Abstract:
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.