A lower bound for the blow-up time to a viscoelastic hyperbolic equation with nonlinear sources

This paper deals with the lower bound for blow-up solutions to a nonlinear viscoelastic hyperbolic equation. An inverse Hölder inequality with the correction constant is employed to overcome the difficulty caused by the failure of the embedding inequality. Moreover, a lower bound for blow-up time is obtained by establishing first-order differential inequality. This result gives an answer to the problem unsolved in our earlier work Sun et al. (2014).