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Multidimensional nonlinear pseudo-differential evolution equation with p-adic spatial variables

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We study the Cauchy problem for p-adic non-linear evolutionary pseudo-differential equations for complex-valued functions of a real positive time variable and p-adic spatial variables. Among the equations under consideration there is the p-adic analog of the porous medium equation (or more generally, the nonlinear filtration equation) which arise in numerous application in mathematical physics and mathematical biology. Our approach is based on the construction of a linear Markov semigroup on a p-adic ball and the proof of m-accretivity of the appropriate nonlinear operator. The latter result is equivalent to the existence and uniqueness of a mild solution of the Cauchy problem of a nonlinear equation of the porous medium type.

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The work by the first- and third-named authors was funded in part under the budget program of Ukraine No. 6541230 “Support to the development of priority research trends”. The third-named author was also supported in part in the framework of the research work “Markov evolutions in real and p-adic spaces” of the Dragomanov National Pedagogical University of Ukraine.

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Correspondence to Anatoly N. Kochubei.

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Antoniouk, A.V., Khrennikov, A.Y. & Kochubei, A.N. Multidimensional nonlinear pseudo-differential evolution equation with p-adic spatial variables. J. Pseudo-Differ. Oper. Appl. 11, 311–343 (2020). https://doi.org/10.1007/s11868-019-00320-3

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  • p-adic numbers
  • Porous medium equation
  • Markov process
  • m-accretive operator

Mathematics Subject Classification

  • Primary 35S10
  • 47J35
  • Secondary 11S80
  • 60J25
  • 76S05