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Basic Theory of Evolutionary Equations

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

Abstract

In the last chapter, we presented a theory describing solutions of a linear evolutionary equation ə t u + Au = 0, where \( {\partial _t}u = \frac{d}{{dt}}u \) on a Banach space W, in terms of C o-semigroups. As we have seen, this theory allows one to construct mild solutions of many linear partial differential equations with constant coefficients. Our objective in this chapter is to generalize this theory so that it applies first to the linear inhomogeneous equation
$$ {\partial _t}u + Au = f(t) $$
(40.1)
and then the nonlinear evolutionary equation
$$ {\partial _t}u + Au = F(u) $$
(40.2)
.

Keywords

Banach Space Weak Solution Evolutionary Equation Strong Solution Mild Solution 
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. J K Hale and S M Verduyn Lunel (1993), Introduction to Functional Differential Equations, Applied Mathematical Sciences, Vol 99, Springer Verlag, New York.Google Scholar
  2. D B Henry (1981), Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, No 840, Springer Verlag, New York.Google Scholar
  3. A Lunardi (1995), Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Boston.zbMATHCrossRefGoogle Scholar
  4. A Pazy (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol 44, Springer Verlag, New York.Google Scholar
  5. V A Pliss and G R Sell (1991), Perturbations of attractors of differential equations,J Differential Equations 92, 100–124.Google Scholar
  6. V A Pliss (1977), Integral Sets of Periodic Systems of Differential Equations, Russian, Izdat Nauka, Moscow.Google Scholar
  7. V A Pliss and G R Sell (1999), Robustness of exponential dichotomies for evolutionary equations,J Dynamics Differential Equations 11, 471–513.Google Scholar
  8. A V Babin and G R Sell (2000), Attractors of nonautonomous parabolic equations and their symmetry properties,JDifferential Equations 160, 1–50.Google Scholar
  9. R J Sacker and G R Sell (1994), Dichotomies for linear evolutionary equations in Banach spaces J Differential Equations 113, 17–67.Google Scholar
  10. W Shen and Y Yi (1996), Ergodicity of minimal sets of scalar parabolic equations J Dynamics Differential Equations 8, 299–323.Google Scholar
  11. W Shen and Y Yi (1998), Almost automorphic and almost periodic dynamics in skew product semiflows Memoirs Am Math Soc, No 647, vol 136.Google Scholar
  12. H Tanabe (1979), Equations of Evolution, Pitman, Lond.zbMATHGoogle Scholar
  13. R Temam (1982), Behaviour at time t = 0 of the solutions of semilinear evolution equations JDifferential Equations 43, 73–92.Google Scholar
  14. R Temam (1988), Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol 68, Springer Verlag, New York.Google 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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