Moore–Gibson–Thompson equation with memory, part II: General decay of energy

Abstract

Abstract. We study a temporally third order (Moore-Gibson-Thompson) equation with a memory term. Previously it is known that, in noncritical regime, the global solutions exist and the energy functionals decay to zero. More precisely, it is known that the energy has exponential decay if the memory kernel decays exponentially. The current work is a generalization of the previous one (Part I) in that it allows the memory kernel to be more general and shows that the energy decays the same way as the memory kernel does, exponentially or not.

Wellposedness and Decay Rates for the Cauchy Problem of the Moore–Gibson–Thompson Equation Arising in High Intensity Ultrasound

“…We also mention the recent paper [15] where the authors consider (1.16) with a memory damping term and show an exponential decay of the energy provided that the kernel is exponentially decaying. This result is generalized in [16], where it is shown that the memory kernel decay determines the solution decay.…”

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

“…This restriction can actually be removed; but to get energy bound, we need to impose condition on g(0). see [22]. Item 3 implies the exponential decay of the kernel.…”

“…Roughly speaking, the memory term creates damping by weakening the total energy in the first place: it borrows energy to initialize the deformation and pays back in the long run. We exploit this point in another paper [22].…”

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.…”

“…We obtain a system of differential equations of four orders with respect to the variable t with constant coefficients and the initial conditions (11); consequently, we get a Cauchy problem of linear differential equations with smooth coefficients that is uniquely solvable. Thus for every N there exists a function u N (x) satisfying (6).…”