This paper is concerned with the existence of global weak solution for a nonlinear viscoelastic problem with internal damping and a logarithmic source term and Dirichlet boundary initial conditions, and with the study of the asymptotic behavior of the problem, involving: a) exponential decay of total energy of solutions for initial data in the set of stability created by the Nehari manifold, b) the exponential growth of the logarithmic source term for negative initial energy. In the existence of global weak solution we employed similar ideas as in the work of S. Cordeiro, J. Ferreira, et al., 2021, where the Faedo-Galerkin method was combined with Aubin- Lions lemmas for the passage to the limit in the nonlinear terms. In the study of the exponential decay of the total energy and in the growth of the logarithmic term of the energy we adapted the perturbed energy methods in a work of Messaoudi & Tatar, 2006 and 2003.

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