Cauchy Problems for Evolutionary Pseudodifferential Equations over p -Adic Field

Abstract

We study a class of evolutionary pseudodifferential equations of the second order in $\mathrm{t,}$ $({\partial }^{\mathrm{2}}u(t,x)/\partial t^{\mathrm{2}}+\mathrm{2}{a}^{\mathrm{2}}{T}^{\alpha /\mathrm{2}}(\partial u(t,x)/\partial t)+b^{\mathrm{2}}{T}^{\alpha }u(t,x)+c^{\mathrm{2}}u(t,x)=q(t,x))$, where $t\in (0,z]$ and $T^{\alpha }$ is pseudodifferential operator in $x\in Q_p$, which defined by Weiyi Su in 1992. We obtained the exact solutions to the equations which belong to mixed classes of real and $p$-adic functions.