New decay estimates for linear damped wave equations and its application to nonlinear problem

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Energy decay estimates for wave equations with a fractional damping (2012, background): “5 below, which was first introduced by Ikehata [11].”
The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case (2020, methodology): “ower bounds of the solution itself with u2 ∈ L2(Rn)∩L1,1(Rn). Before processing these estimates, let us introduce the notation for the integral of f(x) by Pf := R Rn f(x)dx, and recall Lemma 2.1 from [26]. Lemma 3.1. Let us assume f∈ L1,1(Rn). Then, the following estimate holds: |fˆ(ξ)| 6C 1|ξ|kfkL ”
Asymptotic Profiles of Solutions for Structural Damped Wave Equations (2016, methodology): “dξ ≤ Ce−ηt(ku 1k 2 +ku 0k2 H1), with some constant C > 0. This completes the proof of Theorem 1.1. 3 Proof of Theorems 1.2. In this section, let us prove Theorem 1.2 by employing the method due to [8, 9]. We ﬁrst establish the asymptotic proﬁle of the solution in the low frequency region |ξ| ≪ 1,”
Existence and decay of solutions of a viscoelastic equation with a nonlinear source (2006, background): “We refer the reader to [8–11,14] and the references therein for more stabilization results.”