arXiv: Analysis of PDEs | 2019
Study of semi-linear $\\sigma$-evolution equations with frictional and visco-elastic damping.
Abstract
In this article, we study semi-linear $\\sigma$-evolution equations with double damping including frictional and visco-elastic damping for any $\\sigma\\ge 1$. We are interested in investigating not only higher order asymptotic expansions of solutions but also diffusion phenomenon in the $L^p-L^q$ framework, with $1\\le p\\le q\\le \\infty$, to the corresponding linear equations. By assuming additional $L^{m}$ regularity on the initial data, with $m\\in [1,2)$, we prove the global (in time) existence of small data energy solutions and indicate the large time behavior of the global obtained solutions as well to semi-linear equations. Moreover, we also determine the so-called critical exponent when $\\sigma$ is integers.