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.

show all references

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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