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

Abstract In this paper, we derive long time L p − L q 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 , ⋅ ) ‖ M p q as t → ∞ for multipliers of type m ( t , ξ ) = e ± i | ξ | σ t − | ξ | θ t , 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: u t t + ( − Δ ) σ u + ( − Δ ) θ 2 u t = 0 , t ≥ 0 , x ∈ R n , in the so-called non-effective case σ θ .