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The Cauchy problem for a second-order nonlinear hyperbolic equation with initial data on a line of parabolicity

Published online by Cambridge University Press:  17 April 2009

John M.S. Rassias
John M.S. Rassias
Affiliation:
II Dervenakion Str, Daphne, Athens, Greece.
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Abstract

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In this paper we study the Cauchy problem for the second order nonlinear hyperbolic partial differential equation

with initial conditions

where

and |u|, |ux|, |uy| < ∞, y ≥ 0, r = r(x) ∈ C4(·), ν = ν(x) ∈ C4(·).

These conditions on k, H, f, r, and ν are assumed to be satisfied in some sufficiently small neighborhood of the segment I, y = 0, in the upper half-plane y > 0

This paper generalizes the results obtained by N.A. Lar'kin (Differencial'nye Uravnenija 8 (1972), 76–84), who has treated the special case H = H(x, y, u); that is, the quasi-linear hyperbolic equation (*).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 20 , Issue 3 , June 1979 , pp. 345 - 365
Copyright
Copyright © Australian Mathematical Society 1979

References

[1]Березин, И.С. [I.S. Berezin], “О задаче Коши для линейного уравнения второго порядка с начальными данными на линии параболичности” [On Cauchy's problem for linear equations of the second order with initial conditions on a parabolic line], Mat. Sb. 24 (66) (1949), 301–320.Google ScholarPubMed
[2]Bers, L., “On the continuation of a potential gas flow across the sonic line” (NACA Technical Note 2058, 1950).Google Scholar
[3]Birkhoff, Garrett, Rota, Gian-Carlo, Ordinary differential equations (Ginn, Boston, New York, Chicago, Atlanta, Dallas, Palo Alto, Toronto, 1962).Google Scholar
[4]Bitsadze, A.V., Equations of the mixed type (translated by Zador, P.. Macmillan, New York, 1964).Google Scholar
[5]Coddington, Earl A. and Levinson, Norman, Theory of ordinary differential equations (McGraw-Hill, New York, Toronto, London, 1955).Google Scholar
[6]Conti, Roberto, “Sul problema di Cauchy per l'equazione y 2αk 2(x, y)·zxx − zyy = f(x, y, z, zx, zy), con i dati sulla linea parabolica”, Ann. Mat. Pura Appl. 31 (1950), 303–326.CrossRefGoogle Scholar
[7]Courant, Richard and Lax, Peter, “On nonlinear partial differential equations with two independent variables”, Comm. Pure Appl. Math. 2 (1949), 255–273.CrossRefGoogle Scholar
[8]Франкль, Ф.И. [F.I. Frankl'], “ О задаче Коши для уравнений смешанного зллнптико-гиперболического типа с качальными данными на пеоеходиой линии” [On Cauchy's problem for partial differential equations of mixed elliptic-hyperbolic type with initial data on the parabolic line], Bull. Acad. Sci. URSS Ser. Math. [Izv. Akad. Nauk SSSR] 8 (1953), 195–224.Google ScholarPubMed
[9]Germain, P. et Bader, R., “Solutions éléementaires de certaines équations aux dérivées partielles du type mixte”, Bull. Soc. Math. France 81 (1953), 145–174.CrossRefGoogle Scholar
[10]Ларькин, Н.А. [N.A. Lar'kin], “О задаче Коши для квазилинейного гиперболического уравнения второго порядка с начальными данными на линии параболичности” [The Cauchy problem for a second order quasilinear hyperbolic equation with initial data on the curve of parabolicity], Differencial'nye Uravnenija 8 (1972), 76–84.Google ScholarPubMed
[11]Lick, Dale W., “A uniqueness theorem for a singular Cauchy problem”, Ann. Mat. Pura Appl. (4) 72 (1966), 267–274.CrossRefGoogle Scholar
[12]Lick, Dale W., “A quasi-linear singular Cauchy problem”, Ann. Mat. Pura Appl. (4) 74 (1966), 113–128.CrossRefGoogle Scholar
[13]Lick, Dale W., “A singular Cauchy problem”, J. Math. Anal. Appl. 28 (1969), 93–99.CrossRefGoogle Scholar
[14]Мередов, М. [M. Meredov], “ Об однозначной разрешимости задачи Дарбу для одной вырождающейся системы” [The unique solvability of the Darboux problem for a certain degenerate system], Differencial'nye Uravnenija 10 (1974), 89–99.Google ScholarPubMed
[15]Ogawa, Hajimu, “The singular Cauchy problem for a quasi-linear hyperbolic equation of second order”, J. Math. Meeh. 12 (1963), 847–856.Google Scholar
[16]Ogawa, Hajimu, “On the unique solution of the singular Cauchy problem for a quasi-linear equation”, Ann. Mat. Pura Appl. (4) 66 (1964), 391–403.CrossRefGoogle Scholar
[17]Ogawa, Hajimu, “A nonlinear singular Cauchy problem”, J. Math. Anal. Appl. 13 (1966), 527–535.CrossRefGoogle Scholar
[18]Protter, M.H., “The Cauchy problem for a hyperbolic second order equation with data on the parabolic line”, Canad. J. Math. 6 (1954), 542–553.CrossRefGoogle Scholar
[19]Schauder, J., “Der Fixpunktsatz in Funktionalräumen”, Studia Math. 2 (1930), 171–180.CrossRefGoogle Scholar
[20]Singer, Seymour, “The singular Cauchy problem for a nonlinear hyperbolic equation” (PhD thesis, University of California, Berkeley, 1969).Google Scholar
[21]Терсенов, С.А. [S.A. Tersenov], “ К теории гиперболических уравнений с данными на линии вырождения типа” [On the theory of hyperbolic equations with data on a line of degeneration type], Sibirsk. Mat. ž. 2 (1961), 913–935.Google ScholarPubMed