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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
H. Abels, N. Arab, and H. Garcke. Standard Planar Double Bubbles are Stable under Surface Diffusion Flow, arXiv:1505.02979
P. Acquistapace. Zygmund classes with boundary conditions as interpolation spaces, volume 116 of Lecture Notes in Pure and Appl. Math. Dekker, New York, 1989.
Acquistapace P., Terreni B.: Hölder classes with boundary conditions as interpolation spaces. Math. Z. 195(4), 451–471 (1987)
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)
H.W. Alt. Lineare Funktionalanalysis. Springer, 2006.
Amann H.: Global existence for semilinear parabolic systems. J. Reine Angew. Math. 360, 47–83 (1985)
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.
Brauner C., Hulshof J., Lunardi A.: A general approach to stability in free boundary problems. J. Differential Equations 164(1), 16–48 (2000)
D. Depner. Stability Analysis of Geometric Evolution Equations with Triple Lines and Boundary Contact. PhD thesis, Regensburg, 2010.
Depner D., Garcke H.: Linearized stability analysis of surface diffusion for hypersurfaces with triple lines. Hokkaido Math. J. 42(1), 11–52 (2013)
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)
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.
Elliott C.M., Garcke H.: Existence results for diffusive surface motion laws. Adv. Math. Sci. Appl. 7(1), 467–490 (1997)
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.
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.
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)
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)
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)
Hutchings M., Morgan F., Ritoré M., Ros A.: Proof of the double bubble conjecture. Ann. of Math. (2) 155(2), 459–489 (2002)
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)
A. Lunardi. Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995.
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.
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)
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.
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)
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.
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.
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)
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)
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.
About this article
Cite this article
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
Mathematics Subject Classification
- Fully nonlinear parabolic systems
- General nonlinear boundary conditions
- Normally stable
- Surface diffusion flow
- Triple junctions
- Lens-shaped network