Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions

Discrete and Continuous Dynamical Systems - Series B

Volumn 9, Issue 3-4, 2008, Pages 595-633

  Latushkin, Yuri ^a   Prüss, Jan ^b   Schnaubelt, Roland ^c  

^a UNIVERSITY OF MISSOURI   (United States)

^b MARTIN LUTHER UNIVERSITY HALLE WITTENBERG   (Germany)

^c UNIVERSITY OF KARLSRUHE   (Germany)

Author keywords

Anisotropic Slobodetskii spaces; Attractivity; Center manifold reduction; Exponential dichotomy; Extrapolation; Implicit function theorem; Initial boundary value problem; Invariant manifold; Maximal regularity; Nemytskii operators; Parabolic system; Stability

Indexed keywords

EID: 63449092028     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2008.9.595     Document Type: Article

References (22)

1. R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Academic Press, New York, 1975.
2. H. Amann, "Linear and Quasilinear Parabolic Problems, Vol. I. Abstract Linear Theory," Birkhäuser, Boston (MA), 1995.
3. H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.
4. P. Bates and C. Jones, Invariant manifolds for semilinear partial differential equations, In: "U. Kirchgraber and H.-O. Walther (Eds.), Dynamics Reported. A Series in Dynamical Systems and their Applications," 2, Wiley, Chichester, 1989, 1-38.
5. C.-M. Brauner, J. Hulshof and A. Lunardi, A general approach to stability in free boundary problems, J. Differential Equations, 164 (2000), 16-48.
6. G. Da Prato and A. Lunardi, Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations in Banach space, Arch. Rational Mech. Anal., 101 (1988), 115-141.
7. R. Denk, M. Hieber and J. Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Memoirs Amer. Math. Soc, 166 (2003).
8. q estimates for parabolic boundary value roblems with inhomogeneous data, Math. Z., 257 (2007), 193-224.
9. J. Escher, J. Priiss and G. Simonett, Analytic solutions for a Stefan problem with Gibbs-Thomson correction, J. Reine Angew. Math., 563 (2003), 1-52.
10. J. Escher and G. Simonett, A center manifold analysis for the Mullins-Sekerka model, J. Differential Equations, 143 (1998), 267-292.
11. M. Kotschote, "Strong Well-Posedness of a Model for an Ionic Exchange Process," Ph.D. thesis, Martin-Luther-Universität Halle, 2003.
12. Y. Latushkin, J. Prüss and R. Schnaubelt, Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions, J. Evolution Equations, 6 (2006), 537-576.
13. A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, Basel, 1995.
14. A. Mielke, Locally invariant manifolds for quasilinear parabolic equations, Rocky Mountain J. Math., 21 (1991), 707-714.
15. J. Prüss and G. Simonett, Stability of equilibria for the Stefan problem with surface tension, Submitted.
16. M. Renardy, A centre manifold theorem for hyperbolic PDEs, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 363-377.
17. T. Runst and W. Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations," Walter de Gruyter, Berlin, 1996.
18. G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.
19. G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796.
20. H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," Second edition, J. A. Barth, Heidelberg, 1995.
21. A. Vanderbauwhede and S. A. van Gils, Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal., 72 (1987), 209-224.
22. A. Vanderbauwhede and G. Iooss, Center manifolds in infinite dimensions, In: "C.K.R.T. Jones, U. Kirchgraber and H.-O. Walther (Eds.), Dynamics Reported: Expositions in Dynamical Systems (New Series)," 1, Springer-Verlag, Berlin, 1992, 125-163.