Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation

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“…Due to the regularity of A ε , applying the same techniques as in [19,24], it is not hard to show the upper semicontinuity at ε = 0 of the family {A ε }, namely, lim ε→0 δ H 0 ε A ε , A 0 = 0. We now prove the existence of a Lyapunov functional for the semigroup S ε (t).…”

Inertial manifolds of damped semilinear wave equations

“…with the additional condition that u = u t + u 2 satisfies (2 12) It turns out that, for fx € (0, X rt + i), and E small enough, the set of solutions of (2 13), (2 14) which satisfy (2 12) is parametrized by w z (0) e E t Thus, by addmg an initial condition of the form Wj(0) = x 3 we obtam a different mapping u 0^ u for every x E E x By applymg a suitable version of the parametrized contraction theorem, we obtam that, under conditions (2 9) and (2 10) together with max (0,X" + 2£)<cjx^\ n + 1 -2£ 5 ( 2 15) each of these rnappings has a unique fixed point The totality of these frxed points will give us the set M E we are lookmg for, which m fact will be a mamfold parametrized by x E E x Fmally, the behaviour of M z as e -* 0 is also taken care of by our spécifie version of the parametrized contraction theorem on the basis of a previous detailed study of the behaviour as e -+ 0 of the solutions of the non-homogeneous lmear équations (2 13), (2 14) with the additional conditions mentioned above…”

Upper Semicontinuity of Pullback Attractors for Non-Autonomous Generalized 2d Parabolic Equations

“…On the other hand, several authors have studied the upper and lower semicontinuity of attractors of perturbed dynamical systems for the autonomous case [1,4,6] and for the nonautonomous case [2,3,10,13]. This continuous property implies some stability of attractors for the corresponding equations with some perturbations.…”