Blow-up and global solutions for subcritical and critical parabolic equations in ${\mathbb R}^N$

We study the local well-posedness in the framework of the Sobolev space $H^1(\mathbb R^N)$, $N \geq 3$, for a semilinear parabolic equation with asymptotically polynomial nonlinearity up to the critical Sobolev growth. Then we establish the dichotomy between blow-up and global existence for solutions with small energy by means of variational methods and the so-called potential well argument.