Invariant manifolds for one-dimensional parabolic partial differential equations of second order

Mierczyński, Janusz

Colloquium Mathematicae, Tome 78 (1998), p. 285-314 / Harvested from The Polish Digital Mathematics Library

Détail

Publié le : 1998-01-01

Zbl 0927.37056

EUDML-ID : urn:eudml:doc:210543

Bibliographie

[000] [1] H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z. 150 (1976), 281-295. | Zbl 0331.35026

[001] [2] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620-709. | Zbl 0345.47044

[002] [3] H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations (1990), 13-75. | Zbl 0729.35062

[003] [4] S. Angenent, The Morse-Smale property for a semi-linear parabolic equation, J. Differential Equations 62 (1986), 427-442. | Zbl 0581.58026

[004] [5] S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79-96. | Zbl 0644.35050

[005] [6] S. Angenent, Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 91-107. | Zbl 0723.34047

[006] [7] N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), 147-190. | Zbl 0342.46034

[007] [8] P. Brunovský and B. Fiedler, Numbers of zeros on invariant manifolds in reaction-diffusion equations, Nonlinear Anal. 10 (1986), 179-193.

[008] [9] P. Brunovský and B. Fiedler, Connecting orbits in scalar reaction diffusion equations. II: The complete solution, J. Differential Equations 81 (1989), 106-135. | Zbl 0699.35144

[009] [10] P. Brunovský, P. Poláčik and B. Sandstede, Convergence in general periodic parabolic equations in one space dimension, Nonlinear Anal. 18 (1992), 209-215. | Zbl 0796.35009

[010] [11] M. Chen, X.-Y. Chen and J. K. Hale, Structural stability for time-periodic one-dimensional parabolic equations, J. Differential Equations 96 (1992), 355-418. | Zbl 0779.35061

[011] [12] X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equation, ibid. 78 (1989), 160-190. | Zbl 0692.35013

[012] [13] X.-Y. Chen and P. Poláčik, Gradient-like structure and Morse decompositions for time-periodic one-dimensional parabolic equations, J. Dynam. Differential Equations 7 (1995), 73-107.

[013] [14] S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren Math. Wiss. 251, Springer, New York, 1982. | Zbl 0487.47039

[014] [15] S.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite dimensional spaces, J. Differential Equations 94 (1991), 266-291. | Zbl 0749.58043

[015] [16] S.-N. Chow, K. Lu and J. Mallet-Paret, Floquet bundles for scalar parabolic equations, Arch. Rational Mech. Anal. 129 (1995), 245-304. | Zbl 0822.35057

[016] [17] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. | Zbl 0559.47040

[017] [18] C. Foiaş and J.-C. Saut, On the smoothness of the nonlinear spectral manifolds of Navier-Stokes equations, Indiana Univ. Math. J. 33 (1984), 911-926. | Zbl 0572.35081

[018] [19] G. Fusco and W. M. Oliva, Jacobi matrices and transversality, Proc. Roy. Soc. Edinburgh Sect. A 109 (1988), 231-243. | Zbl 0692.58019

[019] [20] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monographs 25, Amer. Math. Soc., Providence, R.I., 1988. | Zbl 0642.58013

[020] [21] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, Berlin, 1981. | Zbl 0456.35001

[021] [22] D. Henry, Some infinite dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations 53 (1985), 401-458.

[022] [23] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser. 247, Longman Sci. Tech., Harlow, 1991. | Zbl 0731.35050

[023] [24] M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math. 383 (1988), 1-53. | Zbl 0624.58017

[024] [25] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math. 583, Springer, Berlin, 1977.

[025] [26] H. Koch, Finite dimensional aspects of semilinear parabolic equations, J. Dynam. Differential Equations 8 (1996), 177-202. | Zbl 0860.35056

[026] [27] H.-H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math. 463, Springer, Berlin, 1975.

[027] [28] S. Lang, Differential Manifolds, Addison-Wesley, Reading, Mass., 1972.

[028] [29] H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ. 18 (1978), 221-227. | Zbl 0387.35008

[029] [30] H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), 401-441. | Zbl 0496.35011

[030] [31] J. Mierczyński, On monotone trajectories, Proc. Amer. Math. Soc. 113 (1991), 537-544. | Zbl 0737.58010

[031] [32] J. Mierczyński, P-arcs in strongly monotone discrete-time dynamical systems, Differential Integral Equations 7 (1994), 1473-1494. | Zbl 0864.58054

[032] [33] M. Miklavčič, Stability for semilinear parabolic equations with noninvertible linear operator, Pacific J. Math. 118 (1985), 199-214. | Zbl 0559.35037

[033] [34] K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleichungen, J. Reine Angew. Math. 211 (1962), 78-94.

[034] [35] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, New York, 1983.

[035] [36] R. R. Phelps, Gaussian null sets and differentiability of Lipschitz maps on Banach spaces, Pacific J. Math. 77 (1978), 523-531. | Zbl 0396.46041

[036] [37] P. 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. | Zbl 0667.35036

[037] [38] W. Shen and Y. Yi, On minimal sets of scalar parabolic equations with skew-product structures, Trans. Amer. Math. Soc. 347 (1995), 4413-4431. | Zbl 0849.35005

[038] [39] A. V. Skorokhod [A. V. Skorohod], Integration in Hilbert Space, translated from the Russian by K. Wickwire, Ergeb. Math. Grenzgeb. 79, Springer, New York, 1974.

[039] [40] J. Smillie, Competitive and cooperative tridiagonal systems of differential equations, SIAM J. Math. Anal. 15 (1984), 530-534. | Zbl 0546.34007

[040] [41] P. Takáč, Convergence to equilibrium on invariant $d$-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl. 148 (1990), 223-244. | Zbl 0744.47037

[041] [42] P. Takáč, Domains of attraction of generic $οmega$-limit sets for strongly monotone discrete-time semigroups, J. Reine Angew. Math. 423 (1992), 101-173. | Zbl 0729.54022

[042] [43] T. I. Zelenyak, Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, Differentsial$^{\prime }$nye Uravneniya 4 (1968), 34-45; English transl.: Differential Equations 4 (1968), 17-22.