In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite different. Our work subsumes all previously known instances of weak convergence of sums of free, identically distributed random variables. In particular, we determine the domains of attraction of stable distributions in the free theory. These freely stable distributions are studied in detail in the appendix, where their unimodality and duality properties are demonstrated.

This paper is concerned with the semilinear strongly damped wave equationThe existence of compact global attractors of optimal regularity is proved for nonlinearities ϕ of critical and supercritical growth.

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We consider the following abstract version of the Moore-Gibson-Thompson equation with memory ∂ ttt u(t) + α∂ tt u(t) + βA∂ t u(t) + γAu(t) − ∫ t 0 g(s)Au(t − s)ds = 0 depending on the parameters α, β, γ > 0, where A is strictly positive selfadjoint linear operator and g is a convex (nonnegative) memory kernel. In the subcritical case αβ > γ, the related energy has been shown to decay exponentially in [19]. Here we discuss the critical case αβ = γ, and we prove that exponential stability occurs if and only if A is a bounded operator. Nonetheless, the energy decays to zero when A is unbounded as well.