On asymptotic properties of solutions to σ-evolution equations with general double damping

Abstract

In this paper, we would like to consider the Cauchy problem for semi-linear σ-evolution equations with double structural damping for any σ⩾1. The main purposes of the present work are to not only study the asymptotic profiles of solutions to the corresponding linear equations but also describe large-time behaviors of globally obtained solutions to the semi-linear equations. We want to emphasize that the new contribution is to find out the sharp interplay of “parabolic like models” corresponding to σ1∈[0,σ/2) and “σ-evolution like models” corresponding to σ2∈(σ/2,σ], which together appear in an equation. In this connection, we understand clearly how each damping term influences the asymptotic properties of solutions.