A closed manifold is called aspherical if it has contractible universal cover. It has been conjectured since the 80s that all closed aspherical manifolds do not admit metrics with positive scalar curvature. In dimensions 3,4,5 this conjecture is solved by works of Schoen—Yau, Gromov—Lawson, Chodosh—Li, and Gromov. We prove for n = 3,4,5 that the connected sum of a closed aspherical n-manifold with an arbitrary non-compact manifold does not admit a complete metric with nonnegative scalar curvature. More generally, for n = 3,4,5, we generalize Chodosh, Li, and Liokumovich's partial classification result of closed PSC n-manifolds to the non-compact case. This is joint work with Jianchun Chu and Jintian Zhu.