Title: Linear and Nonlinear Heat Equations on a p-Adic Ball,
Abstract: 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 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
$D^\alpha_N$ and a nonlinear equation on $B_N$, an analog the classical porous
medium equation.