Semilinear parabolic equation in RN associated with critical Sobolev exponent

We consider the semilinear parabolic equation ut−Δu=|u|p−1u on the whole space RN, N⩾3, where the exponent p=(N+2)/(N−2) is associated with the Sobolev imbedding H1(RN)⊂Lp+1(RN). First, we study the decay and blow-up of the solution by means of the potential-well and forward self-similar transformation. Then, we discuss blow-up in infinite time and classify the orbit.