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

• Autorzy: Antoni Augustynowicz
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

nonlinear equation; deviating argument; analytic solution; Cauchy-Kovalevskaya theorem

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1999
• Tom: 72
• Numer: 2
• Strony: 181-190

Daty

wydano

1999

otrzymano

1998-11-30

poprawiono

1999-04-30

Twórcy

autor

Antoni Augustynowicz

• Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland

Bibliografia

• [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.
• [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.
• [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.
• [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.
• [5] A. Augustynowicz and H. Leszczyński, Periodic solutions to the Cauchy problem for PDEs with retarded variables, submitted.
• [6] A. Augustynowicz, H. Leszczyński and W. Walter, Cauchy-Kovalevskaya theory for equations with deviating variables, Aequationes Math. 58 (1999), 143-156.
• [7] A. Augustynowicz, H. Leszczyński and W. Walter, Cauchy-Kovalevskaya theory for nonlinear equations with deviating variables, Nonlinear Anal., to appear.
• [8] S. von Kowalevsky, Zur Theorie der partiellen Differentialgleichungen, J. Reine Angew. Math. 80 (1875), 1-32.
• [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.
• [10] M. Nagumo, Über das Anfangswertproblem partieller Differentialgleichungen, Japan. J. Math. 18 (1942), 41-47.
• [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.
• [12] W. Walter, An elementary proof of the Cauchy-Kovalevsky Theorem, Amer. Math. Monthly 92 (1985), 115-126.
• [13] W. Walter, Functional differential equations of the Cauchy-Kovalevsky type, Aequationes Math. 28 (1985), 102-113.
• [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.
• [15] T. Yamanaka and H. Tamaki, Cauchy-Kovalevskaya theorem for functional partial differential equations, Comment. Math. Univ. St. Paul. 29 (1980), 55-64.