August  2016, 9(4): 1235-1268. doi: 10.3934/dcdss.2016050

Topological decoupling and linearization of nonautonomous evolution equations

1. 

Institut für Mathematik, Alpen-Adria Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt, Austria

Received  July 2015 Revised  September 2015 Published  August 2016

Topological linearization results typically require solution flows rather than merely semiflows. An exception occurs when the linearization fulfills spectral assumptions met e.g. for scalar reaction-diffusion equations. We employ tools from the geometric theory of nonautonomous dynamical systems in order to extend earlier work by Lu [12] to time-variant evolution equations under corresponding conditions on the Sacker-Sell spectrum of the linear part. Our abstract results are applied to nonautonomous reaction-diffusion and convection equations.
Citation: Christian Pötzsche, Evamaria Russ. Topological decoupling and linearization of nonautonomous evolution equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1235-1268. doi: 10.3934/dcdss.2016050
References:
[1]

B. Aulbach and B. M. Garay, Partial linearization for noninvertible mappings,, Z. Angew. Math. Phys., 45 (1994), 505. doi: 10.1007/BF00991895. Google Scholar

[2]

B. Aulbach and T. Wanner, The Hartman-Grobman theorem for Carathéodory-type differential equations in Banach spaces,, Nonlin. Analysis (TMA), 40 (2000), 91. doi: 10.1016/S0362-546X(00)85006-3. Google Scholar

[3]

P. Bates and K. Lu, A Hartman-Grobman theorem for the Cahn-Hilliard and phase-field equations,, J. Dyn. Differ. Equations, 6 (1994), 101. doi: 10.1007/BF02219190. Google Scholar

[4]

A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences,, 182. Springer, (2013). doi: 10.1007/978-1-4614-4581-4. Google Scholar

[5]

C. Chicone and Y. Latushkin, Center manifolds for infinite dimensional nonautonomous differential equations,, J. Differ. Equations, 141 (1997), 356. doi: 10.1006/jdeq.1997.3343. Google Scholar

[6]

S.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite dimensional spaces,, J. Differ. Equations, 94 (1991), 266. doi: 10.1016/0022-0396(91)90093-O. Google Scholar

[7]

S.-N. Chow and H. Leiva, Dynamical spectrum for skew product flows in Banach spaces,, In J. Henderson (ed.), (1995), 85. Google Scholar

[8]

G. Farkas, A Hartman-Grobman result for retarded functional differential equations with an application to the numerics around hyperbolic equilibria,, Z. Angew. Math. Phys., 52 (2001), 421. doi: 10.1007/PL00001554. Google Scholar

[9]

D. Grobman, Homeomorphism of systems of differential equations,, Doklady Akademii Nauk SSSR, 128 (1959), 880. Google Scholar

[10]

P. Hartman, A lemma in the theory of structural stability of differential equations,, Proc. Am. Math. Soc., 11 (1960), 610. doi: 10.1090/S0002-9939-1960-0121542-7. Google Scholar

[11]

J. Li, K. Lu and P. Bates, Invariant foliations for random dynamical systems,, Discrete and Continuous Dynamical Systems, 34 (2014), 3639. doi: 10.3934/dcds.2014.34.3639. Google Scholar

[12]

K. Lu, A Hartman-Grobman theorem for scalar reaction-diffusion equations,, J. Differ. Equations, 93 (1991), 364. doi: 10.1016/0022-0396(91)90017-4. Google Scholar

[13]

X. Mora and J. Solà-Morales, Existence and nonexistence of finite-dimensional globally attracting invariant manifolds in semilinear damped wave equation,, In S.-N. Chow and J.K. Hale (eds.), 37 (1987), 187. Google Scholar

[14]

N. Van Minh, F. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line,, Integral Equations Oper. Theory, 32 (1998), 332. doi: 10.1007/BF01203774. Google Scholar

[15]

K. J. Palmer, A generalization of Hartman's linearization theorem,, J. Math. Anal. Appl., 41 (1973), 753. doi: 10.1016/0022-247X(73)90245-X. Google Scholar

[16]

C. Pötzsche, Topological decoupling, linearization and perturbation on inhomogeneous time scales,, J. Differ. Equations, 245 (2008), 1210. doi: 10.1016/j.jde.2008.06.011. Google Scholar

[17]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lect. Notes Math. 2002,, Springer, (2010). doi: 10.1007/978-3-642-14258-1. Google Scholar

[18]

C. Pötzsche and E. Russ, Notes on spectrum and exponential decay in nonautonomous evolutionary equations,, Electron. J. Qual. Theory Differ. Equ., (2015). Google Scholar

[19]

E. Russ, On the Dichotomy Spectrum in Infinite Dimensions,, PhD thesis, (2015). Google Scholar

[20]

A. Reinfelds, Partial decoupling for noninvertible mappings,, Differential Equations and Dynamical Systems, 2 (1994), 205. Google Scholar

[21]

A. Reinfelds, The reduction principle for discrete dynamical and semidynamical systems in metric spaces,, Z. Angew. Math. Phys., 45 (1994), 933. doi: 10.1007/BF00952086. Google Scholar

[22]

R. Sacker and G. Sell, A spectral theory for linear differential systems,, J. Differ. Equations, 27 (1978), 320. doi: 10.1016/0022-0396(78)90057-8. Google Scholar

[23]

G. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143., Springer, (2002). doi: 10.1007/978-1-4757-5037-9. Google Scholar

[24]

S. Siegmund, Spektraltheorie, Glatte Faserungen und Normalformen Für Differentialgleichungen vom Carathéodory-Typ,, Dissertation, (1999). Google Scholar

[25]

A. N. Šošitaĭšvili, Bifurcations of topological type at singular points of parametrized vector fields,, Functional Analysis and its Applications, 5 (1972), 169. Google Scholar

[26]

N. Sternberg, A Hartman-Grobman theorem for a class of retarded functional differential equations,, J. Math. Anal. Appl., 176 (1993), 156. doi: 10.1006/jmaa.1993.1206. Google Scholar

[27]

E. M. Wright, A nonlinear difference-differential equation,, J. Reine Angew. Math. 194 (1955), 194 (1955), 66. Google Scholar

show all references

References:
[1]

B. Aulbach and B. M. Garay, Partial linearization for noninvertible mappings,, Z. Angew. Math. Phys., 45 (1994), 505. doi: 10.1007/BF00991895. Google Scholar

[2]

B. Aulbach and T. Wanner, The Hartman-Grobman theorem for Carathéodory-type differential equations in Banach spaces,, Nonlin. Analysis (TMA), 40 (2000), 91. doi: 10.1016/S0362-546X(00)85006-3. Google Scholar

[3]

P. Bates and K. Lu, A Hartman-Grobman theorem for the Cahn-Hilliard and phase-field equations,, J. Dyn. Differ. Equations, 6 (1994), 101. doi: 10.1007/BF02219190. Google Scholar

[4]

A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences,, 182. Springer, (2013). doi: 10.1007/978-1-4614-4581-4. Google Scholar

[5]

C. Chicone and Y. Latushkin, Center manifolds for infinite dimensional nonautonomous differential equations,, J. Differ. Equations, 141 (1997), 356. doi: 10.1006/jdeq.1997.3343. Google Scholar

[6]

S.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite dimensional spaces,, J. Differ. Equations, 94 (1991), 266. doi: 10.1016/0022-0396(91)90093-O. Google Scholar

[7]

S.-N. Chow and H. Leiva, Dynamical spectrum for skew product flows in Banach spaces,, In J. Henderson (ed.), (1995), 85. Google Scholar

[8]

G. Farkas, A Hartman-Grobman result for retarded functional differential equations with an application to the numerics around hyperbolic equilibria,, Z. Angew. Math. Phys., 52 (2001), 421. doi: 10.1007/PL00001554. Google Scholar

[9]

D. Grobman, Homeomorphism of systems of differential equations,, Doklady Akademii Nauk SSSR, 128 (1959), 880. Google Scholar

[10]

P. Hartman, A lemma in the theory of structural stability of differential equations,, Proc. Am. Math. Soc., 11 (1960), 610. doi: 10.1090/S0002-9939-1960-0121542-7. Google Scholar

[11]

J. Li, K. Lu and P. Bates, Invariant foliations for random dynamical systems,, Discrete and Continuous Dynamical Systems, 34 (2014), 3639. doi: 10.3934/dcds.2014.34.3639. Google Scholar

[12]

K. Lu, A Hartman-Grobman theorem for scalar reaction-diffusion equations,, J. Differ. Equations, 93 (1991), 364. doi: 10.1016/0022-0396(91)90017-4. Google Scholar

[13]

X. Mora and J. Solà-Morales, Existence and nonexistence of finite-dimensional globally attracting invariant manifolds in semilinear damped wave equation,, In S.-N. Chow and J.K. Hale (eds.), 37 (1987), 187. Google Scholar

[14]

N. Van Minh, F. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line,, Integral Equations Oper. Theory, 32 (1998), 332. doi: 10.1007/BF01203774. Google Scholar

[15]

K. J. Palmer, A generalization of Hartman's linearization theorem,, J. Math. Anal. Appl., 41 (1973), 753. doi: 10.1016/0022-247X(73)90245-X. Google Scholar

[16]

C. Pötzsche, Topological decoupling, linearization and perturbation on inhomogeneous time scales,, J. Differ. Equations, 245 (2008), 1210. doi: 10.1016/j.jde.2008.06.011. Google Scholar

[17]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lect. Notes Math. 2002,, Springer, (2010). doi: 10.1007/978-3-642-14258-1. Google Scholar

[18]

C. Pötzsche and E. Russ, Notes on spectrum and exponential decay in nonautonomous evolutionary equations,, Electron. J. Qual. Theory Differ. Equ., (2015). Google Scholar

[19]

E. Russ, On the Dichotomy Spectrum in Infinite Dimensions,, PhD thesis, (2015). Google Scholar

[20]

A. Reinfelds, Partial decoupling for noninvertible mappings,, Differential Equations and Dynamical Systems, 2 (1994), 205. Google Scholar

[21]

A. Reinfelds, The reduction principle for discrete dynamical and semidynamical systems in metric spaces,, Z. Angew. Math. Phys., 45 (1994), 933. doi: 10.1007/BF00952086. Google Scholar

[22]

R. Sacker and G. Sell, A spectral theory for linear differential systems,, J. Differ. Equations, 27 (1978), 320. doi: 10.1016/0022-0396(78)90057-8. Google Scholar

[23]

G. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143., Springer, (2002). doi: 10.1007/978-1-4757-5037-9. Google Scholar

[24]

S. Siegmund, Spektraltheorie, Glatte Faserungen und Normalformen Für Differentialgleichungen vom Carathéodory-Typ,, Dissertation, (1999). Google Scholar

[25]

A. N. Šošitaĭšvili, Bifurcations of topological type at singular points of parametrized vector fields,, Functional Analysis and its Applications, 5 (1972), 169. Google Scholar

[26]

N. Sternberg, A Hartman-Grobman theorem for a class of retarded functional differential equations,, J. Math. Anal. Appl., 176 (1993), 156. doi: 10.1006/jmaa.1993.1206. Google Scholar

[27]

E. M. Wright, A nonlinear difference-differential equation,, J. Reine Angew. Math. 194 (1955), 194 (1955), 66. Google Scholar

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