Time-dependent Sobolev inequality along the Ricci flow

In this article, we get a time-dependent Sobolev inequality along the Ricci flow in a more general situation than those in Zhang (A uniform Sobolev inequality under Ricci flow. Int Math Res Not IMRN 2007, no 17, Art ID rnm056, 17 pp), Ye (The logarithmic Sobolev inequality along the Ricci flow. arXiv:0707.2424v2) and Hsu (Uniform Sobolev inequalities for manifolds evolving by Ricci flow. arXiv:0708.0893v1) which also generalizes the results of them. As an application of the time-dependent Sobolev inequality, we get a growth of the ratio of non-collapsing along immortal solutions of Ricci flow.