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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.

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    Now I'm curious, why are you readin$g$ a paper on differential equations without $k$$n$owi$n$g that $n$otatio$n$?2011-08-03

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The notation means that one differentiates the function $\beta_0$ times with respect to $t$ and \beta'_i times with respect to each $x_i$.