The Cauchy problems for evolutionary pseudo-differential equations over p-adic field and the wavelet theory

Abstract

We solve the Cauchy problems for p-adic linear and semi-linear evolutionary pseudo-differential equations (the time-variable t ∈ R and the space-variable x ∈ Q p n ). Among the equations under consideration there are the heat type equation and the Schrödinger type equations (linear and nonlinear). To solve these problems, we develop the “variable separation method” (an analog of the classical Fourier method) which reduces solving evolutionary pseudo-differential equations to solving ordinary differential equations with respect to real variable t. The problem of stabilization for solutions of the Cauchy problems as t → ∞ is also studied. These results give significant advance in the theory of p-adic pseudo-differential equations and can be used in applications.