Author: Z. Kamont 1
View More View Less
  • 1 Institute of Mathematics, University of Gdańsk, Wit Stwosz Street 57, 80-952 Gdańsk, Poland
Restricted access

Abstract

A generalized Cauchy problem for almost linear hyperbolic functional differential systems is considered. A theorem on the global existence of classical solutions is proved. It is shown a result on the differentiability of solutions with respect to initial functions. A method of characteristics and integral functional inequalities are used.

  • [1] Besala, P. 1991 Observations on quasilinear partial differential equations Ann. Polon. Math. 53 267283.

  • [2] Brandi, P., Ceppitelli, R. 1986 Existence, uniqueness and continuous dependence for a first order nonlinear partial differential equations in a hereditary structure Ann. Polon. Math. 47 121135.

    • Search Google Scholar
    • Export Citation
  • [3] Brandi, P., Salvadori, A., Kamont, Z. 2002 Existence of generalized solutions of hyperbolic functional differential equations Nonl. Anal., TMA 50 919940 .

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [4] Cinquini Cibrario, M. 1985 Sopra una class di sistemi de equazioni nonlineari a derivate parziali in piú variabili indipendenti Ann. Mat. Pura et Appl. 140 223253 .

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [5] Cinquini, S. 1979 Sopra i sistemi iperbolici di equazion a derivate parzaili (nonlineari) in piú variabili indipendenti Ann. Mat. Pura et Appl. 120 201214 .

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [6] Crandal, M. G., Evans, L. C., Lions, P. L. 1984 Some properties of viscosity solutions of Hamilton–Jacobi equations Trans. Amer. Math. Soc. 282 487502 .

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [7] Crandal, M. G., Lions, P. L. 1983 Viscosity solutions of Hamilton–Jacobi equations Trans. Amer. Math. Soc. 277 142 .

  • [8] Czernous, W. 2006 Generalized solutions of mixed problems for first order partial functional differential equations Ukrainian Math. Journ. 58 904936 .

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [9] Człapiński, T. 1991 On the existence of generalized solutions of nonlinear first order partial differential functional equations in two independent variables Czechosl. Math. J. 41 490506.

    • Search Google Scholar
    • Export Citation
  • [10] Człapiński, T., Kamont, Z. 1993 Generalized solutions for quasilinear hyperbolic systems of partial differential functional equations J. Math. Anal. Appl. 172 353370 .

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [11] Jaruszewska-Walczak, D. 1990 Existence of solutions of first order partial differential-functional equations Boll. Un. Mat. Ita. 4-B 7 5782.

    • Search Google Scholar
    • Export Citation
  • [12] Kamont, Z. 2009 Classical solutions of hyperbolic functional differential systems Acta Math. Hungar. 124 301319 .

  • [13] Kamont, Z. 1999 Hyperbolic Functional Differential Inequalities Kluwer Acad. Publishers Dordrecht.

  • [14] Leszczyński, H. 1993 On the existence and uniqueness of weak solutions of the Cauchy problem for weakly coupled systems of non-linear partial differential-functional equations of the first order Boll. Un. Mat. Ital. 7-B 7 323354.

    • Search Google Scholar
    • Export Citation
  • [15] Puzniakowska-Gałuch, E. 2010 On the local Cauchy problem for first order partial differential functional equations Ann. Polon. Math. 98 3961 .

  • [16] Szarski, J. 1976 Generalized Cauchy problem for differential functional equations with first order partial derivatives Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 24 575580.

    • Search Google Scholar
    • Export Citation
  • [17] Ważewski, T. 1936 Sur le probléme de Cauchy relatif a une systéme d’équations aux dérivées partielles Ann. Soc. Math. Polon. 15 101127.

    • Search Google Scholar
    • Export Citation
  • [18] Topolski, K. A. 1999 On the existence of viscosity solutions for the functional-differential Cauchy problem Ann. Soc. Math. Polon., Comment. Math. 39 207223.

    • Search Google Scholar
    • Export Citation
  • [19] Topolski, K. A. 2008 On the vanishing viscosity method for first order differential-functional IBVP Czechosl. Math. J. 58 927947 .

  • [20] Turo, J. 1997 Mixed problems for quasilinear hyperbolic systems Nonl. Anal. TMA 30 23292340 .

  • [21] Umanaliev, M. I., Vied’, J. A. 1989 On differential equations with first order partial derivatives and with integral coefficients Differ. Urav. 25 465477 in Russian.

    • Search Google Scholar
    • Export Citation

Monthly Content Usage

Abstract Views Full Text Views PDF Downloads
Oct 2020 0 0 0
Nov 2020 0 0 0
Dec 2020 0 0 0
Jan 2021 0 0 0
Feb 2021 0 0 0
Mar 2021 0 0 0
Apr 2021 0 0 0