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

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