Intrinsic Decay Rate Estimates for Semilinear Abstract Second Order Equations with Memory

Abstract

Semilinear abstract second order equation with a memory is considered.

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

“…However, we are not able to get decay rate out of this continuous version of energy estimate, for it is not in an form applicable to Gronwall inequality. To get what we want, we need two parts: 1. the discretized version of energy estimate developed in [16], which essentially is a generalized version of Gronwall inequality; 2. the iteration technique developed in [17], which can optimize the estimate in finite steps. In the second subsection, we discretize the estimate, based on which we are able to get an "initial" decay estimate.…”

“…For that reason we have to discretize the energy inequality and apply a series of lemmas. The technique has been applied in [17].…”

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

“…As for wave equation, there are hundreds of papers studying either the damping effects caused by the memory terms or the interaction between different damping mechanisms. See [2,3,4,5,6,7,11,12,19,21,24,25,28] to name a few. While a memory term does generate damping in a wave system, the nature of this mechanism seems to be subtle.…”

Global Attractors of the Extensible Plate Equations with Nonlinear Damping and Memory

“…The case of = ≡ 0 in problem (1) has been studied by several authors (see [2,5,[15][16][17][18][19][20][21][22][23] and references therein). For instance, Wu [15] scrutinized the existence of global attractors for the nonlinear plate equation with thermal memory effects due to non-Fourier heat flux laws when ( ) = −Δ .…”