Existence of solutions to a doubly degenerate fourth-order parabolic equation

Abstract

The existence and uniqueness of solutions for a doubly degenerate fourth-order parabolic equation ωt+∇·(ρα|∇Δω|p−2∇Δω)=0 with initial-boundary conditions on a bounded domain are considered. For this purpose, the time variable is discretized for the problem with non-degenerate coefficient case. By solving a series of semi-discrete elliptic problems, the existence and uniqueness of weak solutions for the corresponding parabolic problem are gained by energy method and two limit process. Furthermore, the regularization method is applied to the problem with degenerate coefficient at boundary and the existence and uniqueness can be performed by uniform estimates. Finally, the regularity for weak solutions is also shown.