Some Blowup Problems Of A Semilinear Heat Equation

Refine structures of blowup for non-collapsing maximal solution of a semilinear parabolic equation ut+?u=|u|p-1u+|u|q-1u are studied,where,is a bounded domain in RN,N?3 with smooth boundary and p>1,p>q>1.Firstly,we will give the relationship between energy collapsing and complete blowup.Also,a-priori estimate concerning noncollapsing blowup is estab-lished.So all finite time blow-ups are collapsing and hence complete blowup.That is,the blowup set is empty for non-collapsing blow-up under subcritical case.Secondly,we rewrite the maximal solution as a function of self-similar variables.We can estimate the local energy of the re-scaled solution.We show that finite time non-collapsing blow-up must be refined type ? under critical case.When p>ps?N+2/N-2 for N?3,the Hausdorff dimension of the blowup set for maximal solution whose energy is non-collapsing is shown to be no greater than N-2-4/p-14,which answer a question arose in[7]positively.At the end of the paper,we present some new examples of complete blowups.The main difficulty is that the local energy is no longer monotone in time.Using some techniques in the paper[6],we show the local energy is quasi-monotonicity and the derive the similar results as[8].