Lower and upper bounds for the blow-up time for a thin-film equation with variable exponent in the reaction term

This paper deals with the blow-up phenomena for solutions to a thin-film equation with a variable exponent in the reaction term. We consider the case where the initial energy is positive to obtain a lower bound for blow-up results if blow-up occurs for this case. We give a new blow-up criterion, and under some conditions on $p(\cdot )$ and $u_{0}$ we establish the lifespan $\widehat T$ of the solution. The blow-up time with negative initial energy is also derived.