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.

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We analyze the differential systemdescribing a Timoshenko beam coupled with a temperature evolution of Gurtin-Pipkin type. A necessary and sufficient condition for exponential stability is established in terms of the structural parameters of the equations. In particular, we generalize previously known results on the Fourier-Timoshenko and the Cattaneo-Timoshenko beam models.2010 Mathematics Subject Classification. 35B40, 45K05, 47D03, 74D05, 74F05.

An abstract version of the fourth-order equationsubject to the homogeneous Dirichlet boundary condition is analyzed. Such a model encompasses the Moore-Gibson-Thompson equation with memory in presence of an exponential kernel. The stability properties of the related solution semigroup are investigated. In particular, a necessary and sufficient condition for exponential stability is established, in terms of the values of certain stability numbers depending on the strictly positive parameters α, β, γ, δ, ϱ.

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We consider the Moore-Gibson-Thompson equation with memory of type IIwhere A is a strictly positive selfadjoint linear operator (bounded or unbounded) and α, β, γ > 0 satisfy the relation γ ≤ αβ. First, we prove a well-posedness result without requiring any restriction on the total mass ̺ of g. Then we show that it is always possible to find memory kernels g, complying with the usual mass restriction ̺ < β, such that the equation admits solutions with energy growing exponentially fast. In particular, this provides the answer to a question raised in [2].2000 Mathematics Subject Classification. 35B35, 35G05, 45D05.

We consider the nonclassical diffusion equation with hereditary memory ut − ∆ut − ∞ 0 κ(s)∆u(t − s) ds + f (u) = g on a bounded three-dimensional domain. The main feature of the model is that the equation does not contain a term of the form −∆u, contributing as an instantaneous damping. Setting the problem in the past history framework, we prove that the related solution semigroup possesses a global attractor of optimal regularity. u 0 f (y) dy.

We consider the strongly damped nonlinear wave equation u tt − u t − u + f (u t ) + g(u) = h with Dirichlet boundary conditions, which serves as a model in the description of thermal evolution within the theory of type III heat conduction. In particular, the nonlinearity f acting on u t is allowed to be nonmonotone and to exhibit a critical growth of polynomial order 5. The main focus is the long-term analysis of the related solution semigroup, which is shown to possess the global attractor in the natural weak energy space.