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 consider initial value problems for nonlinear fractional equations where source functions may discontinuous. We obtain the existence and uniqueness of maximal mild solutions of the problem. We also give some appropriate conditions such that mild solutions of the problem blow-up at a finite time. Furthermore, we discuss the continuous dependence of mild solutions of the problem with respect to fractional order.