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Nonlinear Partial Differential Equations

  • George R. Sell
  • Yuncheng You
Part of the Applied Mathematical Sciences book series (AMS, volume 143)

Abstract

In this chapter we turn our attention to the study of the dynamical properties of solutions of nonlinear partial differential equations. We are especially interested here in those nonlinear evolutionary equations which arise in the analysis of two broad classes of partial differential equations: parabolic evolutionary equations and hyperbolic evolutionary equations. While our usage of the terms “parabolic” and “hyperbolic” in this context is motivated by related concepts arising in the basic classification of partial differential equations, we will attribute these terms, instead, to certain dynamical features of the linear ancestry of the underlying nonlinear problems. More precisely, the linear prototypes of the partial differential equations of interest here include the heat equation and the wave equation:
$${\partial _t}u - \nu \Delta u = 0\;and\;\partial _t^2u - \nu \Delta u = 0,$$
on a suitable domain Ω in ℝ d , with various boundary conditions and initial conditions.

Keywords

Strong Solution Mild Solution Global Attractor Nonlinear Partial Differential Equation Nonlinear Wave Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Additional Readings

  1. A V Babin and M I Vishik (1992), Attractors of Evolution Equations, English translation, Studies in Math and its Appl, No 25, North-Holland, Amsterdam.Google Scholar
  2. M L Cartwright (1969), Almost periodic equations and almost periodic flows,J Differential Equations 5 167–181.Google Scholar
  3. S-N Chow and K Lu (1988), Invariant manifolds for flows in Banach spaces, J Differential Equations 74, 285–317.MathSciNetzbMATHCrossRefGoogle Scholar
  4. H Haken (1981), Chaos and order in nature, Chaos and Order in Nature, H Haken, ed., Springer Verlag, Berlin, pp. 2–11.Google Scholar
  5. J K Hale and H Kocak (1991), Dynamics and Bifurcations, Springer Verlag, New York.zbMATHCrossRefGoogle Scholar
  6. J K Hale, X-B Lin, and G Raugel (1988), Upper semicontinuity of attractors and partial differential equations, Math Comp 50, 89–123.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • George R. Sell
    • 1
  • Yuncheng You
    • 2
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of MathematicsUniversity of South FloridaTampaUSA

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