ABSTRACT. We consider positive solutions to the Cauchy problem for quasilinear parabolic equations dtu = Au 171 + u p with max{0,1 -2/N} < m < 1 < p, where N is the space dimension. Putting p^ =m + 2/N, we shall show that if p < p^, then all nontrivial solutions blow up in finite time, and if p > p^, then there are nontrivial global solutions.

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We consider the semilinear parabolic system u_{t}=\triangle u+v^{p} , v_{t}=\triangle v+u^{q} , where x\in R^{N}(N\geq 1) , t>0 and p , q\geq 1 . At t=0 , nonnegative, bounded and continuous initial values (u_{0}(x), v_{0}(x)) are prescribed. The main results are for the case when (u0, v0) have polynomial decay near x=\infty . Assuming u_{0}\sim(\lambda|x|^{-a})^{1/(q+1)} , v_{0}\sim(\lambda|x|^{-a})^{1/(p+1)} with \lambda >0,0 \leq a