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# Article

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Keywords:
singular ordinary differential equation of the second order; time singularities; unbounded domain; asymptotic properties; Kneser solutions; damped solutions; non-oscillatory solutions
Summary:
We investigate an asymptotic behaviour of damped non-oscillatory solutions of the initial value problem with a time singularity $\left( p(t)u^{\prime }(t) \right)^{\prime } + p(t)f ( u(t) )=0$, $u(0)=u_0$, $u^{\prime }(0)=0$ on the unbounded domain $[0,\infty )$. Function $f$ is locally Lipschitz continuous on $\mathbb {R}$ and has at least three zeros $L_0 <0$, $0$ and $L>0$. The initial value $u_0\in (L_0, L)\setminus \lbrace 0\rbrace$. Function $p$ is continuous on $[0,\infty ),$ has a positive continuous derivative on $(0,\infty )$ and $p(0)=0$. Asymptotic formulas for damped non-oscillatory solutions and their first derivatives are derived under some additional assumptions. Further, we provide conditions for functions $p$ and $f$, which guarantee the existence of Kneser solutions.
References:
[1] Abraham, F. F.: Homogeneous Nucleation Theory. Acad. Press, New York, 1974.
[2] Bartušek, M., Cecchi, M., Došlá, Z., Marini, M.: On oscillatory solutions of quasilinear differential equations. J. Math. Anal. Appl. 320 (2006), 108–120. DOI 10.1016/j.jmaa.2005.06.057 | MR 2230460 | Zbl 1103.34016
[3] Bongiorno, V., Scriven, L. E., Davis, H. T.: Molecular theory of fluid interfaces. J. Colloid and Interface Science 57 (1967), 462–475. DOI 10.1016/0021-9797(76)90225-3
[4] Cecchi, M., Marini, M., Villari, G.: On some classes of continuable solutions of a nonlinear differential equation. J. Differential Equations 118 (1995), 403–419. DOI 10.1006/jdeq.1995.1079 | MR 1330834 | Zbl 0827.34020
[5] Cecchi, M., Marini, M., Villari, G.: Comparison results for oscillation of nonlinear differential equations. NoDea 6 (1999), 173–190. DOI 10.1007/s000300050071 | MR 1694795 | Zbl 0927.34023
[6] Derrick, G. H.: Comments on nonlinear wave equations as models for elementary particles. J. Math. Physics 5 (1965), 1252–1254. DOI 10.1063/1.1704233 | MR 0174304
[7] Fife, P. C.: Mathematical Aspects of Reacting and Diffusing Systems. Lecture notes in Biomathematics Springer 28 (1979), 223–224. MR 0527914 | Zbl 0403.92004
[8] Fischer, R. A.: The wave of advance of advantageous genes. Journ. of Eugenics 7 (1937), 355–369. DOI 10.1111/j.1469-1809.1937.tb02153.x
[9] Gouin, H., Rotoli, G.: An analytical approximation of density profile and surface tension of microscopic bubbles for Van der Waals fluids. Mech. Research Communic. 24 (1997), 255–260. DOI 10.1016/S0093-6413(97)00022-0 | Zbl 0899.76064
[10] Ho, L. F.: Asymptotic behavior of radial oscillatory solutions of a quasilinear elliptic equation. Nonlinear Analysis 41 (2000), 573–589. DOI 10.1016/S0362-546X(98)00298-3 | MR 1780633 | Zbl 0962.34019
[11] Jaroš, J., Kusano, T., Tanigawa, T.: Nonoscillatory half-linear differential equations and generalized Karamata functions. Nonlinear Analysis 64 (2006), 762–787. DOI 10.1016/j.na.2005.05.045 | MR 2197094 | Zbl 1103.34017
[12] Kiguradze, I., Chanturia, T.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Acad. Publ., Dordrecht, 1993. MR 1220223 | Zbl 0782.34002
[13] Kulenović, M. R. S., Ljubović, Ć.: All solutions of the equilibrium capillary surface equation are oscillatory. Applied Mathematics Letters 13 (2000), 107–110. DOI 10.1016/S0893-9659(00)00041-0 | MR 1760271
[14] Kusano, T., Manojlović, J. V.: Asymptotic analysis of Emden-Fowler differential equations in the framework of regular variation. Annali di Matematica Pura ed Applicata 190 (2011), 619–644. DOI 10.1007/s10231-010-0166-x | MR 2861062 | Zbl 1245.34039
[15] Kwong, M. K., Wong, J. S. W.: A nonoscillation theorem for sublinear Emden-Fowler equations. Nonlinear Analysis 64 (2006), 1641–1646. DOI 10.1016/j.na.2005.07.015 | MR 2200164 | Zbl 1099.34033
[16] Kwong, M. K., Wong, J. S. W.: A nonoscillation theorem for superlinear Emden–Fowler equations with near-critical coefficients. J. Differential Equations 238 (2007), 18–42. DOI 10.1016/j.jde.2007.03.021 | MR 2334590 | Zbl 1125.34023
[17] Li, W. T.: Oscillation of certain second-order nonlinear differential equations. J. Math. Anal. Appl. 217 (1998), 1–14. DOI 10.1006/jmaa.1997.5680 | Zbl 0893.34023
[18] Lima, P. M., Chemetov, N. V., Konyukhova, N. B., Sukov, A. I.: Analytical–numerical investigation of bubble-type solutions of nonlinear singular problems. J. Comp. Appl. Math. 189 (2006), 260–273. DOI 10.1016/j.cam.2005.05.004 | MR 2202978 | Zbl 1100.65066
[19] Linde, A. P.: Particle Physics and Inflationary Cosmology. Harwood Academic, Chur, Switzerland, 1990.
[20] Ou, C. H., Wong, J. S. W.: On existence of oscillatory solutions of second order Emden–Fowler equations. J. Math. Anal. Appl. 277 (2003), 670–680. DOI 10.1016/S0022-247X(02)00617-0 | MR 1961253 | Zbl 1027.34039
[21] O’Regan, D.: Existence theory for nonlinear ordinary differential equations. Kluwer, Dordrecht, 1997. MR 1449397 | Zbl 1077.34505
[22] Rachůnková, I., Rachůnek, L.: Asymptotic formula for oscillatory solutions of some singular nonlinear differential equation. Abstract and Applied Analysis 2011 (2011), 1–9. Zbl 1222.34034
[23] Rachůnková, I., Tomeček, J.: Bubble-type solutions of nonlinear singular problem. Mathematical and Computer Modelling 51 (2010), 658–669. DOI 10.1016/j.mcm.2009.10.042
[24] Rachůnková, I., Rachůnek, L., Tomeček, J.: Existence of oscillatory solutions of singular nonlinear differential equations. Abstract and Applied Analysis 2011 (2011), 20 pages. MR 2795071 | Zbl 1222.34035
[25] Rachůnková, I., Tomeček, J.: Strictly increasing solutions of a nonlinear singular differential equation arising in hydrodynamics. Nonlinear Analysis 72 (2010), 2114–2118. DOI 10.1016/j.na.2009.10.011 | MR 2577608 | Zbl 1186.34014
[26] Rachůnková, I., Tomeček, J.: Homoclinic solutions of singular nonautonomous second order differential equations. Boundary Value Problems 2009 (2009), 1–21. Zbl 1190.34028
[27] Rohleder, M.: On the existence of oscillatory solutions of the second order nonlinear ODE. Acta Univ. Palacki. Olomuc., Fac. rer. nat., Math. 51, 2 (2012), 107–127. MR 3058877 | Zbl 1279.34050
[28] van der Waals, J. D., Kohnstamm, R.: Lehrbuch der Thermodynamik. 1, Leipzig, 1908.
[29] Wong, J. S. W.: Second–order nonlinear oscillations: A case history. In: Proceedings of the Conference on Differential & Difference Equations and Applications Hindawi (2006), 1131–1138. MR 2309447 | Zbl 1147.34024
[30] Wong, P. J. Y., Agarwal, R. P.: Oscillatory behavior of solutions of certain second order nonlinear differential equations. J. Math. Anal. Appl. 198 (1996), 337–354. DOI 10.1006/jmaa.1996.0086 | MR 1376268 | Zbl 0855.34039
[31] Wong, P. J. Y., Agarwal, R. P.: The oscillation and asymptotically monotone solutions of second order quasilinear differential equations. Appl. Math. Comput. 79 (1996),207–237. MR 1407599

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