$p$-adic evolution pseudo-differential equations and $p$-adic wavelets

V. M. Shelkovich

Izvestiya. Mathematics , Tome 75 (2011) no. 6, pp. 1249-1278

Résumé

In the theory of $p$-adic evolution pseudo-differential equations (with time variable $t\in\mathbb{R}$ and space variable $x\in \mathbb{Q}_p^n$), we suggest a method of separation of variables (analogous to the classical Fourier method) which enables us to solve the Cauchy problems for a wide class of such equations. It reduces the solution of evolution pseudo-differential equations to that of ordinary differential equations with respect to the real variable $t$. Using this method, we solve the Cauchy problems for linear evolution pseudo-differential equations and systems of the first order in $t$, linear evolution pseudo-differential equations of the second and higher orders in $t$, and semilinear evolution pseudo-differential equations. We derive a stabilization condition for solutions of linear equations of the first and second orders as $t\to \infty$. Among the equations considered are analogues of the heat equation and linear or non-linear Schrödinger equations. The results obtained develop the theory of $p$-adic pseudo-differential equations and can be used in applications.

Détail

Keywords: $p$-adic pseudo-differential operator, $p$-adic fractional operator, $p$-adic wavelet bases, $p$-adic pseudo-differential equations.