Previous |  Up |  Next

Article

Keywords:
viscosity solutions; first order equation; parabolic equation; differential functional equations
Summary:
We consider the initial-boundary value problem for first order differential-functional equations. We present the `vanishing viscosity' method in order to obtain viscosity solutions. Our formulation includes problems with a retarded and deviated argument and differential-integral equations.
References:
[1] Alvarez, O., Tourin, A.: Viscosity solutions of nonlinear integro-differential equations. Ann. Inst. H. Poincaré anal. Non Linaire 13 (1996), 293-317. MR 1395674 | Zbl 0870.45002
[2] Bardi, M., Crandall, M. G., Evans, L. C., Soner, H. M., Souganidis, P. E.: Viscosity Solutions and Applications. Springer-Verlag Berlin-Heidelberg-New York (1997).
[3] Brandi, P., Ceppitelli, R.: On the existance of solutions of nonlinear functional partial differential equations of the first order. Atti Sem. Mat. Fis. Univ. Modena 29 (1980), 166-186. MR 0632726
[4] Brandi, P., Ceppitelli, R.: Existence, uniqueness and continuous dependence for a hereditary nonlinear functional partial differential equation of the first order. Ann. Polon. Math. 47 (1986), 121-136. MR 0884930 | Zbl 0657.35124
[5] Brzychczy, S.: Chaplygin's method for a system of nonlinear parabolic differential-functional equations. Differen. Urav. 22 (1986), 705-708 Russian. MR 0843232
[6] Brzychczy, S.: Existence of solutions for non-linear systems of differential-functional equations of parabolic type in an arbitrary domain. Ann. Polon. Math. 47 (1987), 309-317. MR 0927579
[7] Crandall, M. G., Ishii, H., Lions, P. L.: User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992), 1-67. DOI 10.1090/S0273-0979-1992-00266-5 | MR 1118699 | Zbl 0755.35015
[8] Crandall, M. G., Lions, P. L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983), 1-42. DOI 10.1090/S0002-9947-1983-0690039-8 | MR 0690039 | Zbl 0599.35024
[9] Hale, J. K., Lunel, S. M. V.: Introduction to Functional Differential Equations. Springer-Verlag New York (1993). MR 1243878 | Zbl 0787.34002
[10] Ishii, H., Koike, S.: Viscosity solutions of functional differential equations. Adv. Math. Sci. Appl. Gakkotosho, Tokyo 3 (1993/94), 191-218. MR 1287929
[11] Jakobsen, E. R., Karlsen, K. H.: Continuous dependence estimates for viscosity solutions of integro-PDFs. J. Differential Equations 212 (2005), 278-318. DOI 10.1016/j.jde.2004.06.021 | MR 2129093
[12] Kamont, Z.: Initial value problems for hyperbolic differential-functional systems. Boll. Un. Mat. Ital. 171 (1994), 965-984. MR 1315829 | Zbl 0832.35144
[13] Kruzkov, S. N.: Generalized solutions of first order nonlinear equations in several independent variables I. Mat. Sb. 70 (1966), 394-415 II, Mat.Sb. (N.S) 72 (1967), 93-116 Russian. MR 0199543
[14] Ladyzhenskaya, O. A., Solonikov, V. A., Uralceva, N. N.: Linear and Quasilinear Equations of Parabolic Type. Nauka, Moskva (1967), Russian Translation of Mathematical Monographs, Vol. 23, Am. Math. Soc., Providence, R.I. (1968). MR 0241822
[15] Lions, P. L.: Generalized Solutions of Hamilton-Jacobi Equations. Pitman, London (1982). MR 0667669 | Zbl 0497.35001
[16] Sayah, A.: Equations d'Hamilton-Jacobi du premier ordre avec termes integro-differentiales, Parties I & II. Comm. Partial Differential Equations 16 (1991), 1057-1093. DOI 10.1080/03605309108820789 | MR 1116853
[17] Topolski, K.: On the uniqueness of viscosity solutions for first order partial differential-functional equations. Ann. Polon. Math. 59 (1994), 65-75. MR 1270302 | Zbl 0804.35138
[18] Topolski, K.: Parabolic differential-functional inequalities in a viscosity sense. Ann. Polon. Math. 68 (1998), 17-25. MR 1606607
[19] Topolski, K. A.: On the existence of classical solutions for differential-functional IBVP. Abstr. Appl. Anal. 3 (1998), 363-375. DOI 10.1155/S1085337598000608 | MR 1749416
[20] Topolski, K. A.: On the existence of viscosity solutions for the differential-functional Cauchy problem. Comment. Math. 39 (1999), 207-223. MR 1739030 | Zbl 0972.35173
[21] Topolski, Krzysztof A.: On the existence of viscosity solution for the parabolic differential-functional Cauchy problem. Preprint.
Partner of
EuDML logo