The Interplay Between Time-dependent Speed of Propagation and Dissipation in Wave Models

The Influence of Oscillations on Energy Estimates for Damped Wave Models with Time-dependent Propagation Speed and Dissipation

“…The assumptions (B3) and (B4) describe the effective damping case related to a given increasing propagation speed. In particular, (B4) excludes the overdamping case (see [2]). The assumption (B5) allows us to control a certain amount of very fast oscillations in the damping term ρ(t)ω(t)u t .…”

“…This procedure was developed in [15] and [18]. Here we follow some ideas of [2]. However, we shall restrict the considerations to a smaller hyperbolic zone and elliptic zone in the extended phase space to cope with stronger oscillations in ω = ω(t) in our approach.…”

“…to the corresponding model with ω(t) ≡ 1 and effective dissipation case. Note that Z osc (N, ε) ⊂ Z λ hyp (N ) such that we can use the known estimates for E λ = E λ (t, s, ξ) from [2] (see Lemma 3.5 of [2]). This estimate reads as follows:…”

“…We use the decreasing behavior of the function S = S(t, |ξ|) in |ξ|. So, the function S = S(t, |ξ|) takes its maximum for| ξ| satisfying t = t| ξ| , that is, the second integral vanishes in S(t, |ξ|) (see [2]).…”

“…where λ is a monotonously increasing nontrivial shape function in the propagation speed, ρ is a nontrivial shape function in the damping term and ω is a bounded oscillating function in both propagation speed and damping term. Throughout this paper, we restrict ourselves to "effective-like" (due to oscillating term ω = ω(t)) damping ρ(t)ω(t)u t in the sense of [2] and [17]. Here effective means that the solution behaves like that of a corresponding heat equation.…”