In this work, we analyze the well-posedness and the long-time behavior of a wave equation posed in Rn, featuring an energy damping term of the form γEl(t)qut, where El denotes the linear energy of the system. Our main result establishes that, for each coefficient γ 0, the dynamical system (H,Stγ) generated by the mild (weak) solutions possesses a compact global attractor Aγ in the weak phase space topology H=H1(Rn)×L2(Rn). This is achieved by proving that the dynamical system is dissipative and asymptotically compact. To overcome the lack of compactness of Sobolev embeddings in unbounded domains, we combine the compensated compactness criterion with the concept of uniform tail estimates. Finally, we prove the upper semicontinuity of the family of attractors Aγ with respect to the parameter γ 0. To the best of our knowledge, this is the first study addressing the dynamics of a wave equation with energy damping in the whole space Rn.
Freitas et al. (Sun,) studied this question.