Linear and nonlinear heat equations on a $p$ -adic ball

Abstract

We study the Vladimirov fractional differentiation operator $D^{\alpha}_N,\; \alpha > 0,\; N \in Z$, on a $p$-adic ball B$B_N = \{ x \in Q_p : | x|_p \leq p^N\}$. To its known interpretations via the restriction of a similar operator to $Q_p$ and via a certain stochastic process on $B_N$, we add an interpretation as a pseudodifferential 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 of the classical equation of porous medium.