This paper is devoted to the study of a nonlinear pseudoparabolic
equation in an annular with Robin-Dirichlet conditions. At first, by
applying the standard Faedo-Galerkin method, we prove existence and
uniqueness results. Next, using concavity method, we prove blow-up
results for solutions when the initial energy is nonnegative or
negative, then we also establish the lifespan for the equation via
finding the upper bound and the lower bound for the blow-up times.
Finally, we give a sufficient condition for the global existence and
decay of weak solutions.