Critical exponent for a weakly coupled system of semi-linear $\sigma$-evolution equations with frictional damping

TL;DR

This paper investigates the global existence and blow-up of solutions for a weakly coupled system of semi-linear $\sigma$-evolution equations with frictional damping, covering fractional and integer orders.

Contribution

It establishes conditions for global existence of small data solutions and demonstrates blow-up results for Sobolev solutions with fractional $\sigma$.

Findings

Proves global existence of small data solutions for the system.

Shows blow-up of Sobolev solutions when $\sigma$ is fractional.

Analyzes the critical exponent related to the system's behavior.

Abstract

We are interested in studying the Cauchy problem for a weakly coupled system of semi-linear $\sigma$-evolution equations with frictional damping. The main purpose of this paper is two-fold. We would like to not only prove the global (in time) existence of small data energy solutions but also indicate the blow-up result for Sobolev solutions when $\sigma$ is assumed to be any fractional number.