LAUSR

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 semilinear damped σ$$ \sigma $$ ‐evolution equations with a modulus of continuity term in the nonlinearity of derivative type. Moreover, when the latter situation occurs, this work is to report sharp estimates for the lifespan of solutions, which have never appeared in previous literature, by catching the upper and lower bounds simultaneously.