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

Abstract

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 \(J(u_0)\le d\), 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 \(J(u_0) d\), 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 \(t\rightarrow +\infty \) when the initial value belongs to \(\Phi _\alpha \).