Sandwich theorem for PDE's

Question

If $\textbf{v}:\mathbb{R}^n\times [0,\infty]\rightarrow \mathbb{R}^n$ and $\textbf{u} :\mathbb{R}^n\times [0,\infty]\rightarrow \mathbb{R}^n$ are vector field such that $$\dfrac{d}{dt}\textbf{v}\le \dfrac{d}{dt}\textbf{u} $$ for all $t\ge t^\prime$ show that there exist a value $k$ such that $\textbf{v}\le \textbf{u}$ for all $t\ge k\ge t^\prime$. That is if $\textbf{u}$ has more acceleration then $\textbf{v}$ it will eventually surpass $\textbf{v}$ even if $\textbf{u}< \textbf{v}$ at some time $t< t^\prime$. We might need to restrict the derivatives or functions to not be bounded. Here we interpret $$\textbf{f}\le \textbf{g} \Longleftrightarrow f_i\le g_i, ~~~~\forall i\le n $$ I think we can choose $$k = \min\{ t: v_i(x,t)\le u_i(x,t), \forall i\le n \}$$ holds in each component which should depend on when each component passes one another because they will all pass one another at different times.