Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level

Abstract

For a class of semilinear parabolic equations on a bounded domain $\Omega$, we analyze the behavior of the solutions when the initial data varies in the phase space $H^1_0(\Omega)$. We obtain both global solutions and finite time blow-up solutions. Our main tools are the comparison principle and variational methods. Particular attention is paid to initial data at high energy level; to this end, a basic new idea is to exploit the weak dissipativity (respectively antidissipativity) of the semiflow inside (respectively outside) the Nehari manifold.