Study of weak solutions to a nonlinear fourth‐order parabolic equation with boundary degeneracy

Abstract

A double degenerate fourth‐order parabolic equation with a nonlinear second‐order diffusion from thin film theory is introduced, and the existence of weak solutions for the boundary degeneracy problem is studied. For this purpose, a non‐degenerate fourth‐order elliptic problem is constructed, and the corresponding existence and uniqueness are given by the variational method. From it, two kinds of approximate solutions are defined and then the existence and uniqueness of weak solutions for the related non‐degenerate parabolic problem are shown by the iterative estimate, energy method, and compactness argument. Finally, the parabolic problem with boundary degeneracy is solved by applying energy estimate and several limit procedures.