local well-posedness and blow-up of solution for a higher-order wave equation with viscoelastic term and variable-exponent

We investigate in this paper a value problem related to the following nonlinear higher-order wave equation [math formula] firstly, we prove the existence and uniqueness of the local solution under suitable conditions for the relaxation function g and viable-exponent p(.) , using a method, which is a mixture of the faedo-galarkin and banach fixed point theorem, and prove also the solution blows up in finite time. finally, we give a two-dimensional numerical example to illustrate the blow-up result.