In this short paper we find that the Sobolev inequality $$\frac 1{p-2}\left[\left(\int f^{p} dμ\right)^{2/p} - \int f^2 dμ\right] \le C \int |\nabla f|^2 dμ$$ ($p\ge 0$) is equivalent to the exponential convergence of the Markov diffusion semigroup $(P_t)$ to the invariant measure $μ$, in some $Φ$-entropy. We provide the estimate of the exponential convergence in total variation and a bounded perturbation result under the Sobolev inequality. Finally in the one-dimensional case we get some two-sided estimates of the Sobolev constant by means of the generalized Hardy inequality.