This paper is concerned with the long-time behavior of a semilinear hyperbolic coupled system with nonlinear damping and source terms. By using nonlinear semigroups and the theory of monotone operators, we obtain the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data. Under some restrictions on the parameters, we also prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via potential well theory. More precisely, we prove the existence of a unique global weak solution with initial data coming from the ”good” part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin. Finally, a blow-up result for positive energy is proven.
