In this paper, we consider the Robin-Dirichlet problem for a system of nonlinear pseudoparabolic equations with viscoelastic term. By the Faedo-Galerkin method, we first prove the existence and uniqueness of solution for the problem. Next, we give a sufficient condition to get the global existence and decay of the weak solution. Finally, by the concavity method, we prove the blow-up result of the solution when the initial energy is negative. Furthermore, we establish here the lifespan of the solution by finding the upper bound and the lower bound for the blow-up time.

In this paper, we study the Robin-Dirchlet problem (Pn) for a wave equation with the term 1 n Xn i=1 u2( i−1 n , t), n ∈ N. First, for each n ∈ N, under suitable conditions, we prove the local existence and uniqueness of the weak solution un of (Pn). Next, we prove that the sequence of solutions un of (Pn) converges strongly in appropriate spaces to the weak solution u of the problem (P), where (P) is defined by (Pn) by replacing 1 n Xn i=1 u2( i−1 n , t) by Z 1 0 u2(y, t)dy. The main tools used here are the linearization method together with the Faedo-Galerkin method and the weak compact method. We end the paper with a remark related to a similar problem.