On a theorem of Cauchy-Kovalevskaya type for a class of nonlinear PDE's of higher order with deviating arguments

Antoni Augustynowicz

Annales Polonici Mathematici (1999)

  • Volume: 72, Issue: 2, page 181-190
  • ISSN: 0066-2216

Abstract

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We prove an existence theorem of Cauchy-Kovalevskaya type for the equation D t u ( t , z ) = f ( t , z , u ( α ( 0 ) ( t , z ) ) , D z u ( α ( 1 ) ( t , z ) ) , . . . , D z k u ( α ( k ) ( t , z ) ) ) where f is a polynomial with respect to the last k variables.

How to cite

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Augustynowicz, Antoni. "On a theorem of Cauchy-Kovalevskaya type for a class of nonlinear PDE's of higher order with deviating arguments." Annales Polonici Mathematici 72.2 (1999): 181-190. <http://eudml.org/doc/262738>.

@article{Augustynowicz1999,
abstract = {We prove an existence theorem of Cauchy-Kovalevskaya type for the equation $D_t u(t,z) = f(t,z,u(α^\{(0)\}(t,z)), D_z u(α^\{(1)\}(t,z)),...,D_z^k u(α^\{(k)\}(t,z)))$ where f is a polynomial with respect to the last k variables.},
author = {Augustynowicz, Antoni},
journal = {Annales Polonici Mathematici},
keywords = {nonlinear equation; deviating argument; analytic solution; Cauchy-Kovalevskaya theorem; polynomial nonlinearity},
language = {eng},
number = {2},
pages = {181-190},
title = {On a theorem of Cauchy-Kovalevskaya type for a class of nonlinear PDE's of higher order with deviating arguments},
url = {http://eudml.org/doc/262738},
volume = {72},
year = {1999},
}

TY - JOUR
AU - Augustynowicz, Antoni
TI - On a theorem of Cauchy-Kovalevskaya type for a class of nonlinear PDE's of higher order with deviating arguments
JO - Annales Polonici Mathematici
PY - 1999
VL - 72
IS - 2
SP - 181
EP - 190
AB - We prove an existence theorem of Cauchy-Kovalevskaya type for the equation $D_t u(t,z) = f(t,z,u(α^{(0)}(t,z)), D_z u(α^{(1)}(t,z)),...,D_z^k u(α^{(k)}(t,z)))$ where f is a polynomial with respect to the last k variables.
LA - eng
KW - nonlinear equation; deviating argument; analytic solution; Cauchy-Kovalevskaya theorem; polynomial nonlinearity
UR - http://eudml.org/doc/262738
ER -

References

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  1. [1] A. Augustynowicz, Existence and uniqueness of solutions for partial differential-functional equations of the first order with deviating arguments of the derivative of unknown function, Serdica Math. J. 23 (1997), 203-210. Zbl0978.35086
  2. [2] A. Augustynowicz, Analytic solutions to the first order partial differential equations with time delays at the derivatives, Funct. Differ. Equations 6 (1999), 19-29. Zbl1027.35144
  3. [3] A. Augustynowicz and H. Leszczyński, On the existence of analytic solutions of the Cauchy problem for first-order partial differential equations with retarded variables, Comment. Math. Prace Mat. 36 (1996), 11-25. Zbl0877.35130
  4. [4] A. Augustynowicz and H. Leszczyński, On x-analytic solutions to the Cauchy problem for partial differential equations with retarded variables, Z. Anal. Anwendungen 15 (1996), 345-356. Zbl0847.35029
  5. [5] A. Augustynowicz and H. Leszczyński, Periodic solutions to the Cauchy problem for PDEs with retarded variables, submitted. 
  6. [6] A. Augustynowicz, H. Leszczyński and W. Walter, Cauchy-Kovalevskaya theory for equations with deviating variables, Aequationes Math. 58 (1999), 143-156. Zbl0929.35004
  7. [7] A. Augustynowicz, H. Leszczyński and W. Walter, Cauchy-Kovalevskaya theory for nonlinear equations with deviating variables, Nonlinear Anal., to appear. Zbl0989.35132
  8. [8] S. von Kowalevsky, Zur Theorie der partiellen Differentialgleichungen, J. Reine Angew. Math. 80 (1875), 1-32. 
  9. [9] H. Leszczyński, Fundamental solutions to linear first-order equations with a delay at derivatives, Boll. Un. Mat. Ital. A (7) 10 (1996), 363-375. Zbl0865.35137
  10. [10] M. Nagumo, Über das Anfangswertproblem partieller Differentialgleichungen, Japan. J. Math. 18 (1942), 41-47. Zbl0061.21107
  11. [11] R. M. Redheffer and W. Walter, Existence theorems for strongly coupled systems of partial differential equations over Bernstein classes, Bull. Amer. Math. Soc. 82 (1976), 899-902. Zbl0349.35015
  12. [12] W. Walter, An elementary proof of the Cauchy-Kovalevsky Theorem, Amer. Math. Monthly 92 (1985), 115-126. Zbl0576.35002
  13. [13] W. Walter, Functional differential equations of the Cauchy-Kovalevsky type, Aequationes Math. 28 (1985), 102-113. Zbl0576.35003
  14. [14] T. Yamanaka, A Cauchy-Kovalevskaja type theorem in the Gevrey class with a vector-valued time variable, Comm. Partial Differential Equations 17 (1992), 1457-1502. Zbl0816.35013
  15. [15] T. Yamanaka and H. Tamaki, Cauchy-Kovalevskaya theorem for functional partial differential equations, Comment. Math. Univ. St. Paul. 29 (1980), 55-64. Zbl0437.35003

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