A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation

Abstract

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.