  We prove that if $u_1,u_2 : (0,\infty) \times \R^d \to (0,\infty)$ are
sufficiently well-behaved solutions to certain heat inequalities on $\R^d$ then
the function $u: (0,\infty) \times \R^d \to (0,\infty)$ given by
$u^{1/p}=u_1^{1/p_1} * u_2^{1/p_2}$ also satisfies a heat inequality of a
similar type provided $\tfrac{1}{p_1} + \tfrac{1}{p_2} = 1 + \tfrac{1}{p}$. On
iterating, this result leads to an analogous statement concerning $n$-fold
convolutions. As a corollary, we give a direct heat-flow proof of the sharp
$n$-fold Young convolution inequality and its reverse form.
