Asymptotic behavior and blow up of solutions for semilinear parabolic equations at critical energy level

In this paper we study the initial boundary value problem of semilinear parabolic equations with semilinear term f(u). By using the family of potential wells method we prove that if f(u) satisfies some conditions, J(u0) ≤ d and I(u0) > 0, then the solution decays to zero exponentially as t → ∞. On the other hand, if J(u0) ≤ d, I(u0) < 0, then the solution blows up in finite time.