# American Institute of Mathematical Sciences

March  2015, 35(3): 1193-1230. doi: 10.3934/dcds.2015.35.1193

## Center manifolds and attractivity for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions

 1 Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany

Received  February 2014 Revised  June 2014 Published  October 2014

We construct and investigate local invariant manifolds for a large class of quasilinear parabolic problems with fully nonlinear dynamical boundary conditions and study their attractivity properties. In a companion paper we have developed the corresponding solution theory. Examples for the class of systems considered are reaction--diffusion systems or phase field models with dynamical boundary conditions and to the two--phase Stefan problem with surface tension.
Citation: Roland Schnaubelt. Center manifolds and attractivity for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1193-1230. doi: 10.3934/dcds.2015.35.1193
##### References:
 [1] H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence, Math. Z., 202 (1989), 219-250. doi: 10.1007/BF01215256. [2] H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. [3] P. Bates and C. Jones, Invariant manifolds for semilinear partial differential equations, in Dynamics Reported. A Series in Dynamical Systems and their Applications, (eds. U. Kirchgraber and H.-O. Walther), Wiley, 2 (1989), 1-38. [4] R. Denk, M. Hieber and J. Prüss, Optimal $L_p$-$L_q$ estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9. [5] R. Denk, J. Prüss and R. Zacher, Maximal $L_p$-regularity of parabolic problems with boundary dynamics of relaxation type, J. Funct. Anal., 255 (2008), 3149-3187. doi: 10.1016/j.jfa.2008.07.012. [6] J. Escher and G. Simonett, A center manifold analysis for the Mullins-Sekerka model, J. Differential Equations, 143 (1998), 267-292. doi: 10.1006/jdeq.1997.3373. [7] J. Escher, J. Prüss and G. Simonett, Analytic solutions for a Stefan problem with Gibbs-Thomson correction, J. Reine Angew. Math., 563 (2003), 1-52. doi: 10.1515/crll.2003.082. [8] R. Johnson, Y. Latushkin and R. Schnaubelt, Reduction principle and asymptotic phase for center manifolds of parabolic systems with nonlinear boundary conditions, J. Dynam. Differential Equations, 26 (2014), 243-266. doi: 10.1007/s10884-014-9360-7. [9] 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. doi: 10.1007/s00028-006-0272-9. [10] Y. Latushkin, J. Prüss and R. Schnaubelt, Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 595-633. doi: 10.3934/dcdsb.2008.9.595. [11] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. doi: 10.1007/978-3-0348-9234-6. [12] A. Mielke, Locally invariant manifolds for quasilinear parabolic equations, Rocky Mountain J. Math., 21 (1991), 707-714. doi: 10.1216/rmjm/1181072962. [13] M. Meyries and R. Schnaubelt, Maximal regularity with temporal weights for parabolic problems with inhomogeneous boundary conditions, Math. Nachr., 285 (2012), 1032-1051. doi: 10.1002/mana.201100057. [14] K. Palmer, On the stability of the center manifold, Z. Angew. Math. Phys., 38 (1987), 273-278. doi: 10.1007/BF00945412. [15] J. Püss and G. Simonett, Stability of equilibria for the Stefan problem with surface tension, SIAM J. Math. Anal., 40 (2008), 675-698. doi: 10.1137/070700632. [16] J. Prüss, G. Simonett and R. Zacher, On convergence of solutions to equilibria for quasilinear parabolic problems, J. Differential Equations, 246 (2009), 3902-3931. doi: 10.1016/j.jde.2008.10.034. [17] J. Prüss, G. Simonett and R. Zacher, Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension, Arch. Ration. Mech. Anal., 207 (2013), 611-667. doi: 10.1007/s00205-012-0571-y. [18] J. Prüss, M. Wilke and G. Simonett, Invariant foliations near normally hyperbolic equilibria for quasilinear parabolic problems, Adv. Nonlinear Stud., 13 (2013), 231-243. [19] M. Renardy, A centre manifold theorem for hyperbolic PDEs, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 363-377. doi: 10.1017/S0308210500021168. [20] R. Schnaubelt, Stable and unstable manifolds for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions,, submitted, (). [21] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. [22] G. Simonett, Invariant manifolds and bifurcation for quasilinear reaction-diffusion systems, Nonlinear Anal., 23 (1994), 515-544. doi: 10.1016/0362-546X(94)90092-2. [23] G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796. [24] H. Triebel, Interpolation Theory,Function Spaces, Differential Operators, J. A. Barth, Heidelberg, 1995. [25] A. Vanderbauwhede and S. A. van Gils, Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal., 72 (1987), 209-224. doi: 10.1016/0022-1236(87)90086-3. [26] A. Vanderbauwhede and G. Iooss, Center manifolds in infinite dimensions, in Dynamics Reported: Expositions in Dynamical Systems (New Series), (eds. C.K.R.T. Jones, U. Kirchgraber and H.-O. Walther), Springer-Verlag, 1 (1992), 125-163.

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##### References:
 [1] H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence, Math. Z., 202 (1989), 219-250. doi: 10.1007/BF01215256. [2] H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. [3] P. Bates and C. Jones, Invariant manifolds for semilinear partial differential equations, in Dynamics Reported. A Series in Dynamical Systems and their Applications, (eds. U. Kirchgraber and H.-O. Walther), Wiley, 2 (1989), 1-38. [4] R. Denk, M. Hieber and J. Prüss, Optimal $L_p$-$L_q$ estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9. [5] R. Denk, J. Prüss and R. Zacher, Maximal $L_p$-regularity of parabolic problems with boundary dynamics of relaxation type, J. Funct. Anal., 255 (2008), 3149-3187. doi: 10.1016/j.jfa.2008.07.012. [6] J. Escher and G. Simonett, A center manifold analysis for the Mullins-Sekerka model, J. Differential Equations, 143 (1998), 267-292. doi: 10.1006/jdeq.1997.3373. [7] J. Escher, J. Prüss and G. Simonett, Analytic solutions for a Stefan problem with Gibbs-Thomson correction, J. Reine Angew. Math., 563 (2003), 1-52. doi: 10.1515/crll.2003.082. [8] R. Johnson, Y. Latushkin and R. Schnaubelt, Reduction principle and asymptotic phase for center manifolds of parabolic systems with nonlinear boundary conditions, J. Dynam. Differential Equations, 26 (2014), 243-266. doi: 10.1007/s10884-014-9360-7. [9] 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. doi: 10.1007/s00028-006-0272-9. [10] Y. Latushkin, J. Prüss and R. Schnaubelt, Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 595-633. doi: 10.3934/dcdsb.2008.9.595. [11] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. doi: 10.1007/978-3-0348-9234-6. [12] A. Mielke, Locally invariant manifolds for quasilinear parabolic equations, Rocky Mountain J. Math., 21 (1991), 707-714. doi: 10.1216/rmjm/1181072962. [13] M. Meyries and R. Schnaubelt, Maximal regularity with temporal weights for parabolic problems with inhomogeneous boundary conditions, Math. Nachr., 285 (2012), 1032-1051. doi: 10.1002/mana.201100057. [14] K. Palmer, On the stability of the center manifold, Z. Angew. Math. Phys., 38 (1987), 273-278. doi: 10.1007/BF00945412. [15] J. Püss and G. Simonett, Stability of equilibria for the Stefan problem with surface tension, SIAM J. Math. Anal., 40 (2008), 675-698. doi: 10.1137/070700632. [16] J. Prüss, G. Simonett and R. Zacher, On convergence of solutions to equilibria for quasilinear parabolic problems, J. Differential Equations, 246 (2009), 3902-3931. doi: 10.1016/j.jde.2008.10.034. [17] J. Prüss, G. Simonett and R. Zacher, Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension, Arch. Ration. Mech. Anal., 207 (2013), 611-667. doi: 10.1007/s00205-012-0571-y. [18] J. Prüss, M. Wilke and G. Simonett, Invariant foliations near normally hyperbolic equilibria for quasilinear parabolic problems, Adv. Nonlinear Stud., 13 (2013), 231-243. [19] M. Renardy, A centre manifold theorem for hyperbolic PDEs, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 363-377. doi: 10.1017/S0308210500021168. [20] R. Schnaubelt, Stable and unstable manifolds for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions,, submitted, (). [21] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. [22] G. Simonett, Invariant manifolds and bifurcation for quasilinear reaction-diffusion systems, Nonlinear Anal., 23 (1994), 515-544. doi: 10.1016/0362-546X(94)90092-2. [23] G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations, 8 (1995), 753-796. [24] H. Triebel, Interpolation Theory,Function Spaces, Differential Operators, J. A. Barth, Heidelberg, 1995. [25] A. Vanderbauwhede and S. A. van Gils, Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal., 72 (1987), 209-224. doi: 10.1016/0022-1236(87)90086-3. [26] A. Vanderbauwhede and G. Iooss, Center manifolds in infinite dimensions, in Dynamics Reported: Expositions in Dynamical Systems (New Series), (eds. C.K.R.T. Jones, U. Kirchgraber and H.-O. Walther), Springer-Verlag, 1 (1992), 125-163.
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