Lp − Lq estimates for a parameter-dependent multiplier with oscillatory and diffusive components

In this paper, we derive long time Lp−Lq decay estimates, in the full range 1≤p≤q≤∞, for time-dependent multipliers in which an interplay between an oscillatory component and a diffusive component with different scaling appears. We estimate ‖m(t,⋅)‖Mjavax.xml.bind.JAXBElement@5a7a7ba6javax.xml.bind.JAXBElement@50d4f4b1 as t→∞ for multipliers of type m(t,ξ)=e±i|ξ|javax.xml.bind.JAXBElement@974d21t−|ξ|javax.xml.bind.JAXBElement@3d81aa6bt, and suitable perturbations, under the assumption that the scaling of the diffusive component is worse, i.e., θ>σ. These multipliers are, for instance, related to the fundamental solution to the Cauchy problem for the σ-evolution equation with structural damping: utt+(−Δ)σu+(−Δ)[Formula presented]ut=0,t≥0,x∈Rn, in the so-called non-effective case σ