  We give an alternative look at the log-Sobolev inequality (LSI in short) for
log-concave measures by semigroup tools. The similar idea yields a heat flow
proof of LSI under some quadratic Lyapunov condition for symmetric diffusions
on Riemannian manifolds provided the Bakry-Emery's curvature is bounded from
below. Let's mention that, the general $\phi$-Lyapunov conditions were
introduced by Cattiaux-Guillin-Wang-Wu [8] to study functional inequalities,
and the above result on LSI was first proved subject to $\phi(\cdot)=d^2(\cdot,
x_0)$ by Cattiaux-Guillin-Wu [9] through a combination of detective $L^2$
transportation-information inequality $\mathrm{W_2I}$ and the HWI inequality of
Otto-Villani.
  Next, we assert a converse implication that the Lyapunov condition can be
derived from LSI, which means their equivalence in the above setting.
