Abstract
The existence and non-existence of global solutions and theL p blow-up of non-global solutions to the initial value problemu′(t)=Δu(t)+u(t) γ onR n are studied. We consider onlyγ>1. In the casen(γ − 1)/2=1, we present a simple proof that there are no non-trivial global non-negative solutions. Ifn(γ−1)/2≦1, we show under mild technical restrictions that non-negativeL p solutions always blow-up inL p norm in finite time. In the casen(γ−1)/2>1, we give new sufficient conditions on the initial data which guarantee the existence of global solutions.
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