Global existence and blow-up for a fourth order parabolic equation involving the Hessian

This paper deals with a fourth order parabolic equation involving the Hessian, which was studied in Escudero et al. (J Math Pures Appl 103(4):924-957, 2015) recently, where the initial conditions for W-0(2,2) - norm and W-0(1,4)-norm blow-up were got when the initial energy J(u(0)) <= d, where d > 0 is the mountain-pass level. The purpose of this paper is to study two of the open questions proposed in the paper, that is, L-p-norm blow-up and the behavior of the solutions when J(u(0)) > d. For the case of J(u(0)) < 0, we prove the solution blows up in finite time with L-2-norm. Moreover, we estimate the blow-up time and the blow-up rate. For the case of J(u(0)) > d, we find two sets Psi(alpha) and Phi(alpha), and prove that the solution blows up in finite time if the initial value belongs to Psi(alpha), while the solution exists globally and tends to zero as time t -> + infinity when the initial value belongs to Phi(alpha).