Maximal Accretive Operators, Nonlinear Nonexpansive Semigroups, and First-Order Evolution Equations

  • Eberhard Zeidler


In Chapter 30 we considered first-order evolution equations of the form
, with the operators A(t): VV* and b(t) ∈ V* for all t ∈ ]0,T[. In this connection, “VHV*” is an evolution triple.


Manifold Librium Kato 


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References to the Literature

  1. Komura, Y. (1967): Nonlinear semigroups in Hilbert space. J. Math. Soc. Japan 19, 493–507.Google Scholar
  2. Crandall, M. and Pazy, A. (1969): Semigroups of nonlinear contractions and dissipative sets. Israel J. Math. 21, 261–278.Google Scholar
  3. Crandall, M. and Evans, L. (1975): On the relation of the operator d/ds + d/dt to evolution governed by accretive operators. Israel J. Math. 21, 261–278.Google Scholar
  4. Pazy, A. (1983): Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  5. Butzer, P. and Berens, H. (1967): Semigroups of Operators and Approximations. Springer-Verlag, Berlin.CrossRefGoogle Scholar
  6. Nagel, R. et al. (1986): One-Parameter Linear Semigroups of Positive Operators. Springer Lecture Notes in Mathematics, Vol. 1184. Springer-Verlag, Berlin.Google Scholar
  7. Crandall, M. (1986): Nonlinear semigroups and evolution governed by accretive operators. In: Browder, F. [ed.] (1986), Part 1, pp. 305–338.Google Scholar
  8. Brezis, H. (1973): Operateurs maximaux monotones. North-Holland, Amsterdam.MATHGoogle Scholar
  9. Barbu, V. (1976): Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leyden.CrossRefMATHGoogle Scholar
  10. Deimling, K. (1985): Nonlinear Functional Analysis. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  11. Clement, P. et al. (1987): One-Parameter Semigroups. North-Holland, Amsterdam.MATHGoogle Scholar
  12. Pavel, N. (1987): Nonlinear Evolution Operators and Semigroups. Springer-Verlag, New York.MATHGoogle Scholar
  13. Belleni-Morante, A. (1979): Applied Semigroups and Evolution Equations. Clarendon, Oxford.MATHGoogle Scholar
  14. Benilan, P., Crandall, M., and Pazy, A. (1989): Nonlinear Evolution Governed by Accretive Operators (monograph to appear).Google Scholar
  15. Kato, T. (1975): Quasilinear equations of evolution with applications to partial differential equations. Lecture Notes in Mathematics, Vol. 448, pp. 25–70. Springer-Verlag, New York.Google Scholar
  16. Kato, T. (1975a): The Cauchy problem for quasilinear symmetric hyperbolic systems. Arch. Rational Mech. Anal. 58, 181–205.Google Scholar
  17. Kato, T. (1976): Linear and quasilinear equations of evolution of hyperbolic type. In: Hyperbolicity, Centro Internationale Matematico Estivo II Ciclo 1976, Italy, pp. 125–191.Google Scholar
  18. Kato, T. (1985): Abstract Differential Equations and Nonlinear Mixed Problems. Lezione Fermiane, Pisa.Google Scholar
  19. Kato, T. (1986): Nonlinear equations of evolution in B-spaces. In: Browder, F. [ed.] (1986), Part 2, pp. 9–24.Google Scholar
  20. Crandall, M. and Sougandis, P. (1986): Convergence of difference approximations of quasilinear evolution equations. Nonlinear Anal. 10, 425–445.Google Scholar
  21. Amann, H. (1988): Remarks on quasilinear parabolic systems (to appear).Google Scholar
  22. Amann, H. (1988a): Dynamic theory of quasilinear parabolic equations, I, II (to appear).Google Scholar
  23. Hughes, T., Kato, T., and Marsden, J. (1977): Well-posed quasilinear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rational Mech. Anal. 63, 273–294.Google Scholar
  24. Marsden, J. and Hughes, T. (1983): Mathematical Foundations of Elasticity. Prentice- Hall, Englewood Cliffs, NJ.Google Scholar
  25. Majda, A. (1984): Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  26. Christodoulou, D. and Klainerman, S. (1990): Nonlinear Hyperbolic Equations (monograph to appear).Google Scholar
  27. Henry, D. (1981): Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, Vol. 840. Springer-Verlag, Berlin.Google Scholar
  28. Amann, H. (1984): Existence and regularity for semilinear parabolic evolution equations. Ann. Scuola Norm. Sup. Pisa, CI. Sci. Serie IV, 11, 593–676.Google Scholar
  29. Amann, H. (1985): Global existence for semilinear parabolic systems. J. Reine. Angew. Math. 360, 47–83.Google Scholar
  30. Amann, H. (1986): Quasilinear evolution equations and parabolic systems. Trans. Amer. Math. Soc. 293, 191–227.Google Scholar
  31. Amann, H. (1986a): Quasilinear parabolic systems under nonlinear boundary conditions. Arch. Rational Mech. Anal. 92, 153–192.Google Scholar
  32. Amann, H. (1986b): Semigroups and nonlinear evolution equations. Linear Algebra Appl. 84, 3–32.Google Scholar
  33. Amann, H. (1986c): Parabolic evolution equations with nonlinear boundary conditions. In: Browder, F. [ed.] (1986), Part 1, pp. 17–27.Google Scholar
  34. Amann, H. (1988b): Parabolic evolution equations in interpolation and extrapolation spaces. J. Funct. Anal. 78, 233–277.Google Scholar
  35. Amann, H. (1988c): Parabolic equations and nonlinear boundary conditions. J. Differential Equations 72, 201–269.Google Scholar
  36. Babin, A. and Visik, M. (1983): Regular attractors of semigroups and evolution equations. J. Math. Pures Appl. 62, 441–491.Google Scholar
  37. Babin, A. and Visik, M. (1983a): Attractors of evolution equations and estimates for their dimensions. Uspekhi Mat. Nauk 38 (4), 133–187 (Russian).Google Scholar
  38. Babin, A. and Visik, M. (1986): Unstable invariant sets of semigroups of nonlinear operators and their perturbations. Uspekhi Mat. Nauk 41 (4), 3–34 (Russian).Google Scholar
  39. Temam, R. (1986): Infinite-dimensional dynamical systems in fluid dynamics. In: Browder, F. [ed.] (1986), Part 2, pp. 431–445.Google Scholar
  40. Temam, R. (1988): Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  41. Ladyzenskaja, O. (1987): On the construction of minimal global attractors for the Navier-Stokes equations and other partial differential equations. Uspekhi Mat. Nauk 42 (6), 25–60.Google Scholar
  42. Hale, J. (1988): Asymptotic Behavior of Dissipative Systems. Amer. Math. Soc., Providence, RI.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Eberhard Zeidler
    • 1
  1. 1.Sektion MathematikLeipzigGerman Democratic Republic

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