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

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.