# Article

Full entry | PDF   (0.4 MB)
Keywords:
partial differential-functional equations; mixed problem; generalized solutions; local existence; bicharacteristics; successive approximations
Summary:
We consider the mixed problem for the hyperbolic partial differential-functional equation of the first order $D_xz(x,y) = f(x,y,z_{(x,y)}, D_yz(x,y)),$ where $z_{(x,y)} \: [-\tau ,0] \times [0,h] \rightarrow \mathbb{R}$ is a function defined by $z_{(x,y)}(t,s) = z(x+t, y+s)$, $(t,s) \in [-\tau ,0] \times [0,h]$. Using the method of bicharacteristics and the method of successive approximations for a certain integral-functional system we prove, under suitable assumptions, a theorem of the local existence of generalized solutions of this problem.
References:
[1] V. E. Abolina, A. D. Myshkis: Mixed problem for a semilinear hyperbolic system on a plane. Mat. Sb. 50 (1960), 423–442 (Russian).
[2] P. Bassanini: On a boundary value problem for a class of quasilinear hyperbolic systems in two independent variables. Atti Sem. Mat. Fis. Univ. Modena 24 (1975), 343–372. MR 0430543
[3] P. Bassanini: On a recent proof concerning a boundary value problem for quasilinear hyperbolic systems in the Schauder canonic form. Boll. Un. Mat. Ital. (5) 14-A (1977), 325–332. MR 0492913 | Zbl 0355.35059
[4] P. Bassanini: Iterative methods for quasilinear hyperbolic systems. Boll. Un. Mat. Ital. (6) 1-B (1982), 225–250. MR 0654933 | Zbl 0488.35056
[5] P. Bassanini, J. Turo: Generalized solutions of free boundary problems for hyperbolic systems of functional partial differential equations. Ann. Mat. Pura Appl. 156 (1990), 211–230. DOI 10.1007/BF01766980 | MR 1080217
[6] P. Brandi, R. Ceppitelli: Generalized solutions for nonlinear hyperbolic systems in hereditary setting, preprint.
[7] P. Brandi, Z. Kamont, A. Salvadori: Existence of weak solutions for partial differential-functional equations. (to appear).
[8] L. Cesari: A boundary value problem for quasilinear hyperbolic systems in the Schauder canonic form. Ann. Sc. Norm. Sup. Pisa (4) 1 (1974), 311–358. MR 0380132
[9] L. Cesari: A boundary value problem for quasilinear hyperbolic systems. Riv. Mat. Univ. Parma 3 (1974), 107–131. MR 0435616
[10] S. Cinquini: Nuove ricerche sui sistemi di equazioni non lineari a derivate parziali in più variabili indipendenti. Rend. Sem. Mat. Fis. Univ. Milano 52 (1982).
[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), 759–829. MR 0296485 | Zbl 0215.16202
[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. DOI 10.1007/BF01776851 | MR 1553456
[13] T. Człapiński: On the Cauchy problem for quasilinear hyperbolic systems of partial differential-functional equations of the first order. Zeit. Anal. Anwend. 10 (1991), 169–182. DOI 10.4171/ZAA/439
[14] T. Dzłapiński: On the mixed problem for quasilinear partial differential-functional equations of the first order. Zeit. Anal. Anwend. 16 (1997), 463–478. DOI 10.4171/ZAA/773
[15] T. Człapiński: Existence of generalized solutions for hyperbolic partial differential-functional equations with delay at derivatives. (to appear).
[16] Z. Kamont, K. Topolski: Mixed problems for quasilinear hyperbolic differential-functional systems. Math. Balk. 6 (1992), 313–324. MR 1203465
[17] A. D. Myshkis; A. M. Filimonov: Continuous solutions of quasilinear hyperbolic systems in two independent variables. Diff. Urav. 17 (1981), 488–500. (Russian) MR 0610510 | Zbl 1152.35071
[18] A. D. Myshkis, A. M. Filimonov: Continuous solutions of quasilinear hyperbolic systems in two independent variables. Proc. of Sec. Conf. Diff. Equat. and Appl., Rousse (1982), 524–529. (Russian)
[19] 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. MR 0831307 | Zbl 0612.35082
[20] J. Turo: Local generalized solutions of mixed problems for quasilinear hyperbolic systems of functional partial differential equations in two independent variables. Ann. Polon. Math. 49 (1989), 259–278. DOI 10.4064/ap-49-3-259-278 | MR 0997519 | Zbl 0685.35065

Partner of