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Type: Journal article
Title: Normal forms and invariant manifolds for nonlinear, non-autonomous PDEs, viewed as ODEs in infinite dimensions
Author: Hochs, P.
Roberts, A.J.
Citation: Journal of Differential Equations, 2019; 267(12):7263-7312
Publisher: Elsevier
Issue Date: 2019
ISSN: 0022-0396
Statement of
Peter Hochs and A. J. Roberts
Abstract: We prove that a general class of nonlinear, non-autonomous odes in Fréchet spaces are close to odes in a specific normal form, where closeness means that solutions of the normal form ode satisfy the original ode up to a residual that vanishes up to any desired order. In this normal form, the centre, stable and unstable coordinates of the ode are clearly separated, which allows us to define invariant manifolds of such equations in a robust way. The main motivation is the case where the Fréchet space in question is a suitable function space, and the maps involved in an ode in this space are defined in terms of derivatives of the functions, so that the infinite-dimensional ode is a finite-dimensional pde. We show that our methods apply to a relevant class of nonlinear, non-autonomous pdes in this way.
Keywords: Nonlinear Non-autonomous pde Invariant manifold Normal form Differentiable and polynomial maps between Fréchet spaces
Rights: © 2019 Elsevier Inc. All rights reserved.
DOI: 10.1016/j.jde.2019.07.021
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