Estimation of the fundamental solution of a new class for non-Archimedean pseudodifferential equations
DOI:
https://doi.org/10.3842/umzh.v76i10.8687Keywords:
pseudo-differential operator, fundamental solution, Cauchy problem, p-adic analysisAbstract
UDC 517.9
We investigate the equation with the Vladimirov–Taibleson pseudodifferential operator for functions with p-adic time and space variables, which generalizes the p-adic wave equation in the cases where the orders of the time and space derivatives do not coincide. We prove the existence and uniqueness of the solution to the corresponding Cauchy problem. Some properties of this solution are established, including, in particular, the finite domain of dependence, which resembles the behavior of classical hyperbolic equations. We also deduce an L1-estimate for the solution. On the other hand, we prove an estimate for the fundamental solution of the problem, which is an analog of the corresponding estimates for parabolic-type equations with real time and p-adic space variables.
References
S. Albeverio, A. Yu. Khrennikov, V. M. Shelkovich, Theory of p-adic distributions, Cambridge University Press (2010).
A. N. Kochubei, A non-Archimedean wave equation, Pacif. J. Math., 235, № 2, 245–261 (2008). DOI: https://doi.org/10.2140/pjm.2008.235.245
A. N. Kochubei, Radial solutions of non-Archimedean pseudo-differential equations, Pacif. J. Math., 269, 355–369 (2014). DOI: https://doi.org/10.2140/pjm.2014.269.355
A. N. Kochubei, Pseudo-differential equations and stochastics over non-Archimedean fields, Marcel Dekker, New York (2001). DOI: https://doi.org/10.1201/9780203908167
A. N. Kochubei, The Vladimirov–Taibleson operator: inequalities, Dirichlet problem, boundary Hölder regularity, J. Pseudo-Differ. Oper. Appl., 14, № 2, Paper 31 (2023). DOI: https://doi.org/10.1007/s11868-023-00525-7
V. S. Vladimirov, I. V. Volovich, E. I. Zelenov, p-Adic analysis and mathematical physics, World Scientific, Singapore (1994). DOI: https://doi.org/10.1142/1581
W. A. Zuniga-Galindo, Pseudodifferential equations over non-Archimedean spaces, Lecture Notes Math., 2174, (2016). DOI: https://doi.org/10.1007/978-3-319-46738-2
