Abstract
We consider a nonlinear evolution equation for complex-valued functions of a real positive time variable and a p-adic spatial variable. This equation is a non-Archimedean counterpart of the fractional porous medium equation. Developing, as a tool, an \(L^1\)-theory of Vladimirov’s p-adic fractional differentiation operator, we prove m-accretivity of the appropriate nonlinear operator, thus obtaining the existence and uniqueness of a mild solution. We give also an example of an explicit solution of the p-adic porous medium equation.
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Acknowledgements
The second author is grateful to Mathematical Institute, Linnaeus University, for hospitality during his visits to Växjö. The work of the second author was also supported in part by Grant 23/16-18 “Statistical dynamics, generalized Fokker-Planck equations, and their applications in the theory of complex systems” of the Ministry of Education and Science of Ukraine.
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Communicated by Hans G. Feichtinger.
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Khrennikov, A.Y., Kochubei, A.N. p-Adic Analogue of the Porous Medium Equation. J Fourier Anal Appl 24, 1401–1424 (2018). https://doi.org/10.1007/s00041-017-9556-4
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DOI: https://doi.org/10.1007/s00041-017-9556-4
Keywords
- p-adic numbers
- Vladimirov’s p-adic fractional differentiation operator
- p-adic porous medium equation
- Mild solution of the Cauchy problem