- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2011 (2011), Article ID 981401, 9 pages
Asymptotic Formula for Oscillatory Solutions of Some Singular Nonlinear Differential Equation
Department of Mathematics, Faculty of Science, Palacký University, 17. Listopadu 12, 771 46 Olomouc, Czech Republic
Received 28 October 2010; Revised 31 March 2011; Accepted 2 May 2011
Academic Editor: Yuri V. Rogovchenko
Copyright © 2011 Irena Rachůnková and Lukáš Rachůnek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Singular differential equation is investigated. Here is Lipschitz continuous on ℝ and has at least two zeros 0 and . The function is continuous on [0,∞) and has a positive continuous derivative on (0,∞) and . An asymptotic formula for oscillatory solutions is derived.
In this paper, we investigate the equation where satisfies and fulfils Equation (1.1) is a generalization of the equation which arises for and special forms of in many areas, for example: in the study of phase transitions of Van der Waals fluids [1–3], in population genetics, where it serves as a model for the spatial distribution of the genetic composition of a population [4, 5], in the homogeneous nucleation theory , in the relativistic cosmology for the description of particles which can be treated as domains in the universe , in the nonlinear field theory, in particular, when describing bubbles generated by scalar fields of the Higgs type in the Minkowski spaces . Numerical simulations of solutions of (1.1), where is a polynomial with three zeros have been presented in [9–11]. Close problems about the existence of positive solutions can be found in [12–14].
Due to , (1.1) has a singularity at .
2. Existence of Oscillatory Solutions
Here, we will study solutions of (1.1) satisfying the initial conditions with a parameter . Reason is that we focus our attention on damped solutions of (1.1) and that each solution of (1.1) must fulfil (see ).
First, we bring two theorems about the existence of damped and oscillatory solutions.
Theorem 2.2. Assume that (1.2)–(1.6) hold. Further, let there exists such that Then for each problem (1.1), (2.1) has a unique solution . If , then the solution is damped and oscillatory with decreasing amplitudes and
Proof. The assertion follows from Theorems 2.3, 2.10 and 3.1 in .
Example 2.3. The functions (i), , , (ii), , (iii)satisfy (1.5), (1.6), and (2.2).
The functions (i) satisfy (1.5), (1.6), but not (2.2) (the third condition).
The function (i), , , satisfy (1.5), (1.6) but not (2.2) (the second and third conditions).
In the next section, the generalized Matell's theorem which can be found as Theorem 6.5 in the monograph by Kiguradze will be useful. For our purpose, we provide its following special case.
Consider an interval . We write for the set of functions absolutely continuous on and for the set of functions belonging to for each compact interval . Choose and a function matrix which is defined on . Denote by and eigenvalues of . Further, suppose be different eigenvalues of the matrix , and let and be eigenvectors of corresponding to and , respectively.
Theorem 2.5 (see ). Assume that and that there exists such that or Then the differential system has a fundamental system of solutions such that
3. Asymptotic Formula
In order to derive an asymptotic formula for a damped oscillatory solution of problem (1.1), (2.1), we need a little stronger assumption than (2.3). In particular, the function should have a negative derivative at .
Theorem 3.1. Assume that (1.2)–(1.6), and (2.2) hold. Assume, moreover, that there exist and such that Then for each problem (1.1), (2.1) has a unique solution . If , then the solution is damped and oscillatory with decreasing amplitudes such that
Proof. We have the following steps:Step 1 (construction of an auxiliary linear differential system). Choose . By Theorem 2.2, problem (1.1), (2.1) has a unique oscillatory solution with decreasing amplitudes and satisfying (2.4). Having this solution , define a linear differential equation and the corresponding linear differential system Denote By (1.6), (2.4), and (3.1), Eigenvalues of are numbers and , and eigenvectors of are and , respectively. Denote Then eigenvalues of have the form We see that Step 2 (verification of the assumptions of Theorem 2.5). Due to (1.6), (2.4), and (3.1), we can find such that Therefore, by (3.1), and so Further, by (2.2), . Hence, due to (1.6), Since and , we see that (2.8) is satisfied. Using (3.8) we get . This yields for any positive constant . Consequently (2.9) is valid.Step 3 (application of Theorem 2.5). By Theorem 2.5 there exists a fundamental system of solutions of (3.4) such that (2.12) is valid. Hence Using (3.8) and (3.10), we get and, hence, Similarly Therefore, (3.15) implies Step 4 (asymptotic formula). In Step 1, we have assumed that is a solution of (1.1), which means that Consequently and, hence, is also a solution of (3.3). This yields that there are such that . Therefore,
Example 3.4. Consider the initial problem Here and we can check that . Further, all assumptions of Theorems 2.2 and 3.1 are fulfilled. Therefore, by Theorem 2.2, there exists a unique solution of problem (3.24) which is damped and oscillatory and converges to 0. By Theorem 3.1, we have The behaviour of the solution and of the function is presented on Figure 1.
Remark 3.5. Our further research of this topic will be focused on a deeper investigation of all types of solutions defined in Definition 1.2. For example, we have proved in [15, 19] that damped solutions of (1.1) can be either oscillatory or they have a finite number of zeros or no zero and converge to 0. A more precise characterization of behaviour of nonoscillatory solutions are including their asymptotic formulas in as open problem. The same can be said about homoclinic solutions. In  we have found some conditions which guarantee their existence, and we have shown that if is a homoclinic solution of (1.1), then . In order to discover other existence conditions for homoclinic solutions, we would like to estimate their convergence by proper asymptotic formulas.
The authors thank the referees for comments and suggestions. This paper was supported by the Council of Czech Government MSM 6198959214.
- V. Bongiorno, L. E. Scriven, and H. T. Davis, “Molecular theory of fluid interfaces,” Journal of Colloid and Interface Science, vol. 57, pp. 462–475, 1967.
- H. Gouin and G. Rotoli, “An analytical approximation of density profile and surface tension of microscopic bubbles for Van der Waals fluids,” Mechanics Research Communications, vol. 24, pp. 255–260, 1997.
- J. D. Van Der Waals and R. Kohnstamm, Lehrbuch der Thermodynamik, vol. 1, Leipzig, Germany, 1908.
- P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, vol. 28 of Lecture Notes in Biomathematics, Springer, Berlin, Germany, 1979.
- R. A. Fischer, “The wave of advance of advantegeous genes,” Journal of Eugenics, vol. 7, pp. 355–369, 1937.
- F. F. Abraham, Homogeneous Nucleation Theory, Academies Press, New York, NY, USA, 1974.
- A. P. Linde, Particle Physics and Inflationary Cosmology, Harwood Academic, Chur, Switzerland, 1990.
- G. H. Derrick, “Comments on nonlinear wave equations as models for elementary particles,” Journal of Mathematical Physics, vol. 5, pp. 1252–1254, 1964.
- F. Dell'Isola, H. Gouin, and G. Rotoli, “Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations,” European Journal of Mechanics, vol. 15, no. 4, pp. 545–568, 1996.
- G. Kitzhofer, O. Koch, P. Lima, and E. Weinmüller, “Efficient numerical solution of the density profile equation in hydrodynamics,” Journal of Scientific Computing, vol. 32, no. 3, pp. 411–424, 2007.
- P. M. Lima, N. V. Chemetov, N. B. Konyukhova, and A. I. Sukov, “Analytical-numerical investigation of bubble-type solutions of nonlinear singular problems,” Journal of Computational and Applied Mathematics, vol. 189, no. 1-2, pp. 260–273, 2006.
- H. Berestycki, P. L. Lions, and L. A. Peletier, “An ODE approach to the existence of positive solutions for semilinear problems in ,” Indiana University Mathematics Journal, vol. 30, no. 1, pp. 141–157, 1981.
- D. Bonheure, J. M. Gomes, and L. Sanchez, “Positive solutions of a second-order singular ordinary differential equation,” Nonlinear Analysis: Theory, Methods & Appplications, vol. 61, no. 8, pp. 1383–1399, 2005.
- M. Conti, L. Merizzi, and S. Terracini, “Radial solutions of superlinear equations in , part I: a global variational approach,” Archive for Rational Mechanics and Analysis, vol. 153, no. 4, pp. 291–316, 2000.
- I. Rachůnková and J. Tomeček, “Bubble-type solutions of nonlinear singular problems,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 658–669, 2010.
- I. Rachůnková and J. Tomeček, “Strictly increasing solutions of a nonlinear singular differential equation arising in hydrodynamics,” Nonlinear Analysis: Theory, Methods & Appplications, vol. 72, no. 3-4, pp. 2114–2118, 2010.
- I. Rachůnková and J. Tomeček, “Homoclinic solutions of singular nonautonomous second-order differential equations,” Boundary Value Problems, vol. 2009, Article ID 959636, 21 pages, 2009.
- I. Rachůnková, J. Tomeček, and J. Stryja, “Oscillatory solutions of singular equations arising in hydrodynamics,” Advances in Difference Equations, vol. 2010, Article ID 872160, 13 pages, 2010.
- I. Rachůnková, L. Rachůnek, and J. Tomeček, “Existence of oscillatory solutions of singular nonlinear differential equations,” Abstract and Applied Analysis, vol. 2011, Article ID 408525, 20 pages, 2011.
- I. Kiguradze, Some Singular Boundary Value Problems for Ordinary Differential Equations, ITU, Tbilisi, Georgia, 1975.