  Given $p,N>1,$ we prove the sharp $L^p$-log-Sobolev inequality on noncompact
metric measure spaces satisfying the ${\sf CD}(0,N)$ condition, where the
optimal constant involves the asymptotic volume ratio of the space. This proof
is based on a sharp isoperimetric inequality in ${\sf CD}(0,N)$ spaces,
symmetrisation, and a careful scaling argument. As an application we establish
a sharp hypercontractivity estimate for the Hopf-Lax semigroup in ${\sf
CD}(0,N)$ spaces. The proof of this result uses Hamilton-Jacobi inequality and
Sobolev regularity properties of the Hopf-Lax semigroup, which turn out to be
essential in the present setting of nonsmooth and noncompact spaces.
Furthermore, a sharp Gaussian-type $L^2$-log-Sobolev inequality is also
obtained in ${\sf RCD}(0,N)$ spaces. Our results are new, even in the smooth
setting of Riemannian/Finsler manifolds. In particular, an extension of the
celebrated rigidity result of Ni (J. Geom. Anal., 2004) on Riemannian manifolds
will be a simple consequence of our sharp log-Sobolev inequality.
