  We establish a Gagliardo-Nirenberg-type inequality in $\mathbb{R}^n$ for
functions which decay fast as $|x|\to\infty$. We use this inequality to derive
upper bounds for the decay rates of solutions of a degenerate parabolic
equation. Moreover, we show that these upper bounds, hence also the
Gagliardo-Nirenberg-type inequality, are sharp in an appropriate sense.
