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

Annales Polonici Mathematici (1999)

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

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topAugustynowicz, 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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- [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] 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
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- [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] M. Nagumo, Über das Anfangswertproblem partieller Differentialgleichungen, Japan. J. Math. 18 (1942), 41-47. Zbl0061.21107
- [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] W. Walter, An elementary proof of the Cauchy-Kovalevsky Theorem, Amer. Math. Monthly 92 (1985), 115-126. Zbl0576.35002
- [13] W. Walter, Functional differential equations of the Cauchy-Kovalevsky type, Aequationes Math. 28 (1985), 102-113. Zbl0576.35003
- [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] 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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