This paper is devoted to the study of existence, uniqueness, continuous dependence, general decay of solutions of an initial boundary value problem for a viscoelastic wave equation with strong damping and nonlinear memory term. At first, we state and prove a theorem involving local existence and uniqueness of a weak solution. Next, we establish a sufficient condition to get an estimate of the continuous dependence of the solution with respect to the kernel function and the nonlinear terms. Finally, under suitable conditions to obtain the global solution, we prove the general decay property with positive initial energy for this global solution.

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.