Global existence and blow-up of solutions for higher-order viscoelastic wave equation with a nonlinear source term

The initial–boundary value problem for some nonlinear higher-order viscoelastic wave equation with a nonlinear source term in a bounded domain is studied. The existence of global weak solutions for this problem is established by using the Galerkin method. Meanwhile, under suitable conditions on relaxation function g(⋅)g(⋅) and the positive initial energy as well as non-positive initial energy, it is proved that the solution blows up in the finite time and the lifespan estimates of solutions are also given.