Boundary value problem for a classical semilinear parabolic equation

Abstract

In this paper, we study the boundary value problem of the classical semilinear parabolic equations $$u_t-\hat{I}”u=|u{|}^{p-1}u,in\hat{I}\text{c◯}\times (0,T)$$ and $u=0$ on the boundary $\partial \hat{I}\text{c◯}\times [0,T)$ and $u=\ddot{I}•$ at $t=0$, where $\hat{I}\text{c◯}\subset R^n$ is a compact $C^1$ domain, $1<p\le p_S$ is a fixed constant, and $\ddot{I}•\in C_0^2(\hat{I}\text{c◯})$ is a given smooth function. Introducing new idea, we show that there are two sets $\tilde{W}$ and $\tilde{Z}$ such that for $\ddot{I}•\in W$, there is a global positive solution $u(t)\in \tilde{W}$ with $h^1$ omega limit $\{0\}$ and for $\ddot{I}•\in \tilde{Z}$, the solution blows up at finite time.

7 pages