Semilinear parabolic equations involving critical Sobolev exponent: local and asymptotic behavior of solutions

In the framework of the potential well method, we consider the behavior of solutions to the problem (1.1)--(1.3) below with the critical Sobolev exponent. Roughly speaking, in the case where $\Omega$ is star-shaped, time-global solutions which intersect with the stable set at some time converge to zero uniformly for $x \in \Omega$ as $t \to +\infty$ and global solutions which intersect neither the stable nor the unstable sets blow up in infinite time in some sense and further have a property like a $\delta$-function in an appropriate sense as $t \to +\infty$ in the case when $\Omega$ is ball. Furthermore, for a kind of initial data the associated solution blows up at a finite time $T_{m}$, and its energy also satisfies: $J(u(t,\cdot)) = O(\log(T_{m}-t))$ as $t \uparrow T_{m}$.