On the solutions for an extensible beam equation with internal damping and source terms

Abstract

In this manuscript, we consider the nonlinear beam equation with internal damping and source term utt +Δu+M(|∇u|)(−Δu)+ut = |u|r−1u where r > 1 is a constant, M(s) is a continuous function on [0,+∞) . The global solutions are constructed by using the Faedo-Galerkin approximations, taking into account that the initial data is in appropriate set of stability created from the Nehari manifold. The asymptotic behavior is obtained by the Nakao method.