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

Hale, Jack K.; Raugel, Geneviève

Ann. Mat. Pura Appl., IV. Ser. 154, 281-326 (1989).

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.

MSC:

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

Keywords:

family of semigroups on a Banach space; local attractors; gradient system with hyperbolic equilibria; lower-semicontinuous at zero; ordinary and functional differential equations; parabolic partial differential equations; space and time discretizations; Hausdorff distance

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