PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2014 | Vol. 34, no. 2 | 425--441
Tytuł artykułu

On the Tonelli method for the degenerate parabolic Cauchy problem with functional argument

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The degenerate parabolic Cauchy problem is considered. A functional argument in the equation is of the Hale type. As a limit of piecewise classical solutions we obtain a viscosity solution of the main problem. Presented method is an adaptation of Tonelli's constructive method to the partial differential-functional equation. It is also shown that this approach can be improved by the vanishing viscosity method and regularisation process.
Wydawca

Rocznik
Strony
425--441
Opis fizyczny
Bibliogr. 24 poz.
Twórcy
  • University of Gdansk Institute of Mathematics Wit Stwosz Street 57, 80-952 Gdansk, matkt@mat.ug.edu.pl
Bibliografia
  • [1] M. Bardi, M.G. Crandall, L.C. Evans, H.M. Soner, P.E. Souganidis, Viscosity Solutions and Applications, Springer-Verlag, Berlin Heidelberg New York, 1997.
  • [2] E. Baiada, C. Vinti, Un teorema d’esistenza della soluzione per un’equazione alle derivate parziali, del 1 ordine, Ann. Scuola. Norm. Sup. Pisa 9 (1955), 115–140.
  • [3] S. Cinquini, Tonelli’s constructive method in the study of partial differential equations, [in:] Contributions to Modern Calculus of Variations, L. Cesari (ed.), Pitman Research Notes in Mathematics Series 148, New York 1987, 54–79.
  • [4] M. Cinquini Cibrario, Teoremi di esistenza per sistemi semilineari di equazioni quasi lineari a derivate parziali in piu variabili independenti, Ann. Mat. Pura Appl. (4) 44 (1957), 357–418.
  • [5] R. Conti, Sul problema du Darboux per l’equazione zxy = f(x; y; z; zx; zy), Ann. Univ. Ferrara 2 (1953), 129–140.
  • [6] M.G. Crandall, H. Ishii, P.L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 1–67.
  • [7] M.G. Crandall, P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1–42.
  • [8] M.G. Crandall, P.L. Lions, On the existence and uniqueness of solutions of Hamilton-Jacobi equations, Nonlinear Anal. 10 (1986), 353–370.
  • [9] L.C. Evans, Partial Differential Equations, AMS, 1998.
  • [10] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., 1964.
  • [11] J.K Hale, S.M.V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
  • [12] H. Ishii, S. Koike, Viscosity solutions of functional differential equations, Adv. Math.Sci. Appl. Gakkotosho, Tokyo, 3 (1993/94), 191–218.
  • [13] Z. Kamont, Existence of solutions to Hamilton-Jackobi functional differential equations, Nonl. Anal. T.M.& A. 73 (2010), 767–778.
  • [14] A. Karpowicz, The maximum principle for viscosity solutions of elliptic differential functional equations, Opusc. Math. 33 (2013) 1, 99–105.
  • [15] O.A. Ladyzhenskaya, V.A. Solonikov, N.N. Uralceva, Linear and Quasilinear Equation of Parabolic Type, Nauka, Moskva, 1967 [in Russian]. (Translation of Mathematical Monographs, Vol.23, Am. Math. Soc., Providence, R.I., 1968.)
  • [16] J. Szarski, Sur un systèm non linéaire d’inéqualités différentialles paraboliques contenant des functionnelles, Colloq. Math. 16 (1967), 141–145.
  • [17] L. Tonelli, Sulle equazioni integrali di Voltera. Bull. Calcutta. Math. Soc. 20 (1928), 31–48. [Opere scelte, 4, 198-212.]
  • [18] K. Topolski, On the uniqueness of viscosity solutions for first order partial differential-functional equations, Ann. Polon. Math. 59 (1994), 65–75.
  • [19] K.A. Topolski, Parabolic differential-functional inequalities in a viscosity sense, Ann. Polon. Math. 68 (1998), 17–25.
  • [20] K.A. Topolski, On the classical solutions for parabolic differential-functional Cauchy problem, Comment. Math. Prace Mat. 44 (2004) 2, 217–226.
  • [21] K.A. Topolski, On the vanishing viscosity method for first order differential-functional IBVP, Czechoslovak Math. J. 58 (133) 4 (2008), 927–947.
  • [22] K.A. Topolski, On the existence of viscosity solutions for the parabolic differential--functional Cauchy problem, Acta Math. Hungar. 129 (2010) 3, 277–296.
  • [23] P.K.C. Wang, Asymptotic stability of a time-delayed diffusion system, J. Appl. Mech. 30 (1963), 500–504.
  • [24] J. Wu, Theory and Applications of Partial Functional Differential Equations,Springer-Verlag New York, Inc., 1996.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-1e420cde-dba4-4173-a50b-526fa83d7e42
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.