Cauchy problems for evolutionary pseudodifferential equations over \(p\)-adic field. (English) Zbl 1471.35353

Summary: We study a class of evolutionary pseudodifferential equations of the second order in \(t\), \((\partial^2 u(t,x)/\partial t^2+2a^2 T^{\alpha/2}(\partial u(t,x)/\partial t)+b^2 T^\alpha u(t,x)+c^2u (t,x)=q(t,x))\), where \(t\in (0,z]\) and \(T^\alpha\) is pseudodifferential operator in \(x\in Q_p\), which defined by W. Su [Sci. China, Ser. A 35, No. 7, 826–836 (1992; Zbl 0774.43006)]. We obtain the exact solutions to the equations which belong to mixed classes of real and \(p\)-adic functions.


35S10 Initial value problems for PDEs with pseudodifferential operators
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
30G06 Non-Archimedean function theory
35C05 Solutions to PDEs in closed form


Zbl 0774.43006
Full Text: DOI


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