A comparison of the v-number of a monomial ideal and its integral closure

Let $I$ be a monomial ideal in a standard graded polynomial ring and let $\overline{I}$ denote its integral closure. We study the relationship between $\mathrm{v}(I)$ and $\mathrm{v}(\overline{I})$. We prove that $\mathrm{v}(\overline{I}) \leq \mathrm{v}(I)$ for monomial ideals in two variables, for equigenerated monomial ideals in three variables and for several special classes of monomial ideals, while providing examples showing that this inequality does not hold in general. For the edge ideal $I(G)$ of a connected graph $G$, we show that $\mathrm{v}(I(G)^k)=\mathrm{v}(\overline{I(G)^k}) = 2k-1$ for all $k \geq 1+|E(G)|$. Moreover, when $G$ is disconnected, we prove that $\mathrm{v}(\overline{I(G)^k})\leq\mathrm{v}({I(G)^k})$ for all sufficiently large $k$.