On a Robin-Dirichlet problem for a system of nonlinear pseudoparabolic equations with the viscoelastic term

We consider a Robin-Dirichlet problem for a system of nonlinear pseudoparabolic equations with the viscoelastic term. Based on the Faedo-Galerkin method, we first prove existence and uniqueness. Next, we give a sufficient condition for the global existence and decay of weak solutions. Finally, using concavity method, we prove blow-up results for solutions when the initial energy is nonnegative or negative. Furthermore, we establish here the lifespan for the equation via finding the upper bound and the lower bound for the blow-up times.