Large patterns of elliptic systems in infinite cylinders

Journal des Mathematiques Pures et Appliquees

Volumn 77, Issue 9, 1998, Pages 879-907

  Fiedler, Bernold ^a   Scheel, Arnd ^a   Vishik, Mark I ^b  

^a FREIE UNIVERSITÄT BERLIN   (Germany)

^b INSTITUTE FOR INFORMATION TRANSMISSION PROBLEMS   (Russian Federation)

Author keywords

Attractors; Conley index; Elliptic systems; Traveling waves

Indexed keywords

EID: 0032210477     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0021-7824(01)80002-7     Document Type: Article

References (42)

1. [1] S. ANGENENT and R. van der VORST, A superquadratic indefinite elliptic system and its Morse-Conley-Floer homology, Preprint 1997.
2. [2] J. APPELL and P. P. ZABREJKO, Nonlinear Superposition Operators, Cambridge University Press, Cambridge, 1990.
3. [3] A. V. BABIN, Inertial manifolds for traveling-wave solutions of reaction-diffusion systems, Comm. Pure Appl. Math., 48, 1995, pp. 167-198.
4. [4] A. V. BABIN, Attractor of a semigroup of multi-valued mappings corresponding to an elliptic equation, Izv. Ross. Acad. Nauk Ser. Mat., 58, 1994, pp. 3-18.
5. [5] A. V. BABIN and M. I. VISHIK, Attractors of Evolution Equations, North-Holland, 1992.
6. [6] H. BERESTYCKI and L. NIRENBERG, Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, P. H. Rabinowitz, E. Zehnder (ed.), Analysis, et cetera, Res. Papers in Honor of J. Moser's 60th Birthd. Academic Press, Boston, 1990, pp. 115-164.
7. [7] A. CALSINA, X. MORA and J. SOLÀ-MORALES, The dynamical approach to elliptic problems in cylindrical domains and a study of their parabolic singular limit, J. Diff. Eqns., 102, 1993, pp. 244-304.
8. [8] V. V. CHEPYZHOV and M. I. VISHIK, Trajectory attractors for evolution equations, C. R. Acad. Sci. Paris, Série 1, Math., 231, 1995, pp. 1309-1314.
9. [9] V. V. CHEPYZHOV and M. I. VISHIK, Evolution equations and their trajectory attractors, J. Math. Pures. Appl., 76, No. 10, 1997, pp. 913-964.
10. [10] C. C. CONLEY, Isolated Invariant Sets and the Morse Index, Conf. Board Math. Science, 38, Amer. Math. Soc, Providence, 1978.
11. [11] C. C. CONLEY and J. SMOLLER, Bifurcation and stability of stationary solutions of the Fitz-Hugh-Nagumo equations, J. Diff. Eqns., 63, 1986, pp. 389-405.
12. [12] H. O. CORDES, Über die eindeutige Bestimmtheit der Lösungen elliptischer Differentialgleichungen durch Anfangsvorgaben, Nachr. Akad. Wiss. Göttingen, Math.-Phys.-Kl., 11, 1956, pp. 239-258.
13. [13] B. FIEDLER and C. ROCHA, Heteroclinic orbits of semilinear parabolic equations, J. Diff. Eqns., 125, 1996, pp. 239-281.
14. [14] B. FIEDLER and C. ROCHA, Orbit equivalence of global attractors of semilinear parabolic equations, Trans. Amer. Math. Soc., 1998, to appear.
15. [15] G. FISCHER, Zentrumsmannigfaltigkeiten bei elliptischen Differentialgleichungen, Math. Nachr., 115, 1984, pp. 137-157.
16. [16] A. FLOER, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys., 120, 1989, pp. 575-611.
17. [17] R. FRANZOSA, The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc., 311, 1989, pp. 561-592.
18. [18] R. FRANZOSA and C. MISCHAIKOW, The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces, J. Diff. Eqns., 71, 1988, pp. 270-287.
19. [19] R. GARDNER, Existence of multidimensional traveling wave solutions of an initial boundary value problem, J. Diff. Eqns., 61, 1986, pp. 335-379.
20. [20] I. C. GOHBERG and M. G. KREIN, An Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., 18, Amer. Math. Soc., Providence, 1969.
21. [21] J. HALE, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs, 25, Amer. Math. Soc., Providence, 1988.
22. [22] J. HADAMARD, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale University Press, New Haven, 1923.
23. [23] S. HEINZE, Traveling Waves for Semilinear Parabolic Partial Differential Equations in Cylindrical Domains, Doctoral Thesis, Heidelberg, 1989.
24. [24] H. HOFER and E. ZEHNDER, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser-Verlag, Basel, 1994.
25. [25] K. KIRCHGÄSSNER, Wave solutions of reversible systems and applications, J. Diff. Eqns., 45, 1982, pp. 113-127.
26. [26] O. LADYZHENSKAYA, Attractors for Semigroups and Evolution Equations, Cambridge Univ. Press, 1991.
27. [27] A. S. MARKUS and V. I. MATSAEV, Comparison theorems for the spectra of linear operators and spectral asymptotics, Tr. Mask. Math. O., 45, 1982, pp. 133-181.
28. [28] A. S. MARKUS and V. I. MATSAEV, A theorem on comparison of spectra, and the spectral asymptotics for a Keldysh pencil, Math. Sbornik, 123, 1984, pp. 391-406.
29. [29] A. MIELKE, Essential manifolds for an elliptic problem in an infinite strip, J. Diff. Eqns., 110, 1994, pp. 322-355.
30. [30] K. MISCHAIKOW, Conley Index Theory, R. Johnson (ed.), Coll. Dynamical Systems, Montecatini Terme 1995, Lecture Notes in Math., 1609, 1995, pp. 119-207.
31. [31] K. P. RYBAKOWSKI, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin, 1987.
32. [32] D. A. SALAMON, Morse theory, the Conley index, and Floer homology, Bull London Math. Soc., 22, 1990, pp. 113-140.
33. [33] A. SCHEEL, Existence of fast traveling waves for some parabolic equations - a dynamical systems approach, J. Dyn. Diff. Eqns., 8, 1996, pp. 469-548.
34. [34] B.-W. SCHULZE, M. I. VISHIK, I. VITT and S. V. ZELIK, Trajectory attractors for the nonlinear elliptic system in the cylindrical domain with piece-wise smooth boundary, Preprint 1997.
35. [35] L. SCHWARTZ, Cours d'Analyse Mathématique, Herman, Paris, 1967.
36. [36] J. SMOLLER, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.
37. [37] R. TEMAM, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci., 68, Springer-Verlag, 1988.
38. [38] D. TERMAN, Directed graphs and traveling waves, Trans. Amer. Math. Soc., 289, 1985, pp. 809-847.
39. [39] M. I. VISHIK and S. V. ZELIK, The trajectory attractor for a nonlinear elliptic system in a cylindrical domain, Mat. Sb., 187, 1996, pp. 21-56.
40. [40] M. I. VISHIK and S. V. ZELIK, Attractors of nonlinear elliptic systems in cylindrical domains and their approximations, Preprint 1997.
41. [41] A. I. VOLPERT, V. A. VOLPERT and V. A. VOLPERT, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr., 140, Amer. Math. Soc., Providence, 1994.
42. [42] S. V. ZELIK, Boundedness of solutions for nonlinear elliptic systems in cylindrical domains, Mat. Zam., 61, 1997, pp. 447-450.