Damped solutions of singular IVPS with Φ-Laplacian

• Autorzy: Stryja J.

Warianty tytułu

CS

Tlumená řešení singulární úlohy s Φ-Laplaciánem

• Języki publikacji: EN

Abstrakty

EN

We study analytical properties of a singular nonlinear ordinary differential equation with a Φ-Laplacian. We investigate solutions of the initial value problem (p(t)Φ(u’(t)))’ + p(t)f(Φ(u(t))) = 0, u(0) = uₒ ϵ [Lₒ,L], u’(0) = 0 on the half-line [0,∞). Here, f is a continuous function with three zeros, function p is positive on (0,∞) and p(0) = 0. The integral ∫₀¹dsds/p(s) may be divergent which yields the time singularity at t = 0. Our equation generalizes equations which appear in hydrodynamics or in the nonlinear field theory.

CS

Budeme se zabývat chováním rešení singulární obycejné diferenciální rovnice druhého rádu s Φ-Laplaciánem (p(t)Φ(u’(t)))’ + p(t)f(Φ(u(t))) = 0, u(0) = uₒ ϵ [Lₒ,L], u’(0) = 0 na poloprímce [0, ∞) za pocátecních podmínek u(0) = uₒ ϵ [Lₒ,L], u’(0) = 0 Funkce f je spojitá a má tri nulové body, funkce p je kladná na (0,1)a dále platí p(0) = 0. Integrál ∫₀¹ds/p(s)) muže být divergentní, což zpusobuje singularitu v t = 0. Naše rovnice zobecnuje rovnice vyskytující se v modelech napríklad v hydrodynamice nebo v nelineární teorii pole.

Słowa kluczowe

EN

second order ODE; time singularity; existence and uniqueness; Φ-Laplacian; damped solution

PL

równanie różniczkowe drugiego rzędu; osobliwość czasu; istnienie i jednoznaczność; Φ-Laplacian

• Wydawca: STE GROUP
• Czasopismo: Systemy Wspomagania w Inżynierii Produkcji
• Rocznik: 2017
• Tom: Vol. 6, iss. 4
• Strony: 306--313
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Stryja J.

• VŠB - Technical University of Ostrava, Department of Mathematics and Descriptive Geometry, 17. listopadu 15, 708 33, Ostrava, Czech Republic, tel.: [phone]

Bibliografia

• 1. J. Burkotová, M. Hubner, I. Rachunková, E. B. Weinmüller. “Asymptotic properties of Kneser solutions to nonlinear second order ODEs with regularly varying coefficients”, J. Appl. Math. Comp., Vol. 274, 2015, p. 65–82.
• 2. J. Burkotová, I. Rachunková, M. Rohleder, J. Stryja. “Existence and uniqueness of damped solutions of singular IVPs with _-Laplacian”, Electron. J. Qual. Theory Differ. Equ., Vol. 2016, No. 121, 2016, p. 1–28.
• 3. J. Burkotová, M. Rohleder, J. Stryja. “On the existence and properties of three types of solutions of singular IVPs”, Electron. J. Qual. Theory Differ. Equ. Vol. 2015, No. 29, 2015, p. 1–25.
• 4. Z. Došlá, M. Marini, S. Matucci. “A boundary value problem on a half-line for differential equations with indefinite weight”, Commun. Appl. Anal., Vol. 15, 2011, p. 341–352.
• 5. J. Jaroš, T. Kusano, J. V. Manojlovic. “Asymptotic analysis of positive solutions of generalized Emden-Fowler differential equations in the framework of regular variation”, Cent. Eur. J. Mathc., Vol. 11, No. 12, 2013, p. 2215–2233.
• 6. S. Matucci, P. Rehák. “Asymptotics of decreasing solutions of coupled p-Laplacian systems in the framework of regular variation”, Ann. Mat. Pura Appl., Vol. 193, 2014, p. 837–858.
• 7. I. Rachunková, L. Rachunek, J. Tomecek. “Existence of oscillatory solutions of singular nonlinear differential equations”, Abstr. Appl. Anal., Vol. 2011, p. 1–20.
• 8. I. Rachunková, J. Tomecek. “Homoclinic solutions of singular nonautonomous second order differential equations”, Bound. Value Probl., Vol. 2009, p. 1–21.
• 9. I. Rachunková, J. Tomecek. “Strictly increasing solutions of a nonlinear singular differential equation arising in hydrodynamics”, Nonlinear Anal., Vol. 72, 2010, p. 2114–2118.
• 10. I. Rachunková, J. Tomecek, J. Stryja. “Oscillatory solutions of singular equations arising in hydrodynamics”, Adv. Difference Equ., Vol. 2010, p. 1–13.
• 11. M. Rohleder. “On the existence of oscillatory solutions of the second order nonlinear ODE” Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. Vol. 51, No. 2, 2012, p. 107–127.
• 12. J. Vampolová. “On existence and asymptotic properties of Kneser solutions to singular second order ODE”, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., Vol 52, No. 1, 2013, p. 135–152.