Invariant manifolds of partial functional differential equations

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• 作者：Van Minh ; Nguyen ; Wu ; Jianhong
• 关键词：primary 34K19 ; 37L10 ; secondary 35B40 ; 34G20 ; Partial functional differential equation ; Evolutionary process ; Invariant manifold ; Smoothness
• 刊名：Journal of Differential Equations
• 出版年：2004
• 出版时间：April 10, 2004
• 年：2004
• 卷：198
• 期：2
• 页码：381-421
• 全文大小：426 K

文摘

This paper is concerned with the existence, smoothness and attractivity of invariant manifolds for evolutionary processes on general Banach spaces when the nonlinear perturbation has a small global Lipschitz constant and locally C^k-smooth near the trivial solution. Such a nonlinear perturbation arises in many applications through the usual cut-off procedure, but the requirement in the existing literature that the nonlinear perturbation is globally C^k-smooth and has a globally small Lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut-off function. Our general results are illustrated by and applied to partial functional differential equations for which the phase space C([−r,0],X) (where r>0 and X being a Banach space) has no smooth inner product structure and for which the validity of variation-of-constants formula is still an interesting open problem.