# A remark on semi-linear damped $\sigma$-evolution equations with a modulus of continuity term in nonlinearity

## TL;DR

This paper investigates semi-linear damped sigma-evolution equations with a modulus of continuity in the nonlinearity, establishing conditions for either global existence of small solutions or finite-time blow-up.

## Contribution

It provides new criteria under which solutions either exist globally or blow up, considering the effects of a modulus of continuity in the nonlinearity.

## Key findings

- Global existence of small Sobolev solutions under certain assumptions.
- Finite-time blow-up of solutions under different conditions.
- Clarifies the role of the modulus of continuity in solution behavior.

## Abstract

In this article, we indicate that under suitable assumptions of a modulus of continuity we obtain either the global (in time) existence of small data Sobolev solutions or the blow-up result of local (in time) Sobolev solutions to semi-linear damped $\sigma$-evolution equations with a modulus of continuity term in nonlinearity.