Global Existence and Blow-Up Results for a Classical Semilinear Parabolic Equation

The author studies the boundary value problem of the classical semilinear parabolic equations $$u_t - \Delta u = \left| u \right|^{p - 1} u in \Omega \times (0,{\rm T}),$$ , and u = 0 on the boundary ?Ω × 0, T) and u = φ at t = 0, where Ω ? ? ⁿ is a compact C ¹ domain, 1 < p ≤ pS is a fixed constant, and φ ∈ C ₁ ⁰ (Ω) is a given smooth function. Introducing a new idea, it is shown that there are two sets $\tilde W$ and $\tilde Z$ , such that for $\varphi \in \tilde W$ , there is a global positive solution $u(t) \in \tilde W$ with H ¹ omega limit 0 and for $\varphi \in \tilde Z$ , the solution blows up at finite time.