Classical solutions of mixed problems for quasilinear first order PFDEs on a cylindrical domain

• Autorzy: Czernous W.

Identyfikatory

DOI

http://dx.doi.org/10.7494/OpMath.2014.34.2.291

• Języki publikacji: EN

Abstrakty

EN

We abandon the setting of the domain as a Cartesian product of real intervals, customary for first order PFDEs (partial functional differential equations) with initial boundary conditions. We give a new set of conditions on the possibly unbounded domain Ω with Lipschitz differentiable boundary. Well-posedness is then reliant on a variant of the normal vector condition. There is a neighbourhood of ∂Ω with the property that if a characteristic trajectory has a point therein, then its every earlier point lies there as well. With local assumptions on coefficients and on the free term, we prove existence and Lipschitz dependence on data of classical solutions on (0,c)×Ω to the initial boundary value problem, for small c. Regularity of solutions matches this domain, and the proof uses the Banach fixed-point theorem. Our general model of functional dependence covers problems with deviating arguments and integro-differential equations.

Słowa kluczowe

EN

partial functional differential equations; classical solutions; local existence; characteristcs; cylindrical domain; a priori estimates

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2014
• Tom: Vol. 34, no. 2
• Strony: 291--310
• Opis fizyczny: Bibliogr. 29 poz., rys.

Twórcy

autor

Czernous W.

• University of Warmia and Mazury Faculty of Mathematics and Computer Science Sloneczna 54, 10-710 Olsztyn, Poland

Bibliografia

• [1] R.A. Adams, J.J.F. Fournier, Sobolev Spaces, Pure and Applied Mathematics 140, Academic Press, Amsterdam, 2003.
• [2] P. Bassanini, M.C. Galaverni, Un problema ai limiti per sistemi integrodifferenziali nonlineari di tipo iperbolico, Boll. Un. Mat. Ital. 7 (1981), 785–798, [in Italian].
• [3] P. Brandi, R. Ceppitelli, Existence, uniqueness and continuous dependence for a hereditary nonlinear functional partial differential equations, Ann. Polon. Math. 47 (1986), 121–136.
• [4] P. Brandi, Z. Kamont, A. Salvadori, Existence of generalized solutions of hyperbolic functional differential equations, Nonl. Anal. TMA 50 (2002), 919–940.
• [5] C.J. Chyan, G.F. Webb, A model of proliferating cell populations with correlation of mother – daughter mitotic times, Ann. Mat. Pura Appl. 157 (1991), 1–11.
• [6] S. Cinquini, On hyperbolic systems of (nonlinear) partial differential equations in several independent variables, Ann. Mat. Pura Appl. 120 (1979), 201–214, [in Italian].
• [7] M. Cinquini Cibrario, Teoremi di esistenza per sistemi di equazioni quasi lineari a derivate parziali in piú variabili indipendenti, Ann. Mat. Pura Appl. 75 (1967), 1–46 [in Italian].
• [8] M. Cinquini Cibrario, A class of systems of partial differential equations in many independent variables, Rend. Mat. 2 (1982) 3, 499–522, [in Italian, English summary].
• [9] W. Czernous, Infinite systems of first order PFDEs with mixed conditions, Ann. Polon. Math. 94 (2008) 3, 209–230.
• [10] W. Czernous, Classical solutions of hyperbolic IBVPs with state dependent delays,Neliniini Koliv. 13 (2010) 4, 556–573.
• [11] W. Czernous, Semilinear hyperbolic functional differential problem on a cylindrical domain, Bull. Belg. Math. Soc. Simon Stevin 19O (2012), 1–17.
• [12] W. Czernous, Classical solutions of hyperbolic IBVPs with state dependent delays on a cylindrical domain Nonlinear Analysis Series A: Theory, Methods & Applications 75 (2012) 17, 6325–6342.
• [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.
• [14] T. Człapiński, On the existence of generalized solutions of nonlinear first order partial differential functional equations in two independent variables, Czechosl. Math. Journ. 41 (1992), 490–506.
• [15] J.L. Doob, Measure Theory, Graduate Texts in Mathematics 140, Springer, New York, 1994.
• [16] W. Eichhorn, W. Gleissner, On a functional differential equation arising in the theory of distribution of wealth, Aequat. Math. 28 (1985), 190–198.
• [17] M. El-Doma, Analysis of nonlinear integro differential equations arising in age dependent epidemic models, Nonl. Anal. TMA 11 (1978), 913–937.
• [18] M.E. Gurtin, L.F. Murphy, On the optimal harvesting of persistent age – structured population, J. Math. Biol. 13 (1981), 131–148.
• [19] G.A. Kamenskii, A.D. Myshkis, Mixed functional-differential equations, Nonl. Anal TMA 30 (1997), 2577–2584.
• [20] Z. Kamont, Hyperbolic Functional Differential Inequalities and Applications, Kluwer Acad. Publ., Dordrecht, 1999.
• [21] Z. Kamont, Existence of solutions of first order partial differential functional equations, Ann. Soc. Math. Polon., Comm. Math. 25 (1985), 249–263.
• [22] A. Lasota, M.C. Mackey, M. Ważewska-Czyżewska, Minimizing therapeutically induced anemia, J. Math. Biol. 13 (1981), 149–158.
• [23] A. Lorenzi, An identification problem related to a nonlinear hyperbolic integro differential equation, Nonl. Anal. TMA 22 (1994), 21–44.
• [24] A.D. Myshkis, A.S. Slopak, A mixed problem for systems of differential functional equations with partial derivatives and with operators of the Volterra type, Mat. Sb. (N.S.) 41 (1957), 239–256 [in Russian].
• [25] E. Puźniakowska-Gałuch, Classical solutions of quasilinear functional differential systems on the Haar pyramid, Differ. Equ. Appl. 1 (2009) 2, 179–197.
• [26] K. Topolski, Generalized solutions of first order partial differential functional equations in two independent variables, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), 669–684.
• [27] J. Turo, Global solvability of the mixed problem for first order functional partial differential equations, Ann. Polon. Math. 52 (1991), 231–238.
• 28] T. Ważewski, Sur le probleème de Cauchy relatif a un système d’équations aux derives partielles, Ann. Soc. Math. Polon. 15 (1936), 101–127.
• [29] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, Berlin, 1996.