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The Cauchy problems for evolutionary pseudo-differential equations over p-adic field and the wavelet theory |
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| Authors: | S Albeverio AYu Khrennikov |
| Institution: | a Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany b SFB 611, Bonn University, Germany c IZKS, Bonn University, Germany d BiBoS (Bielefeld-Bonn), Germany e Dip. Matematika, Universita di Trento, Italy f International Center for Mathematical Modelling in Physics and Cognitive Sciences MSI, Växjö University, SE-351 95, Växjö, Sweden g Department of Mathematics, St.-Petersburg State Architecture and Civil Engineering University, 2 Krasnoarmeiskaya 4, 190005, St. Petersburg, Russia |
| 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  ). 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. |
| Keywords: | p-Adic pseudo-differential equation p-Adic pseudo-differential operator Fractional operator p-Adic Lizorkin space p-Adic wavelets Variable separation method |
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