Bounds for the blow-up time of solution to a nonlinear viscoelastic equation with fractional damping

The upper bound of blow-up time guarantees the blowing up of the solution, and the lower bound may provide us a safe time interval for the operation if we use the problem to model a physical process. In this paper, we deduce the lower and upper bound of blow-up time of solution to a hyperbolic equation with fractional damping term using differential inequality technique under some sufficient conditions on the variable exponents and initial data. The local existence of solutions is proved in suitable Orlicz–Sobolev spaces using Galerkin approximation technique with suitable conditions on the variable exponents and variable coefficients.