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

TL;DR

This paper establishes the global existence of small data solutions for a weakly coupled system of semi-linear visco-elastic damped $\sigma$-evolution models with different nonlinearities and $\sigma$ values, using advanced $L^q$ estimates.

Contribution

It introduces a novel approach combining $L^m igcap L^q$ and $L^q$ estimates to handle different nonlinearities and $\sigma$ values in coupled visco-elastic models.

Findings

Proves global existence of solutions for the system.

Develops new $L^q$ and $L^m$ estimate techniques.

Allows relaxation of restrictions on admissible exponents $p$.

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 $p$.