Moore–Gibson–Thompson equation with memory, part I: exponential decay of energy

Abstract

Abstract. We are interested in the Moore-Gibson-Thompson(MGT) equation with memoryWe first classify the memory into three types. Then we study how a memory term creates damping mechanism and how the memory causes energy decay.

A note on the Moore–Gibson–Thompson equation with memory of type II

“…In the case of memory of type I, Theorem 3.7 therein proves the exponential decay of solutions in the subcritical case. This result has been already shown for finite memory of type I in [8] (see also [9] for more general relaxation kernels leading to uniform but not exponential decays). In the case of memory of type II, the same [1, Theorem 3.7] establishes the exponential decay of the energy in the subcritical case, but under strong "smallness" type restrictions imposed on the mass of kernel ̺.…”

“…In the case of memory of type II, the same [1, Theorem 3.7] establishes the exponential decay of the energy in the subcritical case, but under strong "smallness" type restrictions imposed on the mass of kernel ̺. Here, again, this is an extension to infinite memory of the results obtained in [8]. In short, this "smallness" condition requires a rather fast decay of the kernel g with respect to the strictly positive value αβ − γ.…”

“…For the memory of type I, the picture is quite understood. As shown in [8], in the subcritical case and under proper decay assumptions on the kernel g all the solutions converge exponentially to zero. Instead, in the critical case the decay is only strong, with a counterexample to exponential stability if the operator A is unbounded [2].…”

“…Coming instead to the MGT equation with memory of type II in the critical case, it is still unknown whether exponential stability could be achieved with suitably calibrated relaxation kernel (fast decay and small mass). Positive results are available in the literature (see [1,8]), but only in the subcritical regime.…”

Galerkin method for nonlocal mixed boundary value problem for the Moore‐Gibson‐Thompson equation with integral condition

“…See in this regard the works by Kaltenbacher, Lasiecka, and Marchand, Marchand, McDevitt, and Triggiani, and Kaltenbacher, Lasiecka, and Pospieszalska . Later on, Irena Lasiecka and Xiaojun Wang showed a general decay result for the MGT equation with memory.…”