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Linear degenerate parabolic equations of arbitrary order with a finite region of dependence

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The Cauchy problem is considered for equations of the form ut-Lu=0, where Lu=L(t, x1, ..., xn, ∂/∂x1, ..., ∂xn)u is an elliptic differential expression of arbitrary order which is degenerate for certain values of the arguments in the first order differential expression. Conditions are stated on the nature of the degeneracy which are sufficient for a solution of this problem to have a finite region of dependence.

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Literature cited

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    M. I. Freidlin, “The formulation of boundary value problems for degenerate elliptic equations,” Dokl. Akad. Nauk SSSR,170, No. 2, 282–285 (1966).

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    Yu. N. Blagoveshchenskii and M. I. Freidlin, “Some properties of diffusion processes which depend on a parameter,” Dokl. Akad. Nauk SSSR,138, No. 3, 508–511 (1961).

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    O. A. Oleinik, “The smoothness of solutions of degenerate elliptic and parabolic equations,” Dokl. Akad. Nauk SSSR,163, No. 3, 577–580 (1965).

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Translated from Matematicheskie Zametki, Vol. 6, No. 3, pp. 289–294, September, 1969.

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Kalashnikov, A.S. Linear degenerate parabolic equations of arbitrary order with a finite region of dependence. Mathematical Notes of the Academy of Sciences of the USSR 6, 630–633 (1969). https://doi.org/10.1007/BF01119681

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  • Cauchy Problem
  • Parabolic Equation
  • Arbitrary Order
  • Degenerate Parabolic Equation
  • Finite Region