This is a notation I see in page 8 of Guy Barles and Espen R. Jakobsen, namely \partial_t^{\beta_0}D^{\beta'}\phi(x,t) where $\phi: \mathbb{R}^n\times[0,T]\longrightarrow \mathbb{R}$ is smooth, $\beta_0\in \mathbb{N}$ , \beta'=(\beta'_i)_i \in \mathbb{N}^n.
I want to know the precise definition of the notation.