On the local Cauchy problem for nonlinear hyperbolic functional differential equations

Tomasz Człapiński

Annales Polonici Mathematici (1997)

  • Volume: 67, Issue: 3, page 215-232
  • ISSN: 0066-2216

Abstract

top
We consider the local initial value problem for the hyperbolic partial functional differential equation of the first order (1) D ₓ z ( x , y ) = f ( x , y , z ( x , y ) , ( W z ) ( 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).

How to cite

top

Tomasz Człapiński. "On the local Cauchy problem for nonlinear hyperbolic functional differential equations." Annales Polonici Mathematici 67.3 (1997): 215-232. <http://eudml.org/doc/270287>.

@article{TomaszCzłapiński1997,
abstract = {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).},
author = {Tomasz Człapiński},
journal = {Annales Polonici Mathematici},
keywords = {functional differential equations; weak solutions; bicharacteristics; successive approximations; method of bicharacteristics; local existence of weak solutions},
language = {eng},
number = {3},
pages = {215-232},
title = {On the local Cauchy problem for nonlinear hyperbolic functional differential equations},
url = {http://eudml.org/doc/270287},
volume = {67},
year = {1997},
}

TY - JOUR
AU - Tomasz Człapiński
TI - On the local Cauchy problem for nonlinear hyperbolic functional differential equations
JO - Annales Polonici Mathematici
PY - 1997
VL - 67
IS - 3
SP - 215
EP - 232
AB - 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).
LA - eng
KW - functional differential equations; weak solutions; bicharacteristics; successive approximations; method of bicharacteristics; local existence of weak solutions
UR - http://eudml.org/doc/270287
ER -

References

top
  1. [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. Zbl0355.35059
  2. [2] P. Bassanini, Iterative methods for quasilinear hyperbolic systems, Boll. Un. Mat. Ital. B (6) 1 (1982), 225-250. Zbl0488.35056
  3. [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. Zbl0716.35088
  4. [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. Zbl0476.35073
  5. [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. Zbl0657.35124
  6. [6] P. Brandi, Z. Kamont and A. Salvadori, Existence of weak solutions for partial differential-functional equations, to appear. Zbl0737.35134
  7. [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. [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. Zbl0307.35063
  9. [9] L. Cesari, A boundary value problem for quasilinear hyperbolic systems, Riv. Mat. Univ. Parma 3 (1974), 107-131. Zbl0342.35036
  10. [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. Zbl0599.35028
  11. [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. Zbl0215.16202
  12. [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. [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. Zbl0797.35159
  14. [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. Zbl0763.35055
  15. [15] D. Jaruszewska-Walczak, Existence of solutions of first order partial differential-functional equations, Boll. Un. Mat. Ital. B (7) 4 (1990), 57-82. Zbl0705.35147
  16. [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. Zbl0629.35110
  17. [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. Zbl0819.35003
  18. [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. [19] O. A. Oleĭnik, Discontinuous solutions of nonlinear differential equations, Uspekhi Mat. Nauk 12 (3) (1957), 3-73 (in Russian). 
  20. [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. [21] J. Szarski, Characteristics and Cauchy problems for nonlinear partial differential functional equations of first order, Univ. Kansas, Lawrence, Kan., 1959. 
  22. [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. Zbl0612.35082
  23. [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. Zbl62.1283.03

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.