PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1997 | 67 | 3 | 215-232
Tytuł artykułu

On the local Cauchy problem for nonlinear hyperbolic functional differential equations

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We consider the local initial value problem for the hyperbolic partial functional differential equation of the first order
(1) $Dₓz(x,y) = f(x,y,z(x,y),(Wz)(x,y),D_y z(x,y))$ on E,
(2) z(x,y) = ϕ(x,y) on [-τ₀,0]×[-b,b],
where E is the Haar pyramid and τ₀ ∈ ℝ₊, b = (b₁,...,bₙ) ∈ ℝⁿ₊. Using the method of bicharacteristics and the method of successive approximations for a certain functional integral system we prove, under suitable assumptions, a theorem on the local existence of weak solutions of the problem (1),(2).
Rocznik
Tom
67
Numer
3
Strony
215-232
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-11-20
poprawiono
1997-03-10
Twórcy
  • Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland
Bibliografia
  • [1] P. Bassanini, On a recent proof concerning a boundary value problem for quasilinear hyperbolic systems in the Schauder canonic form, Boll. Un. Mat. Ital. A (5) 14 (1977), 325-332.
  • [2] P. Bassanini, Iterative methods for quasilinear hyperbolic systems, Boll. Un. Mat. Ital. B (6) 1 (1982), 225-250.
  • [3] P. Bassanini and J. Turo, Generalized solutions of free boundary problems for hyperbolic systems of functional partial differential equations, Ann. Mat. Pura Appl. 156 (1990), 211-230.
  • [4] P. Brandi and R. Ceppitelli, On the existence of solutions of a nonlinear functional partial differential equation of the first order, Atti Sem. Mat. Fis. Univ. Modena 29 (1980), 166-186.
  • [5] P. Brandi and R. Ceppitelli, Existence, uniqueness and continuous dependence for a hereditary nonlinear functional partial differential equations, Ann. Polon. Math. 47 (1986), 121-136.
  • [6] P. Brandi, Z. Kamont and A. Salvadori, Existence of weak solutions for partial differential-functional equations, to appear.
  • [7] M. G. Cazzani-Nieri, Un problema ai limiti per sistemi integrodifferenziali non lineari di tipo iperbolico, Ann. Mat. Pura Appl. 157 (1994), 351-387.
  • [8] L. Cesari, A boundary value problem for quasilinear hyperbolic systems in the Schauder canonic form, Ann. Scuola Norm. Sup. Pisa (4) 1 (1974), 311-358.
  • [9] L. Cesari, A boundary value problem for quasilinear hyperbolic systems, Riv. Mat. Univ. Parma 3 (1974), 107-131.
  • [10] M. Cinquini Cibrario, Nuove ricerche sui sistemi di equazioni non lineari a derivate parziali in più variabili indipendenti, Rend. Sem. Mat. Fis. Univ. Milano 52 (1982), 531-550.
  • [11] M. Cinquini Cibrario, Teoremi di esistenza per sistemi di equazioni non lineari a derivate parziali in più variabili indipendenti, Rend. Ist. Lombardo 104 (1970), 795-829.
  • [12] M. Cinquini Cibrario, Sopra una classe di sistemi di equazioni non lineari a derivate parziali in più variabili indipendenti, Ann. Mat. Pura Appl. 140 (1985), 223-253.
  • [13] T. Człapiński, On the existence of generalized solutions of nonlinear first order partial differential-functional equations in two independent variables, Czechoslovak Math. J. 41 (1991), 490-506.
  • [14] T. Człapiński, On the Cauchy problem for quasilinear hyperbolic systems of partial differential-functional equations of the first order, Z. Anal. Anwendungen 10 (1991), 169-182.
  • [15] D. Jaruszewska-Walczak, Existence of solutions of first order partial differential-functional equations, Boll. Un. Mat. Ital. B (7) 4 (1990), 57-82.
  • [16] Z. Kamont and S. Zacharek, On the existence of weak solutions of nonlinear first order partial differential equations in two independent variables, Boll. Un. Mat. Ital. B (6) 5 (1986), 851-879.
  • [17] T. Kiguradze, Some boundary value problems for systems of linear partial differential equations of hyperbolic type, Mem. Differential Equations Math. Phys. 1 (1994), 1-144.
  • [18] A. D. Myshkis and A. S. Slopak, A mixed problem for systems of differential-functional equations with partial derivatives and with operators of Volterra type, Mat. Sb. 41 (1957), 239-256 (in Russian).
  • [19] O. A. Oleĭnik, Discontinuous solutions of nonlinear differential equations, Uspekhi Mat. Nauk 12 (3) (1957), 3-73 (in Russian).
  • [20] A. Salvadori, Sul problema di Cauchy per una struttura ereditaria di tipo iperbolico. Esistenza, unicità e dipendenza continua, Atti Sem. Mat. Fis. Univ. Modena 32 (1983), 329-356.
  • [21] J. Szarski, Characteristics and Cauchy problems for nonlinear partial differential functional equations of first order, Univ. Kansas, Lawrence, Kan., 1959.
  • [22] J. Turo, On some class of quasilinear hyperbolic systems of partial differential-functional equations of the first order, Czechoslovak Math. J. 36 (1986), 185-197.
  • [23] T. Ważewski, Sur l'appréciation du domaine d'existence des intégrales de l'équation aux dérivées partielles du premier ordre, Ann. Soc. Polon. Math. 14 (1935), 149-177.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-apmv67z3p215bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.