Boundary value problem for a classical semilinear parabolic equation

In this paper, we study the boundary value problem of the classical semilinear parabolic equations $$u_t-\Delta u=|u|^{p-1}u, \ \ in \ \ \Omega\times (0,T)$$ and $u=0$ on the boundary $\partial\Omega\times [0,T)$ and $u=\phi$ at $t=0$, where $\Omega\subset R^n$ is a compact $C^1$ domain, $1<p\leq p_S$ is a fixed constant, and $\phi\in C^2_0(\Omega)$ is a given smooth function. Introducing new idea, we show that there are two sets $\tilde{W}$ and $\tilde{Z}$ such that for $\phi\in W$, there is a global positive solution $u(t)\in \tilde{W}$ with $h^1$ omega limit $\{0\}$ and for $\phi\in \tilde{Z}$, the solution blows up at finite time.