A note on a weakly coupled system of semi-linear visco-elastic damped $\sigma$-evolution models with different power nonlinearities and different $\sigma$ values

Abstract

In this article, we prove the global (in time) existence of small data solutions from energy spaces basing on $L^q$ spaces, with $q\in (1,\infty )$, to the Cauchy problems for a weakly coupled system of semi-linear visco-elastic damped $\sigma$-evolution models. Here we consider different power nonlinearities and different $\sigma$ values in the comparison between two single equations. To do this, we use $(L^m\cap L^q)-L^q$ and $L^q-L^q$ estimates, i.e., by mixing additional $L^m$ regularity for the data on the basis of $L^q-L^q$ estimates for solutions, with $m\in [1,q)$, to the corresponding linear Cauchy problems. In addition, allowing loss of decay and the flexible choice of parameters $\sigma$, $m$ and $q$ bring some benefits to relax the restrictions to the admissible exponents $p$.

arXiv commentarXiv admin note: text overlap with arXiv:1809.06744