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∂tttu(t)+α∂ttu(t)+βA∂tu(t)+γAu(t)−∫0tg(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 . 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.
Journal of Differential Equations
Dell'Oro, F., Lasiecka, I., & Pata, V. (2016). The Moore–Gibson–Thompson equation with memory in the critical case. Journal of Differential Equations, 261 (7), 4188-4222. https://doi.org/10.1016/j.jde.2016.06.025