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

Abstract

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 Hölder setting which allows to deal with nonlocal terms including highest order point evaluation. In this direction, some theorems concerning the linearized systems are 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.

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Correspondence to Helmut Abels.

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Abels, H., Arab, N. & Garcke, H. On convergence of solutions to equilibria for fully nonlinear parabolic systems with nonlinear boundary conditions. J. Evol. Equ. 15, 913–959 (2015). https://doi.org/10.1007/s00028-015-0287-1

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Mathematics Subject Classification

  • 35B35
  • 35K55
  • 35K50
  • 37L10
  • 53C44
  • 35B65

Keywords

  • Fully nonlinear parabolic systems
  • General nonlinear boundary conditions
  • Normally stable
  • Surface diffusion flow
  • Triple junctions
  • Lens-shaped network