Well-posedness and finite-time singularity of solutions in a 4th-order parabolic equation

This paper deals with a 4th-order parabolic PDE involving $ p $-Laplacian in a bounded domain, subject to Neumann boundary conditions. Firstly, we show the proof of the local existence and uniqueness of weak solutions and the energy identity. The blow-up time estimates are discussed for the corresponding blow-up solutions. Secondly, we define different Nehari manifolds and the energy functional and then determine the existence of blow-up and global solutions in the subcritical, critical, super critical energy cases. Thirdly, we combine some ordinary differential inequalities and different auxiliary functions to obtain the conditions on the existence of extinction and non-extinction solutions with extinction rate and time.