# American Institute of Mathematical Sciences

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 and 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), 505-542. doi: 10.1007/BF00991895. [2] B. Aulbach and T. Wanner, The Hartman-Grobman theorem for Carathéodory-type differential equations in Banach spaces, Nonlin. Analysis (TMA), 40 (2000), 91-104. doi: 10.1016/S0362-546X(00)85006-3. [3] P. Bates and K. Lu, A Hartman-Grobman theorem for the Cahn-Hilliard and phase-field equations, J. Dyn. Differ. Equations, 6 (1994), 101-145. doi: 10.1007/BF02219190. [4] A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4. [5] C. Chicone and Y. Latushkin, Center manifolds for infinite dimensional nonautonomous differential equations, J. Differ. Equations, 141 (1997), 356-399. doi: 10.1006/jdeq.1997.3343. [6] S.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite dimensional spaces, J. Differ. Equations, 94 (1991), 266-291. doi: 10.1016/0022-0396(91)90093-O. [7] S.-N. Chow and H. Leiva, Dynamical spectrum for skew product flows in Banach spaces, In J. Henderson (ed.), Boundary value problems for functional differential equations, 85-105. World Scientific, Singapore etc., 1995. [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), 421-432. doi: 10.1007/PL00001554. [9] D. Grobman, Homeomorphism of systems of differential equations, Doklady Akademii Nauk SSSR, 128 (1959), 880-881. [10] P. Hartman, A lemma in the theory of structural stability of differential equations, Proc. Am. Math. Soc., 11 (1960), 610-620. doi: 10.1090/S0002-9939-1960-0121542-7. [11] J. Li, K. Lu and P. Bates, Invariant foliations for random dynamical systems, Discrete and Continuous Dynamical Systems, 34 (2014), 3639-3666. doi: 10.3934/dcds.2014.34.3639. [12] K. Lu, A Hartman-Grobman theorem for scalar reaction-diffusion equations, J. Differ. Equations, 93 (1991), 364-394. doi: 10.1016/0022-0396(91)90017-4. [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.), Dynamics of Infinite Dimensional Systems, Springer, New York etc., 37 (1987), 187-210. [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), 332-353. doi: 10.1007/BF01203774. [15] K. J. Palmer, A generalization of Hartman's linearization theorem, J. Math. Anal. Appl., 41 (1973), 753-758. doi: 10.1016/0022-247X(73)90245-X. [16] C. Pötzsche, Topological decoupling, linearization and perturbation on inhomogeneous time scales, J. Differ. Equations, 245 (2008), 1210-1242. doi: 10.1016/j.jde.2008.06.011. [17] C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lect. Notes Math. 2002, Springer, Berlin etc., 2010. doi: 10.1007/978-3-642-14258-1. [18] C. Pötzsche and E. Russ, Notes on spectrum and exponential decay in nonautonomous evolutionary equations, Electron. J. Qual. Theory Differ. Equ., accepted, 2015. [19] E. Russ, On the Dichotomy Spectrum in Infinite Dimensions, PhD thesis, Alpen-Adria Universität Klagenfurt, 2015. [20] A. Reinfelds, Partial decoupling for noninvertible mappings, Differential Equations and Dynamical Systems, 2 (1994), 205-215. [21] A. Reinfelds, The reduction principle for discrete dynamical and semidynamical systems in metric spaces, Z. Angew. Math. Phys., 45 (1994), 933-955. doi: 10.1007/BF00952086. [22] R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Differ. Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8. [23] G. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143. Springer, Berlin etc., 2002. doi: 10.1007/978-1-4757-5037-9. [24] S. Siegmund, Spektraltheorie, Glatte Faserungen und Normalformen Für Differentialgleichungen vom Carathéodory-Typ, Dissertation, Universität Augsburg, Germany, 1999. [25] A. N. Šošitaĭšvili, Bifurcations of topological type at singular points of parametrized vector fields, Functional Analysis and its Applications, 5 (1972), 169-170. [26] N. Sternberg, A Hartman-Grobman theorem for a class of retarded functional differential equations, J. Math. Anal. Appl., 176 (1993), 156-165. doi: 10.1006/jmaa.1993.1206. [27] E. M. Wright, A nonlinear difference-differential equation, J. Reine Angew. Math. 194 (1955), 66-87.

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##### References:
 [1] B. Aulbach and B. M. Garay, Partial linearization for noninvertible mappings, Z. Angew. Math. Phys., 45 (1994), 505-542. doi: 10.1007/BF00991895. [2] B. Aulbach and T. Wanner, The Hartman-Grobman theorem for Carathéodory-type differential equations in Banach spaces, Nonlin. Analysis (TMA), 40 (2000), 91-104. doi: 10.1016/S0362-546X(00)85006-3. [3] P. Bates and K. Lu, A Hartman-Grobman theorem for the Cahn-Hilliard and phase-field equations, J. Dyn. Differ. Equations, 6 (1994), 101-145. doi: 10.1007/BF02219190. [4] A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4. [5] C. Chicone and Y. Latushkin, Center manifolds for infinite dimensional nonautonomous differential equations, J. Differ. Equations, 141 (1997), 356-399. doi: 10.1006/jdeq.1997.3343. [6] S.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite dimensional spaces, J. Differ. Equations, 94 (1991), 266-291. doi: 10.1016/0022-0396(91)90093-O. [7] S.-N. Chow and H. Leiva, Dynamical spectrum for skew product flows in Banach spaces, In J. Henderson (ed.), Boundary value problems for functional differential equations, 85-105. World Scientific, Singapore etc., 1995. [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), 421-432. doi: 10.1007/PL00001554. [9] D. Grobman, Homeomorphism of systems of differential equations, Doklady Akademii Nauk SSSR, 128 (1959), 880-881. [10] P. Hartman, A lemma in the theory of structural stability of differential equations, Proc. Am. Math. Soc., 11 (1960), 610-620. doi: 10.1090/S0002-9939-1960-0121542-7. [11] J. Li, K. Lu and P. Bates, Invariant foliations for random dynamical systems, Discrete and Continuous Dynamical Systems, 34 (2014), 3639-3666. doi: 10.3934/dcds.2014.34.3639. [12] K. Lu, A Hartman-Grobman theorem for scalar reaction-diffusion equations, J. Differ. Equations, 93 (1991), 364-394. doi: 10.1016/0022-0396(91)90017-4. [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.), Dynamics of Infinite Dimensional Systems, Springer, New York etc., 37 (1987), 187-210. [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), 332-353. doi: 10.1007/BF01203774. [15] K. J. Palmer, A generalization of Hartman's linearization theorem, J. Math. Anal. Appl., 41 (1973), 753-758. doi: 10.1016/0022-247X(73)90245-X. [16] C. Pötzsche, Topological decoupling, linearization and perturbation on inhomogeneous time scales, J. Differ. Equations, 245 (2008), 1210-1242. doi: 10.1016/j.jde.2008.06.011. [17] C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lect. Notes Math. 2002, Springer, Berlin etc., 2010. doi: 10.1007/978-3-642-14258-1. [18] C. Pötzsche and E. Russ, Notes on spectrum and exponential decay in nonautonomous evolutionary equations, Electron. J. Qual. Theory Differ. Equ., accepted, 2015. [19] E. Russ, On the Dichotomy Spectrum in Infinite Dimensions, PhD thesis, Alpen-Adria Universität Klagenfurt, 2015. [20] A. Reinfelds, Partial decoupling for noninvertible mappings, Differential Equations and Dynamical Systems, 2 (1994), 205-215. [21] A. Reinfelds, The reduction principle for discrete dynamical and semidynamical systems in metric spaces, Z. Angew. Math. Phys., 45 (1994), 933-955. doi: 10.1007/BF00952086. [22] R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Differ. Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8. [23] G. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143. Springer, Berlin etc., 2002. doi: 10.1007/978-1-4757-5037-9. [24] S. Siegmund, Spektraltheorie, Glatte Faserungen und Normalformen Für Differentialgleichungen vom Carathéodory-Typ, Dissertation, Universität Augsburg, Germany, 1999. [25] A. N. Šošitaĭšvili, Bifurcations of topological type at singular points of parametrized vector fields, Functional Analysis and its Applications, 5 (1972), 169-170. [26] N. Sternberg, A Hartman-Grobman theorem for a class of retarded functional differential equations, J. Math. Anal. Appl., 176 (1993), 156-165. doi: 10.1006/jmaa.1993.1206. [27] E. M. Wright, A nonlinear difference-differential equation, J. Reine Angew. Math. 194 (1955), 66-87.
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