The Moore-Gibson-Thompson equation with memory in the critical case

We consider the following abstract version of the Moore-Gibson-Thompson equation with memory partial derivative(ttt)u(t) + alpha partial derivative(tt)u(t) + beta A partial derivative(t)u(t) + gamma Au(t) - integral(t) g(s)Au(t - s)ds = 0 depending on the parameters alpha, beta, gamma > 0, where A is strictly positive selfadjoint linear operator and g is a convex (nonnegative) memory kernel. In the subcritical case alpha beta > gamma, the related energy has been shown to decay exponentially in [19]. Here we discuss the critical case alpha beta = gamma, 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. (C) 2016 Elsevier Inc. All rights reserved.