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An example of Floquet operator with purely point spectrum and energy instability is presented. In the unperturbed energy eigenbasis its eigenfunctions are exponentially localized.

In this work we prove continuity of solutions with respect to initial conditions and couple parameters and we prove joint upper semicontinuity of a family of global attractors for the problem    ∂us ∂t (t) − div(Ds|∇us| ps(x)−2 ∇us) + |us| ps(x)−2 us = B(us(t)), t > 0, us(0) = u 0s , under homogeneous Neumann boundary conditions, u 0s ∈ H := L 2 (Ω), Ω ⊂ R n (n ≥ 1) is a smooth bounded domain, B : H → H is a globally Lipschitz map with Lipschitz constant L ≥ 0, Ds ∈ [1, ∞), ps(•) ∈ C(Ω), p − s := ess inf ps ≥ p, p + s := ess sup ps ≤ a, for all s ∈ N, when ps(•) → p in L ∞ (Ω) and Ds → ∞ as s → ∞, with a, p > 2 positive constants.

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