The $\text{v}$-function of powers of sums of ideals

Let $K$ be a field, $I\subset R=K[x_1,\dots ,x_n]$ and $J\subset T=K[y_1,\dots ,y_m]$ be graded ideals. Set $S=R{\otimes }_{K}T$ and let $L=IS+JS$. The behaviour of the $\text{v}$-function $\text{v}(L^k)$ in terms of the $\text{v}$-functions $\text{v}(I^k)$ and $\text{v}(J^k)$ is investigated. When $I$ and $J$ are monomial ideals, we describe $\text{v}(L^k)$, giving an explicit formula involving $\text{v}(I^k)$ and $\text{v}(J^k)$.