Lower semicontinuity of attractors of gradient systems and applications. (English) Zbl 0712.47053

Summary: Let \(0\leq \epsilon \leq \epsilon_ 0\), let \(T_{\epsilon}(t)\), \(t\geq 0\), be a family of semigroups on a Banach space X with local attractors \(A_{\epsilon}\). Under the assumptions that \(T_ 0(t)\) is a gradient system with hyperbolic equilibria and \(T_{\epsilon}(t)\) converges to \(T_ 0(t)\) in an appropriate sense, it is shown that the attractors \(\{A_{\epsilon}\), \(0\leq \epsilon \leq \epsilon_ 0\}\) are lower- semicontinuous at zero. Applications are given to ordinary and functional differential equations, parabolic partial differential equations and their space and time discretizations. We also give an estimate of the Hausdorff distance between \(A_{\epsilon}\) and \(A_ 0\), in some examples.


47H20 Semigroups of nonlinear operators
47D06 One-parameter semigroups and linear evolution equations
34G10 Linear differential equations in abstract spaces
34G20 Nonlinear differential equations in abstract spaces
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI


[1] Babin, A. V.; Vishik, M. I., Regular attractors of semigroups and evolution equations, J. Math. Pures Appl., 62, 441-491 (1983) · Zbl 0565.47045
[2] A. V.Babin - M. I.Vishik,Attracteurs maximaux dans les équations aux dérivées partielles (Notes rédigées par A. Haraux), Collège de France, 1984, Pittman (1985).
[3] Babin, A. V.; Vishik, M. I., Unstable invariant sets of semigroups of nonlinear operators and their perturbations, Uspekhi Mat. Nauk, 41, 3-34 (1986) · Zbl 0624.47065
[4] Brunovsky, P.; Chow, S.-N., Generic properties of stationary solutions of reaction-diffusion equations, J. Diff. Equat., 53, 1-23 (1984) · Zbl 0544.34019
[5] Cooperman, G., α-condensing maps and dissipative systems, Ph. D. Thesis (1978), Providence, R.I.: Brown University, Providence, R.I.
[6] Gidaglia, J. M.; Témam, R., Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66, 273-319 (1987) · Zbl 0572.35071
[7] J. K.Hale,Functional Differential Equations, Appl. Math. Sciences, Vol.3, Springer-Verlag, 1977. · Zbl 0352.34001
[8] J.K.Hale,Asymptotic behavior and dynamics in infinite dimensions, inNonlinear Differential Equations, J. K.Hale and P.Martinez-Amores, Eds., Pittman,132 (1985).
[9] Hale, J. K., Asymptotic Behavior of Dissipative Systems, Mathematical Surveys,25 (1988), Providence, R. I.: Am. Math. Soc., Providence, R. I. · Zbl 0642.58013
[10] Hale, J. K.; Lasalle, J. P.; Slemrod, M., Theory of a general class of dissipative processes, J. Math. Anal. Appl., 39, 177-191 (1972) · Zbl 0238.34098
[11] Hale, J. K.; Lin, X. B.; Raugel, Gr., Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. of Computation, 50, 89-123 (1988) · Zbl 0666.35013
[12] J. K.Hale - L.Magalhães - W.Oliva,An Introduction to Infinite Dimensional Dynamical Systems, Applied Math. Sciences, Vol. 47, Springer-Verlag (1984). · Zbl 0533.58001
[13] Hale, J. K.; Raugel, G., Upper-semicontinuous of the attractor for a singularity perturbed hyperbolic equation, J. Diff. Equat., 73, 197-214 (1988) · Zbl 0666.35012
[14] J. K.Hale - G.Raugel,Lower semicontinuity of the attractor for a singularity perturbed hyperbolic equation, to appear in Dynamics and Differential Equations. · Zbl 0752.35034
[15] Hale, J. K.; Rybakowski, K., On a gradient-like integro-differential equation, Proc. Roy Soc. Edinburgh, 92A, 77-85 (1982) · Zbl 0512.45011
[16] Haraux, A.; Lions, J. L., Two remarks on dissipative hyperbolic problems, Séminaire du Collège de France (1985), Boston: Pittman, Boston · Zbl 0579.35057
[17] D.Henry,Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol.840, Springer-Verlag (1981). · Zbl 0456.35001
[18] Henry, D., Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Diff. Equat., 59, 165-205 (1985) · Zbl 0572.58012
[19] D.Henry,Generic properties of equilibrium solutions by perturbation of the boundary, Preprint of the « Centre de Recerca Matematica Institut d’Estudis Catalans »,37 (1986).
[20] M.Hirsch - C.Pugh - M.Shub,Invariant Manifolds, Lecture Notes in Math., Vol.583’ Springer-Verlag. · Zbl 0226.58009
[21] P. E.Kloeden - J.Lorenz,Stable attracting sets in dynamical systems and in their one-step discretizations, SIAM J. Num. Anal. · Zbl 0613.65083
[22] X. B.Lin - G.Raugel,Approximation of attractors of Morse-Smale systems given by parabolic equations (in preparation).
[23] J.Palis - W.de Melo,Geometric Theory of Dynamical Systems, Springer-Verlag (1982). · Zbl 0491.58001
[24] Rocha, C., Generic properties of equilibria of reaction-diffusion equations with variable diffusion, Proc. Roy. Soc. Edinburgh, 101A, 45-56 (1985) · Zbl 0601.35053
[25] Smoller, J.; Wasserman, A., Generic bifurcation of steady-state solutions, J. Diff. Equat., 52, 432-438 (1984) · Zbl 0488.58015
[26] Wells, J. C., Invariant manifolds of nonlinear operators, Pacific J. Math., 62, 285-293 (1976) · Zbl 0343.58010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.