initial data, but it may also be arbitrarily close to the minimal length π