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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.

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
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