Abstract

In this article, we study semi-linear σ-evolution equations with double damping including frictional and visco-elastic damping for any σ≥1. We are interested in investigating not only higher order asymptotic expansions of solutions but also diffusion phenomenon in the Lp−Lq framework, with 1≤p≤q≤∞, to the corresponding linear equations. By assuming additional Lm regularity on the initial data, with m∈[1,2), we prove the global (in time) existence of small data energy solutions and indicate the large time behavior of global obtained solutions as well to semi-linear equations. Moreover, we also determine the so-called critical exponent when σ is integers.