# 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 & Continuous Dynamical Systems - A, 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.  doi: 10.1007/BF01215256.  Google Scholar [2] H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13.   Google Scholar [3] P. Bates and C. Jones, Invariant manifolds for semilinear partial differential equations,, in Dynamics Reported. A Series in Dynamical Systems and their Applications, 2 (1989), 1.   Google Scholar [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.  doi: 10.1007/s00209-007-0120-9.  Google Scholar [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.  doi: 10.1016/j.jfa.2008.07.012.  Google Scholar [6] J. Escher and G. Simonett, A center manifold analysis for the Mullins-Sekerka model,, J. Differential Equations, 143 (1998), 267.  doi: 10.1006/jdeq.1997.3373.  Google Scholar [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.  doi: 10.1515/crll.2003.082.  Google Scholar [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.  doi: 10.1007/s10884-014-9360-7.  Google Scholar [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.  doi: 10.1007/s00028-006-0272-9.  Google Scholar [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.  doi: 10.3934/dcdsb.2008.9.595.  Google Scholar [11] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Birkhäuser, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar [12] A. Mielke, Locally invariant manifolds for quasilinear parabolic equations,, Rocky Mountain J. Math., 21 (1991), 707.  doi: 10.1216/rmjm/1181072962.  Google Scholar [13] M. Meyries and R. Schnaubelt, Maximal regularity with temporal weights for parabolic problems with inhomogeneous boundary conditions,, Math. Nachr., 285 (2012), 1032.  doi: 10.1002/mana.201100057.  Google Scholar [14] K. Palmer, On the stability of the center manifold,, Z. Angew. Math. Phys., 38 (1987), 273.  doi: 10.1007/BF00945412.  Google Scholar [15] J. Püss and G. Simonett, Stability of equilibria for the Stefan problem with surface tension,, SIAM J. Math. Anal., 40 (2008), 675.  doi: 10.1137/070700632.  Google Scholar [16] J. Prüss, G. Simonett and R. Zacher, On convergence of solutions to equilibria for quasilinear parabolic problems,, J. Differential Equations, 246 (2009), 3902.  doi: 10.1016/j.jde.2008.10.034.  Google Scholar [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.  doi: 10.1007/s00205-012-0571-y.  Google Scholar [18] J. Prüss, M. Wilke and G. Simonett, Invariant foliations near normally hyperbolic equilibria for quasilinear parabolic problems,, Adv. Nonlinear Stud., 13 (2013), 231.   Google Scholar [19] M. Renardy, A centre manifold theorem for hyperbolic PDEs,, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 363.  doi: 10.1017/S0308210500021168.  Google Scholar [20] R. Schnaubelt, Stable and unstable manifolds for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions,, submitted, ().   Google Scholar [21] G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer-Verlag, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar [22] G. Simonett, Invariant manifolds and bifurcation for quasilinear reaction-diffusion systems,, Nonlinear Anal., 23 (1994), 515.  doi: 10.1016/0362-546X(94)90092-2.  Google Scholar [23] G. Simonett, Center manifolds for quasilinear reaction-diffusion systems,, Differential Integral Equations, 8 (1995), 753.   Google Scholar [24] H. Triebel, Interpolation Theory,Function Spaces, Differential Operators,, J. A. Barth, (1995).   Google Scholar [25] A. Vanderbauwhede and S. A. van Gils, Center manifolds and contractions on a scale of Banach spaces,, J. Funct. Anal., 72 (1987), 209.  doi: 10.1016/0022-1236(87)90086-3.  Google Scholar [26] A. Vanderbauwhede and G. Iooss, Center manifolds in infinite dimensions,, in Dynamics Reported: Expositions in Dynamical Systems (New Series), 1 (1992), 125.   Google Scholar

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##### References:
 [1] H. Amann, Dynamic theory of quasilinear parabolic systems. III. Global existence,, Math. Z., 202 (1989), 219.  doi: 10.1007/BF01215256.  Google Scholar [2] H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13.   Google Scholar [3] P. Bates and C. Jones, Invariant manifolds for semilinear partial differential equations,, in Dynamics Reported. A Series in Dynamical Systems and their Applications, 2 (1989), 1.   Google Scholar [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.  doi: 10.1007/s00209-007-0120-9.  Google Scholar [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.  doi: 10.1016/j.jfa.2008.07.012.  Google Scholar [6] J. Escher and G. Simonett, A center manifold analysis for the Mullins-Sekerka model,, J. Differential Equations, 143 (1998), 267.  doi: 10.1006/jdeq.1997.3373.  Google Scholar [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.  doi: 10.1515/crll.2003.082.  Google Scholar [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.  doi: 10.1007/s10884-014-9360-7.  Google Scholar [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.  doi: 10.1007/s00028-006-0272-9.  Google Scholar [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.  doi: 10.3934/dcdsb.2008.9.595.  Google Scholar [11] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Birkhäuser, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar [12] A. Mielke, Locally invariant manifolds for quasilinear parabolic equations,, Rocky Mountain J. Math., 21 (1991), 707.  doi: 10.1216/rmjm/1181072962.  Google Scholar [13] M. Meyries and R. Schnaubelt, Maximal regularity with temporal weights for parabolic problems with inhomogeneous boundary conditions,, Math. Nachr., 285 (2012), 1032.  doi: 10.1002/mana.201100057.  Google Scholar [14] K. Palmer, On the stability of the center manifold,, Z. Angew. Math. Phys., 38 (1987), 273.  doi: 10.1007/BF00945412.  Google Scholar [15] J. Püss and G. Simonett, Stability of equilibria for the Stefan problem with surface tension,, SIAM J. Math. Anal., 40 (2008), 675.  doi: 10.1137/070700632.  Google Scholar [16] J. Prüss, G. Simonett and R. Zacher, On convergence of solutions to equilibria for quasilinear parabolic problems,, J. Differential Equations, 246 (2009), 3902.  doi: 10.1016/j.jde.2008.10.034.  Google Scholar [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.  doi: 10.1007/s00205-012-0571-y.  Google Scholar [18] J. Prüss, M. Wilke and G. Simonett, Invariant foliations near normally hyperbolic equilibria for quasilinear parabolic problems,, Adv. Nonlinear Stud., 13 (2013), 231.   Google Scholar [19] M. Renardy, A centre manifold theorem for hyperbolic PDEs,, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 363.  doi: 10.1017/S0308210500021168.  Google Scholar [20] R. Schnaubelt, Stable and unstable manifolds for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions,, submitted, ().   Google Scholar [21] G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer-Verlag, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar [22] G. Simonett, Invariant manifolds and bifurcation for quasilinear reaction-diffusion systems,, Nonlinear Anal., 23 (1994), 515.  doi: 10.1016/0362-546X(94)90092-2.  Google Scholar [23] G. Simonett, Center manifolds for quasilinear reaction-diffusion systems,, Differential Integral Equations, 8 (1995), 753.   Google Scholar [24] H. Triebel, Interpolation Theory,Function Spaces, Differential Operators,, J. A. Barth, (1995).   Google Scholar [25] A. Vanderbauwhede and S. A. van Gils, Center manifolds and contractions on a scale of Banach spaces,, J. Funct. Anal., 72 (1987), 209.  doi: 10.1016/0022-1236(87)90086-3.  Google Scholar [26] A. Vanderbauwhede and G. Iooss, Center manifolds in infinite dimensions,, in Dynamics Reported: Expositions in Dynamical Systems (New Series), 1 (1992), 125.   Google Scholar
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