Linear and Nonlinear Heat Equations on a p-Adic Ball

We study the Vladimirov fractional differentiation operator $D_N^{\alpha }$, $\alpha >0,N\in \mathbb{Z}$, 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 similar operator on ${\mathbb{Q}}_p$ and via a certain stochastic process on $B_N$, we add an interpretation as a pseudo-differential operator in terms of the Pontryagin duality on the additive group of $B_N$. We investigate the Green function of $D_N^{\alpha }$ and a nonlinear equation on $B_N$, an analog the classical porous medium equation.