Global existence and blow-up results for a classical semilinear parabolic equation

Abstract

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 ⊂ ℝⁿ is a compact C ¹ domain, 1 < p ≤ pS is a fixed constant, and φ ∈ C ⁰₁ (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.