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.

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.