p-adic Laplacian in local fields

In this paper a family of multi-dimensional fractional differential operators T-alpha and their corresponding pseudo-differential equations over p-adic fields are investigated. The test function class D(Q(p)(n)) and distribution class D'(Q(p)(n)) are invariant under the actions of these operators. The p-adic Laplacian Delta(p) and a fundamental solution of the Laplace equation are constructed. We study the spectral properties of the Laplacian Delta(p), and obtain an orthonormal basis of the eigen-functions of this operator in L-2(Q(p)(n)). Furthermore, the Cauchy problems for the wave and heat equations on the p-adic fields related to Delta(p) are also studied. (C) 2016 Elsevier Ltd. All rights reserved.