On weak solutions for a nonlocal model with nonlocal damping term

In this paper, we consider a linear bond-based peridynamic nonlocal evolution problem with the nonlocal viscoelastic damping term, the existence, uniqueness, and continuous dependence upon datum of a weak solution are proved by Galerkin methods. Together with energy inequality, we establish the regularity results of weak solutions in time. In addition, we also briefly analyze the limit behavior of the weak solution as δ→0, and find that its limit function solves the corresponding classical local evolution problem exactly in the sense of distributions. Under more stronger regularity conditions for solutions, the solution of the nonlocal evolution problem strongly converges to the solution of the corresponding classical local problem only in the interior of domain Ω.