A NOTE ON L1-L1 ESTIMATES FOR SOLUTIONS TO VISCO-ELASTIC DAMPED σ-EVOLUTION MODELS

DOI: 10.18173/2354-1059.2023-0017

Abstract

This note is to conclude some L1 − L1 estimates for solutions to the following Cauchy problem for visco-elastic damped σ-evolution models: (u utt (0+ ( , x) = −∆) u0σ(ux+ ( ), u −∆) t(0σ, x ut) = = 0u, x 1(x), x ∈ ∈ R Rn n, t > , 0, (0.1) where σ > 1, in all space dimensions n ≥ 1. The novelty of the paper is that we have succeeded in deriving the L1 − L1 estimates for solutions to (0.1) in general cases of σ > 1 without any constraint condition to space dimensions, i.e. our results are valid for any n ≥ 1. To establish this, the application of the theory of modified Bessel functions linked to Faa di Bruno’s formula comes into play for small frequencies, meanwhile, for both large frequencies and middle frequencies we will rigorously carry out some useful estimates to obtain the exponential decay.