Evolution models with time-dependent coefficients in friction and viscoelastic damping terms

2025/06/30 by Aslan, Halit Sevki, Reissig, Michael
#35B40 #35L05 #35L15 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.24058

Abstract

We study the following Cauchy problem for the linear wave equation with both time-dependent friction and time-dependent viscoelastic damping: \begincases u_tt- Δu + b(t)u_t - g(t)Δu_t=0, amp;(t,x) ∈ (0,∞) × ℝⁿ,
u(0,x)= u₀(x), u_t(0,x)= u₁(x), amp;x ∈ ℝⁿ. \endcases Our goal is to derive decay estimates for higher order energy norms of solutions to this problem. We focus on the interplay between the time-dependent coefficients in both damping terms and their influence on the qualitative behavior of solutions. The analysis is based on a classification of the damping mechanisms, frictional damping b(t)u_t and viscoelastic damping -g(t)Δu_t as well, and employs the WKB-method in the extended phase space.

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