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A non-Archimedean wave equation. (English) Zbl 1190.35235
Summary: Let $K$ be a non-Archimedean local field with the normalized absolute value $|\cdot |$. It is shown that a “plane wave” $f(t+\omega_1\times 1 + \dots + \omega_n\times n)$, where $f$ is a Bruhat-Schwartz complex-valued test function on $K$ with $(t,x_1,\dots ,x_n) \in K^{n+1}$ and $\text{max}_{1\leq j\leq n}|\omega_j|=1$, satisfies, for any $f$, a certain homogeneous pseudodifferential equation, an analogue of the classical wave equation. A theory of the Cauchy problem for this equation is developed.

35S10Initial value problems for pseudodifferential operators
35L99Hyperbolic equations and systems
46S10Functional analysis over fields (not $\Bbb R$, $\Bbb C$, $\Bbb H$or quaternions)
11S80Other analytic theory of local fields
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