Upper Semicontinuity of Attractors for Approximations of Semigroups and Partial Differential Equations

Abstract

Abstract.Suppose a given evolutionary equation has a compact attractor and the evolutionary equation is approximated by a finite-dimensional system. Conditions are given to ensure the approximate system has a compact attractor which converges to the original one as the approximation is refined. Applications are given to parabolic and hyperbolic partial differential equations. Introduction.Suppose A" is a Banach space and T(t), t > 0, is a Crsemigroup on X with r > 0; that is, T(t), t > 0, is a semigroup with T(t) continuous in t, x together with the derivatives in x up through the order r.Following standard terminology (see, for instance, Hale [12]), a set B C X is said to attract a set C C X under the semigroup T(t) if, for any e > 0, there is a t0 = t0(B,C,e) such that T(t)C C M(B,e) for t > t0, where M(B,e) denotes the

Lower Semicontinuity of a Non-Hyperbolic Attractor

“…As is known (see [1,6,10]), continuity of the map S. (t, •): O x Hi -* Hi and the uniform compactness of the attractors imply (0.6).…”

“…Relations (0.6) and (0.7) are usually referred to as the upper and lower semicontinuity of the attractor 90^ in 0. Problems of this sort for different classes of equations and different types of perturbations have been studied in a number of papers (see [1,2,5,6,4,10,11,15]). Equalities like (0.6) hold for a very general class of semigroups including the ones just introduced.…”

Computation of Morse decompositions for semilinear elliptic PDEs

“…Using the (computable) one-to-one correspondence between the problems then determines the set E h (λ) of solutions u h ∈ S h that approximates the set of solutions E(λ) ⊂ H 1 0 (Ω). The question of whether the finite element problem Au h + g[u h , λ] = 0 ∈ S * h has a global attractor A h (λ) that is an accurate representation of the global attractor A(λ) of the original problem has been addressed by Hale, Lin, and Raugel [9] and Hale and Raugel [10]. The property of upper semi-continuity is a common feature of parameter-dependent attractors, and this is shown in [8] for problems of the type considered here.…”

“…It is the special structure of gradient systems that leads to the property of lower semi-continuity. In [9] it is shown that the attractor A h (λ) for the approximate problem converges to the attractor A(λ) of the continuous problem, assuming E(λ) consists of a finite number of hyperbolic rest points and the discretizations have typical approximation properties. Together these results imply that E h (λ) → E(λ), as h → 0 + , under the assumptions just stated.…”

Approximation of the bifurcation function for elliptic boundary value problems

“…In this case, it is possible for the intersection property to be lost, although it is shown to be retained in some important special cases. Investigating the same question from a rather different perspective, Hale, Lin, and Raugel [11] and Hale and Raugel [12] have shown that, in the case of gradient systems, the attractors for the approximate problems converge to the attractor for the continuous problem, assuming that S(λ) consists of a finite number of hyperbolic rest points and the discretizations have typical approximation properties. Their results apply to the types of problems considered here and imply that S h (λ) → S(λ), as h → 0 + , under the assumptions just stated.…”