Evolutionary Equations

Part of the Springer Monographs in Mathematics book series (SMM)


This chapter provides preliminary material dealing with evolutionary abstract equations that serve as a prototype for von Karman evolutions discussed in subsequent chapters. Special attention is paid to the existence and uniqueness of solutions to nonlinear evolutionary equations. The methods used are based on monotone operator theory and their adaptation to nonmonotone problems. In what follows we provide a brief exposition of nonlinear semigroup theory and related concepts in maximal-monotone operator theory. We restrict our attention to single-valued operators, although many results stated below remain true in the multivalued setting.

We also state several results pertinent to linear plate equations in a form convenient for applications in subsequent chapters.


Generalize Solution Weak Solution Evolutionary Equation Strong Solution Free Boundary Condition 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

  1. 9.
    G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic plates with free boundary conditions and without mechanical dissipation, SIAM J. Math. Anal., 29 (1998), 155–182.MATHCrossRefMathSciNetGoogle Scholar
  2. 18.
    V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.MATHGoogle Scholar
  3. 34.
    H. Brézis, Opérateurs Maximaux Monotones, North-Holland, Amsterdam, 1973.MATHGoogle Scholar
  4. 62.
    I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Commun. Partial Diff. Eqs., 27 (2002), 1901–1951.MATHCrossRefMathSciNetGoogle Scholar
  5. 75.
    I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Memoirs of AMS, vol.195, no. 912, AMS, Providence, RI, 2008.Google Scholar
  6. 76.
    I. Chueshov and I. Lasiecka, Attractors and long time behavior of von Karman thermoelastic plates, Appl. Math. Optim., 58 (2008), 195–241.MATHCrossRefMathSciNetGoogle Scholar
  7. 121.
    H. Gajewski, K. Greger, and K. Zacharias, Nichtlineare Operator Gleichungen und Operator Differential Gleichungen, Akademic-Verlar, Berlin, 1974.Google Scholar
  8. 123.
    J.M. Ghidaglia and R. Temam, Regularity of the solutions of second order evolution equations and their attractors, Ann. della Scuola Norm. Sup. Pisa, 14 (1987), 485–511.MATHMathSciNetGoogle Scholar
  9. 164.
    H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity - full von Karman systems. In: Evolution Equations, Semigroups and Functional Analysis, W.R. Neumann, A. Lorenzi, and A. Lorenzi (Eds.), (Progress in Nonlinear Differential Equations and their Applications, vol.50), Birkhäuser, Basel, 2002, 197–216.Google Scholar
  10. 177.
    J. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback, J. Diff. Eqs., 91 (1991), 355–388.MATHCrossRefMathSciNetGoogle Scholar
  11. 178.
    J. Lagnese and J.L. Lions Modeling, Analysis and Control of Thin Plates, Masson, Paris, 1988.Google Scholar
  12. 179.
    I. Lasiecka. Existence and uniqueness of solutions to second order nonlinear and nonmonotone boundary conditions, Nonlin. Anal. TMA, 24 (1994), 797–823.CrossRefMathSciNetGoogle Scholar
  13. 193.
    I. Lasiecka, J.L. Lions and R. Triggiani, Non homogenous boundary value problems for second order hyperbolic operators, J.Math. Pures Appl., 65 (1986), 149–192.MATHMathSciNetGoogle Scholar
  14. 201.
    I. Lasiecka and R. Triggiani, Regularity theory of hyperbolic equations with non-homo-genous Neumann boundary conditions II: General boundary data, J. Diff. Eqs., 94 (1991), 112–164.MATHCrossRefMathSciNetGoogle Scholar
  15. 210.
    I. Lasiecka and R. Triggiani, A sharp trace result on a thermo-elastic plate equation with coupled hinged/Neumann boundary conditions, Discr. Cont. Dyn. Syst., 5 (1999), 585–598.MATHCrossRefMathSciNetGoogle Scholar
  16. 211.
    I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equations with controls in the Dirichlet and Neumann boundary conditions: A non-conservative case, SIAM J. Control, 27 (1989), 330–373.MATHCrossRefMathSciNetGoogle Scholar
  17. 212.
    I. Lasiecka and R. Triggiani. Sharp regularity ntheory for elastic and thermoelastic Kirchoff equations with free boundary conditions, Rocky Mount. J., 30 (2000), 981–1024.MATHCrossRefMathSciNetGoogle Scholar
  18. 214.
    I. Lasiecka and R. Triggiani, Optimal regularity of elastic and thermoelastic Kirchoff plates with clamped boundary control. In: Optimal Control of Complex Structures, K. Hoffmann et al., (Eds.) (ISNM, vol. 139), Birkhäuser, Basel, 2002, 171–182.Google Scholar
  19. 215.
    I. Lasiecka and R. Triggiani, Factor spaces and implications on Kirchhoff equations with clamped boundary conditions, Abstr. Appl. Anal., 6 (2001), 441–488.MATHCrossRefMathSciNetGoogle Scholar
  20. 216.
    I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations, Cambridge University Press, Cambridge, 2000.Google Scholar
  21. 221.
    J. L. Lions, Controlabilité Exacte, Perturbations et Stabilization des Systèms Distribués, Masson, Paris, 1988.Google Scholar
  22. 222.
    J. L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer, New York, 1972.Google Scholar
  23. 241.
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1986.Google Scholar
  24. 260.
    R. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, AMS Providence, 1997Google Scholar
  25. 275.
    H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators, North-Holland, Amsterdam, 1978.Google Scholar

Copyright information

© Springer Science+Business Media 2010

Authors and Affiliations

  1. 1.Department of Mechanics & MathematicsKharkov National UniversityKharkovUkraine
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

Personalised recommendations