Global existence for weakly coupled systems of semi-linear structurally damped $\sigma$-evolution models with different power nonlinearities

TL;DR

This paper establishes the global existence of small data Sobolev solutions for weakly coupled semi-linear structurally damped $\sigma$-evolution systems with different power nonlinearities, under certain initial data regularity conditions.

Contribution

It introduces new global existence results for coupled damped evolution models with varying nonlinearities using advanced linear estimates and initial data regularity assumptions.

Findings

Proves global existence of solutions for small initial data.

Uses $(L^m igcap L^2)- L^2$ and $L^2- L^2$ estimates for linear problems.

Extends previous results to systems with different power nonlinearities.

Abstract

In this paper, we study the Cauchy problems for weakly coupled systems of semi-linear structurally damped $\sigma$-evolution models with different power nonlinearities. By assuming additional $L^m$ regularity on the initial data, with $m \in [1,2)$, we use $(L^m \cap L^2)- L^2$ and $L^2- L^2$ estimates for solutions to the corresponding linear Cauchy problems to prove the global (in time) existence of small data Sobolev solutions to the weakly coupled systems of semi-linear models from suitable function spaces.