A Comparison between Two Theories for Multi-Valued Semiflows and Their Asymptotic Behaviour

Set-Valued Analysis

Volumn 11, Issue 3, 2003, Pages 297-322

  Caraballo, T ^a   Marin Rubio P ^a   Robinson, J C ^b  

^a UNIVERSITY OF SEVILLE   (Spain)

^b UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

Differential inclusions; Generalized and multi valued semiflows; Partial differential equations without uniqueness

Indexed keywords

EID: 1642581471     PISSN: 09276947     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1024422619616     Document Type: Article

References (31)

1. Arnold, L.: Random Dynamical Systems, Monographs Math., Springer-Verlag, Berlin, 1998.
2. Aubin, J. P. and Cellina, A.: Differential Inclusions. Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984.
3. Aubin, J. P. and Frankowska, H.: Set-Valued Analysis, Birkhäuser Boston Inc., Boston, 1990.
4. Babin, A. V. and Vishik, M. I.: Attractors of Evolution Equations, Stud. Math. Appl. 25, North-Holland Publishing Co., Amsterdam, 1992.
5. Ball, J. M.: Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci. 7(5) (1997), 475-502.
6. Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Bucuresti Romania, Noordhoff International Publishing, The Netherlands, 1976.
7. Bridgland, T. F.: Contributions to the theory of generalized differential equations I, Math. Systems Theory 3 (1969), 17-50.
8. Bushaw, D.: Dynamical polysystems and optimization, Contrib. Differential Equations 2 (1963), 351-365.
9. Chepyzhov, V. V. and Vishik, M. I.: A Hausdorff dimension estimate for kernel sections of nonautonomous evolution equations, Indiana Univ. Math. J. 42(3) (1993), 1057-1076.
10. Crauel, H., Debussche, A. and Flandoli, F.: Random attractors, J. Dynamics Differential Equations 9(2) (1997), 307-341.
11. Crauel, H. and Flandoli, F.: Attractors for random dynamical systems, Probab. Theory Related Fields 100(3) (1994), 365-393.
12. Elmounir, A. O. and Simondon, F.: Attracteurs compacts pour des problèmes d'évolution sans unicité, Ann. Fac. Sci. Toulouse Math. (6) 9(4) (2000), 631-654.
13. Hale, J. K.: Asymptotic Behavior of Dissipative Systems, Math. Surveys Monographs 25, American Mathematical Society, Providence, RI, 1988.
14. Hartman, P.: Ordinary Differential Equations, John Wiley & Sons Inc., New York, 1964.
15. Henry, D.: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, Berlin, 1981.
16. Kapustian, A. V. and Valero, J.: Attractors of multi-valued semiflows generated by differential inclusions and their approximations, Abstr Appl. Anal. 5(1) (2000), 33-46.
17. Kapustyan, O. V.: An attractor of a semiflow generated by a system of phase-field equations without uniqueness of the solution, Ukraïn. Mat. Zh. 51(7) (1999), 1006-1009.
18. Kloeden, P. E.: General control systems without backwards extension, In: P. L. E. Roxin and R. Sternberg (eds), Differential Games and Control Theory, Marcel-Dekker, 1974, pp. 49-58.
19. Kloeden, P. E.: General control systems, In: W. A. Coppel (ed.), Mathematical Control Theory, Lecture Notes in Math. 680, Springer-Verlag, 1978, pp. 119-138.
20. Kloeden, P. E. and Schmalfuß, B.: Nonautonomous systems, cocycle attractors and variable time-step discretization, Numer Algorithms 14(1-3) (1997), 141-152. Dynamical numerical analysis (Atlanta, GA, 1995).
21. Kloeden, P. E. and Stonier, D. J.: Cocycle attractors in nonautonomously perturbed differential equations, Dynam. Contin. Discrete Impuls. Systems 4(2) (1998), 211-226.
22. Ladyzhenskaya, O.: Attractorsfor Semigroups and Evolution Equations, Lincei Lectures 25, Cambridge University Press, Cambridge, 1991.
23. Lamba, H.: Dynamical systems and adaptive timestepping in ODE solvers, BIT 40(2) (2000), 314-335.
24. Lamba, H. and Stuart, A. M.: Convergence results for the MATLAB ODE23 routine, BIT 38(4) (1998), 751-780.
25. Melnik, V. S. and Valero, J.: On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal. 6(1) (1998), 83-111.
26. Roxin, E. O.: On generalized dynamical systems defined by contingent equations, J. Differential Equations 1 (1965), 188-205.
27. Roxin, E. O.: Stability in general control systems, J. Differential Equations 1 (1965), 115-150.
28. Sell, G.: On the fundamental theory of ordinary differential equations, J. Differential Equations 1 (1965), 370-392.
29. Smirnov, G. V.: Introduction to the Theory of Differential Inclusions, Amer. Math. Soc., Providence, 2002.
30. Szegö, G. P. and Treccani, G.: Semigruppi di Trasformazioni Multivoche, Lecture Notes in Math. 101, Springer-Verlag, 1969.
31. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. 68, second edn, Springer-Verlag, New York, 1997.