  We show that the sharp Young's inequality for convolutions first obtained by
Bechner and Brascamp-Lieb can be derived from the monotone in time evolution of
a Lyapunov functional of the convolution of two solutions to the heat equation,
with different diffusion coefficients, first introduced by Bennett and Bez. Our
proof is based on a suitable adaptation of an old idea of Stam and Blachman,
used to obtain Shannon's entropy power inequality.
