Journal of Evolution Equations

, Volume 15, Issue 4, pp 913–959 | Cite as

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

  • Helmut AbelsEmail author
  • Nasrin Arab
  • Harald Garcke


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.


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

Mathematics Subject Classification

35B35 35K55 35K50 37L10 53C44 35B65 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Abels, N. Arab, and H. Garcke. Standard Planar Double Bubbles are Stable under Surface Diffusion Flow, arXiv:1505.02979
  2. 2.
    P. Acquistapace. Zygmund classes with boundary conditions as interpolation spaces, volume 116 of Lecture Notes in Pure and Appl. Math. Dekker, New York, 1989.Google Scholar
  3. 3.
    Acquistapace P., Terreni B.: Hölder classes with boundary conditions as interpolation spaces. Math. Z. 195(4), 451–471 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math. 17, 35–92 (1964)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    H.W. Alt. Lineare Funktionalanalysis. Springer, 2006.Google Scholar
  6. 6.
    Amann H.: Global existence for semilinear parabolic systems. J. Reine Angew. Math. 360, 47–83 (1985)zbMATHMathSciNetGoogle Scholar
  7. 7.
    O. Baconneau and A. Lunardi. Smooth solutions to a class of free boundary parabolic problems. Trans. Amer. Math. Soc., 356(3):987–1005 (electronic), 2004.Google Scholar
  8. 8.
    Brauner C., Hulshof J., Lunardi A.: A general approach to stability in free boundary problems. J. Differential Equations 164(1), 16–48 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    D. Depner. Stability Analysis of Geometric Evolution Equations with Triple Lines and Boundary Contact. PhD thesis, Regensburg, 2010.Google Scholar
  10. 10.
    Depner D., Garcke H.: Linearized stability analysis of surface diffusion for hypersurfaces with triple lines. Hokkaido Math. J. 42(1), 11–52 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Depner D., Garcke H., Kohsaka Y.: Mean curvature flow with triple junctions in higher space dimensions. Arch. Rational Mech. Anal. 211(1), 301–334 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    S. D. Eidelman and N. V. Zhitarashu. Parabolic boundary value problems, volume 101 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1998. Translated from the Russian original by Gennady Pasechnik and Andrei Iacob.Google Scholar
  13. 13.
    Elliott C.M., Garcke H.: Existence results for diffusive surface motion laws. Adv. Math. Sci. Appl. 7(1), 467–490 (1997)zbMATHMathSciNetGoogle Scholar
  14. 14.
    J. Escher, U. F. Mayer, and G. Simonett. The surface diffusion flow for immersed hypersurfaces. SIAM J. Math. Anal., 29(6):1419–1433 (electronic), 1998.Google Scholar
  15. 15.
    H. Garcke, K. Ito, and Y. Kohsaka. Linearized stability analysis of stationary solutions for surface diffusion with boundary conditions. SIAM J. Math. Anal., 36(4):1031–1056 (electronic), 2005.Google Scholar
  16. 16.
    Garcke H., Ito K., Kohsaka Y.: Surface diffusion with triple junctions: a stability criterion for stationary solutions. Adv. Differential Equations 15(5–6), 437–472 (2010)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Garcke H., Novick-Cohen A.: A singular limit for a system of degenerate Cahn–Hilliard equations. Adv. Differential Equations 5(4-6), 401–434 (2000)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Geymonat G., Grisvard P.: Alcuni risultati di teoria spettrale per i problemi ai limiti lineari ellittici. Rend. Sem. Mat. Univ. Padova 38, 121–173 (1967)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Hutchings M., Morgan F., Ritoré M., Ros A.: Proof of the double bubble conjecture. Ann. of Math. (2) 155(2), 459–489 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Latushkin Y., Prüss J., Schnaubelt R.: Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions. J. Evol. Equ. 6(4), 537–576 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    A. Lunardi. Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995.Google Scholar
  22. 22.
    A. Lunardi. Chapter 5: Nonlinear parabolic equations and systems. In C.M. Dafermos and E. Feireisl, editors, Handbook of Differential Equations: Evolutionary Equations, volume 1, pages 385–436. North-Holland, 2002.Google Scholar
  23. 23.
    Lunardi A., Sinestrari E., von Wahl W.: A semigroup approach to the time dependent parabolic initial-boundary value problem. Differential Integral Equations 5(6), 1275–1306 (1992)zbMATHMathSciNetGoogle Scholar
  24. 24.
    C. B. Morrey, Jr. Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, Band 130. Springer-Verlag New York, Inc., New York, 1966.Google Scholar
  25. 25.
    Prüss J., Simonett G., Zacher R.: On convergence of solutions to equilibria for quasilinear parabolic problems. Journal of Differential Equations 246(10), 3902–3931 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    J. Prüss, G. Simonett, and R. Zacher. On normal stability for nonlinear parabolic equations. Discrete Contin. Dyn. Syst., (Dynamical Systems, Differential Equations and Applications. 7th AIMS Conference, suppl.):612–621, 2009.Google Scholar
  27. 27.
    T. Runst and W. Sickel. Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, volume 3 of de Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyter & Co., Berlin, 1996.Google Scholar
  28. 28.
    Schnürer O., Azouani A., Georgi M., Hell J., Jangle N., Koeller A., Marxen T., Ritthaler S., Sáez M., Schulze F., Smith B.: Evolution of convex lens-shaped networks under the curve shortening flow. Trans. Amer. Math. Soc. 363(5), 2265–2294 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Solonnikov V.A.: On boundary value problems for linear parabolic systems of differential equations of general form. Trudy Mat. Inst. Steklov. 83, 3–163 (1965)MathSciNetGoogle Scholar
  30. 30.
    E. Zeidler. Nonlinear functional analysis and its applications. I. Springer-Verlag, New York, 1986. Fixed-point theorems, Translated from the German by Peter R. Wadsack.Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany

Personalised recommendations