Abstract: We introduce a potential well argument for a semilinear parabolic equation involving critical Sobolev exponent. The potential well structure is a kind of energy structure which gives a very precise view for the analysis and the classification of the asymptotic behavior of solutions. In the subcritical case, together with the compactness of the Sobolev embedding, this structure gives a very precise result. On the other hand, in the critical case, the analysis of the asymptotics is not yet complete due to the lack of compactness. In this talk, the outline of the potential well argument and some open questions in the critical case will be given.