An Lp theory of invariant manifolds of parabolic partial differential equations of ℝd

Kazuo Kobayashi

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    Abstract

    We study the problem about the existence of finite-dimensional invariant manifolds for nonlinear heat equations of the form ∂u/∂τ = △u + F(u, ∇u) on ℝd × [1, ∞). We show that in spite of the fact that the linearized equation has continuous spectrum extending from negative infinity to zero, there exist finite dimensional invariant manifolds which control the long time asymptotics of solutions. We consider the problem for these equations in the framework of weighted Sobolev spaces of Lp type. The Lp theory of this problem gives the L estimate of the long-time asymptotics of solutions under natural assumptions on the nonlinear term F and their initial data.

    Original languageEnglish
    Pages (from-to)195-212
    Number of pages18
    JournalJournal of Differential Equations
    Volume179
    Issue number1
    DOIs
    Publication statusPublished - 2002 Feb 10

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    ASJC Scopus subject areas

    • Analysis

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