This study focuses on differentiating an explosive bubble process from a unit root process with a partial quadratic trend on an unknown break date, denoted as a URQ process herein. We propose a DBF test for differentiating these two similar processes. We develop the corresponding limiting distributions under the local-to-unity hypothesis which covers the null of URQ, the stationary and explosive root alternatives.Simulations suggest that the tests have a good finite sample performance and confirm our proposed test can differentiate URQ processes and explosive bubble processes. The DBF test is applied to the stock prices of Kweichow Moutai and Apple, which exhibit striking price rises during their respective sample periods, similar to an explosive bubble process. However, the empirical results suggest that an improvement in the fundamental value rather than irrational exuberance mainly drives such drastic price rises.