On monotone trajectories

Mierczyński, Janusz

doi:10.1090/S0002-9939-1991-1056682-1

On monotone trajectories
HTML articles powered by AMS MathViewer

by Janusz Mierczyński PDF

Proc. Amer. Math. Soc. 113 (1991), 537-544 Request permission

Abstract:

In this paper ${C^1}$ strongly monotone dynamical systems are investigated. It is proved that the set of points with precompact orbits which converge to a not unstable equilibrium but whose trajectories are not eventually strongly monotone is nowhere dense. This improves on and extends a recent result by P. Poláčik [13].

References

Herbert Amann, Dynamic theory of quasilinear parabolic equations. I. Abstract evolution equations, Nonlinear Anal. 12 (1988), no. 9, 895–919. MR 960634, DOI 10.1016/0362-546X(88)90073-9
Herbert Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations 3 (1990), no. 1, 13–75. MR 1014726
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244, DOI 10.1007/BFb0089647
Daniel B. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations 59 (1985), no. 2, 165–205. MR 804887, DOI 10.1016/0022-0396(85)90153-6
Morris W. Hirsch, The dynamical systems approach to differential equations, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 1–64. MR 741723, DOI 10.1090/S0273-0979-1984-15236-4
Morris W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math. 383 (1988), 1–53. MR 921986, DOI 10.1515/crll.1988.383.1
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173, DOI 10.1007/BFb0092042
P. D. Lax, A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Comm. Pure Appl. Math. 9 (1956), 747–766. MR 86991, DOI 10.1002/cpa.3160090407
H. Matano, Strongly order-preserving local semidynamical systems—theory and applications, Semigroups, theory and applications, Vol. I (Trieste, 1984) Pitman Res. Notes Math. Ser., vol. 141, Longman Sci. Tech., Harlow, 1986, pp. 178–185. MR 876941
H. Matano, Strong comparison principle in nonlinear parabolic equations, Nonlinear parabolic equations: qualitative properties of solutions (Rome, 1985) Pitman Res. Notes Math. Ser., vol. 149, Longman Sci. Tech., Harlow, 1987, pp. 148–155. MR 901104
Xavier Mora, Semilinear parabolic problems define semiflows on $C^{k}$ spaces, Trans. Amer. Math. Soc. 278 (1983), no. 1, 21–55. MR 697059, DOI 10.1090/S0002-9947-1983-0697059-8
Peter Poláčik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations, J. Differential Equations 79 (1989), no. 1, 89–110. MR 997611, DOI 10.1016/0022-0396(89)90115-0
Peter Poláčik, Domains of attraction of equilibria and monotonicity properties of convergent trajectories in parabolic systems admitting strong comparison principle, J. Reine Angew. Math. 400 (1989), 32–56. MR 1013724, DOI 10.1515/crll.1989.400.32
Hal L. Smith, Systems of ordinary differential equations which generate an order preserving flow. A survey of results, SIAM Rev. 30 (1988), no. 1, 87–113. MR 931279, DOI 10.1137/1030003