Nonlinear damped wave equation: Existence and blow-up

In this paper, we consider the following nonlinear wave equation with variable exponents: utt−Δu+aut|ut|m(⋅)−2=bu|u|p(⋅)−2,where a,b are positive constants. By using the Faedo-Galerkin method, the existence of a unique weak solution is established under suitable assumptions on the variable exponents m and p. We also prove the finite time blow-up of solutions and give a two-dimension numerical example to illustrate the blow up result.