On convergence of solutions to equilibria for fully nonlinear parabolic problems with nonlinear boundary conditions

Helmut Abels, Nasrin Arab, Harald Garcke

Published 2014-03-18, updated 2014-08-20Version 3

Convergence to stationary solutions in fully nonlinear parabolic systems with general nonlinear boundary conditions is shown in situations where the set of stationary solutions creates a $C^2$-manifold of finite dimension which is normally stable. We apply the parabolic H\"older setting which allows to deal with nonlocal terms including highest order point evaluation. In this direction some theorems concerning the linearized systems is also extended. As an application of our main result we prove that the lens-shaped networks generated by circular arcs are stable under the surface diffusion flow.