Abstract and Applied Analysis

VolumeΒ 2011, Article IDΒ 981401, 9 pages

http://dx.doi.org/10.1155/2011/981401

## 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.Β Rogovchenko

Copyright Β© 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.

#### Abstract

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.

#### 1. Introduction

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 [6], in the relativistic cosmology for the description of particles which can be treated as domains in the universe [7], in the nonlinear field theory, in particular, when describing bubbles generated by scalar fields of the Higgs type in the Minkowski spaces [8]. 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 .

*Definition 1.1. * A function which satisfies (1.1) for all is called a * solution* of (1.1).

*Definition 1.2. * Let be a solution of (1.1) and let be of (1.2). Denote . If ( or ), then is called a * damped* solution (a * bounding homoclinic* solution or an * escape* solution).

These three types of solutions have been investigated in [15–19]. In particular, the existence of damped oscillatory solutions which converge to 0 has been proved in [19].

The main result of this paper is contained in Section 3 in Theorem 3.1, where we provide an asymptotic formula for damped oscillatory solutions of (1.1).

#### 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 [19]).

First, we bring two theorems about the existence of damped and oscillatory solutions.

Theorem 2.1 (see [19]). *Assume that (1.2)–(1.6) hold. Then for each problem (1.1), (2.1) has a unique solution. This solution is damped.*

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 [19].

*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).

*Example 2.4. * Let . (i)The function
satisfies (1.2) with , (1.3), (1.4) with and (2.3). (ii)The function
satisfies (1.2) with , (1.3), (1.4) with but not (2.3) (the second condition).

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 [20]). *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,

*Remark 3.2. *Due to (2.2) and (3.2), we have for a solution of Theorem 3.1

*Example 3.3. *Let . (i)The functions and satisfy all assumptions of Theorem 3.1. (ii) The functions and satisfy (1.2)–(1.4) but not (3.1) (the second condition).

*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 [17] 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.

#### Acknowledgments

The authors thank the referees for comments and suggestions. This paper was supported by the Council of Czech Government MSM 6198959214.

#### References

- V. Bongiorno, L. E. Scriven, and H. T. Davis, βMolecular theory of fluid interfaces,β
*Journal of Colloid and Interface Science*, vol. 57, pp. 462β475, 1967. View at Publisher Β· View at Google Scholar - H. Gouin and G. Rotoli, βAn analytical approximation of density profile and surface tension of microscopic bubbles for Van der Waals fluids,β
*Mechanics Research Communications*, vol. 24, pp. 255β260, 1997. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - 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, βThe wave of advance of advantegeous genes,β
*Journal of Eugenics*, vol. 7, pp. 355β369, 1937. View at Publisher Β· View at Google Scholar - 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, βComments on nonlinear wave equations as models for elementary particles,β
*Journal of Mathematical Physics*, vol. 5, pp. 1252β1254, 1964. View at Publisher Β· View at Google Scholar - F. Dell'Isola, H. Gouin, and G. Rotoli, βNucleation of spherical shell-like interfaces by second gradient theory: numerical simulations,β
*European Journal of Mechanics*, vol. 15, no. 4, pp. 545β568, 1996. View at Google Scholar Β· View at Zentralblatt MATH - G. Kitzhofer, O. Koch, P. Lima, and E. WeinmΓΌller, βEfficient numerical solution of the density profile equation in hydrodynamics,β
*Journal of Scientific Computing*, vol. 32, no. 3, pp. 411β424, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - P. M. Lima, N. V. Chemetov, N. B. Konyukhova, and A. I. Sukov, βAnalytical-numerical investigation of bubble-type solutions of nonlinear singular problems,β
*Journal of Computational and Applied Mathematics*, vol. 189, no. 1-2, pp. 260β273, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H. Berestycki, P. L. Lions, and L. A. Peletier, βAn ODE approach to the existence of positive solutions for semilinear problems in ${\mathrm{\beta \x84\x9d}}^{N}$,β
*Indiana University Mathematics Journal*, vol. 30, no. 1, pp. 141β157, 1981. View at Publisher Β· View at Google Scholar - D. Bonheure, J. M. Gomes, and L. Sanchez, βPositive solutions of a second-order singular ordinary differential equation,β
*Nonlinear Analysis: Theory, Methods & Appplications*, vol. 61, no. 8, pp. 1383β1399, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - M. Conti, L. Merizzi, and S. Terracini, βRadial solutions of superlinear equations in ${\mathrm{\beta \x84\x9d}}^{N}$, part I: a global variational approach,β
*Archive for Rational Mechanics and Analysis*, vol. 153, no. 4, pp. 291β316, 2000. View at Publisher Β· View at Google Scholar - I. RachΕ―nkovΓ‘ and J. TomeΔek, βBubble-type solutions of nonlinear singular problems,β
*Mathematical and Computer Modelling*, vol. 51, no. 5-6, pp. 658β669, 2010. View at Google Scholar Β· View at Zentralblatt MATH - I. RachΕ―nkovΓ‘ and J. TomeΔek, βStrictly increasing solutions of a nonlinear singular differential equation arising in hydrodynamics,β
*Nonlinear Analysis: Theory, Methods & Appplications*, vol. 72, no. 3-4, pp. 2114β2118, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - I. RachΕ―nkovΓ‘ and J. TomeΔek, βHomoclinic solutions of singular nonautonomous second-order differential equations,β
*Boundary Value Problems*, vol. 2009, Article ID 959636, 21 pages, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - I. RachΕ―nkovΓ‘, J. TomeΔek, and J. Stryja, βOscillatory solutions of singular equations arising in hydrodynamics,β
*Advances in Difference Equations*, vol. 2010, Article ID 872160, 13 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - I. RachΕ―nkovΓ‘, L. RachΕ―nek, and J. TomeΔek, βExistence of oscillatory solutions of singular nonlinear differential equations,β
*Abstract and Applied Analysis*, vol. 2011, Article ID 408525, 20 pages, 2011. View at Publisher Β· View at Google Scholar - I. Kiguradze,
*Some Singular Boundary Value Problems for Ordinary Differential Equations*, ITU, Tbilisi, Georgia, 1975.