Constructing weak solutions in a direct variational method and an application of varifold theory (Q1275319)

The paper deals with initial boundary value problem for 4th-order parabolic equation \[ \partial u/ \partial t+\Delta^2u-\text{div} F_p (x,u, \nabla u)+F_y(x,u, \nabla u)=0, \] with \(F=F(x,y,p)\) a \(C^1\)-function, \(\Omega\) a bounded domain and Dirichlet boundary conditions. This is essentially the equation for the Morse semiflow of the \(C^1\)-functional \(J: W^{2,2}(\Omega)\to\mathbb{R}: v\mapsto\int_\Omega {1\over 2}|\Delta v|^2+ F(x,v, \nabla v)dx\). Approximation by Rothe method (i.e. semidiscretization in time) is made and then convergence of approximate solutions and thus existence of weak solutions is proved by using the varifold theory which is briefly presented, too. The difficulty lies in the lack of estimates for \(\partial u/\partial t\) and thus compactness for \(\nabla u\).