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We study the Vladimirov fractional differentiation operator $D^\alpha_N$,$\alpha >0, N\in \mathbb Z$, on a $p$-adic ball $B_N=\{ x\in \mathbb Q_p:\|x|_p\le p^N\}$. To its known interpretations via restriction from a similaroperator on $\mathbb Q_p$ and via a certain stochastic process on $B_N$, we addan interpretation as a pseudo-differential operator in terms of the Pontryaginduality on the additive group of $B_N$. We investigate the Green function of$D^\alpha_N$ and a nonlinear equation on $B_N$, an analog the classical porousmedium equation.

Linear and Nonlinear Heat Equations on a p-Adic Ball