Asymptotic Formula for Oscillatory Solutions of Some Singular Nonlinear Differential Equation

Abstract

Singular differential equation (𝑝(𝑡)𝑢^)^=𝑝(𝑡)𝑓(𝑢) is investigated. Here 𝑓 is Lipschitz continuous on ℝ and has at least two zeros 0 and 𝐿>0 . The function 𝑝 is continuous on [0,∞) and has a positive continuous derivative on (0,∞) and 𝑝(0)=0. An asymptotic formula for oscillatory solutions is derived.

1. Introduction

In this paper, we investigate the equation𝑝(𝑡)𝑢^^=𝑝(𝑡)𝑓(𝑢),𝑡∈(0,∞),(1.1) where 𝑓 satisfies𝑓∈𝐿𝑖𝑝∃_loc(ℝ),𝑓(0)=𝑓(𝐿)=0,𝑓(𝑥)<0,𝑥∈(0,𝐿),(1.2)𝐹𝐵∈(−∞,0):𝑓(𝑥)>0,𝑥∈𝐵,0,(1.3)𝐵=𝐹(𝐿),where𝐹(𝑥)=−^𝑥0𝑓(𝑧)d𝑧,𝑥∈ℝ,(1.4) and 𝑝 fulfils𝑝𝑝∈𝐶[0,∞)∩𝐶¹(0,∞),𝑝(0)=0,(1.5)_(𝑡)>0,𝑡∈(0,∞),lim_𝑡→∞𝑝^(𝑡)𝑝(𝑡)=0.(1.6) Equation (1.1) is a generalization of the equation𝑢^+𝑘−1𝑡𝑢^=𝑓(𝑢),𝑡∈(0,∞),(1.7) which arises for 𝑘>1 and special forms of 𝑓 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 𝑓 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 𝑝(0)=0, (1.1) has a singularity at 𝑡=0.

Definition 1.1. A function 𝑢∈𝐶¹[0,∞)∩𝐶²(0,∞) which satisfies (1.1) for all 𝑡∈(0,∞) is called a solution of (1.1).

Definition 1.2. Let 𝑢 be a solution of (1.1) and let 𝐿 be of (1.2). Denote 𝑢_sup=sup{𝑢(𝑡):𝑡∈[0,∞)}. If 𝑢_sup<𝐿 (𝑢_sup=𝐿 or 𝑢_sup>𝐿), then 𝑢 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 conditions𝑢(0)=𝐵,𝑢^(0)=0,(2.1) with a parameter 𝐵≤𝐿. Reason is that we focus our attention on damped solutions of (1.1) and that each solution 𝑢 of (1.1) must fulfil 𝑢^(0)=0 (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 𝐵∈[𝐵,𝐿) 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 𝑘₀∈(0,∞) such that 𝑝∈𝐶²(0,∞),limsup_𝑡→∞||||𝑝^(𝑡)𝑝^(𝑡)||||<∞,liminf^𝑡→∞𝑝(𝑡)𝑡^𝑘0lim_𝑥→0+∈(0,∞],(2.2)𝑓(𝑥)𝑥<0,lim^𝑥→0−𝑓(𝑥)𝑥<0.(2.3) Then for each 𝐵∈[𝐵,𝐿) problem (1.1), (2.1) has a unique solution 𝑢. If 𝐵≠0, then the solution 𝑢 is damped and oscillatory with decreasing amplitudes and lim_𝑡→∞𝑢(𝑡)=0.(2.4)

Proof. The assertion follows from Theorems 2.3, 2.10 and 3.1 in [19].

Example 2.3. The functions (i)𝑝(𝑡)=𝑡^𝑘, 𝑝(𝑡)=𝑡^𝑘ln(𝑡^ℓ+1), 𝑘,ℓ∈(0,∞), (ii)𝑝(𝑡)=𝑡+𝛼sin𝑡, 𝛼∈(−1,1), (iii)𝑝(𝑡)=𝑡^𝑘/(1+𝑡^ℓ),𝑘,ℓ∈(0,∞),ℓ<𝑘satisfy (1.5), (1.6), and (2.2).
The functions (i)𝑝(𝑡)=ln(𝑡+1),𝑝(𝑡)=arctan𝑡,𝑝(𝑡)=𝑡^𝑘/(1+𝑡^𝑘),𝑘∈(0,∞) satisfy (1.5), (1.6), but not (2.2) (the third condition).
The function (i)𝑝(𝑡)=𝑡^𝑘+𝛼sin𝑡^𝑘, 𝛼∈(−1,1), 𝑘∈(1,∞), satisfy (1.5), (1.6) but not (2.2) (the second and third conditions).

Example 2.4. Let 𝑘∈(0,∞). (i)The function 𝑓(𝑥)=−𝑥𝑘(𝑥𝑥,−1),ffoorr𝑥𝑥≤>00,,(2.5) satisfies (1.2) with 𝐿=1, (1.3), (1.4) with 𝐵=−(3𝑘)^−1/2 and (2.3). (ii)The function 𝑓(𝑥)=𝑘𝑥²𝑥(𝑥,−1),ffoorr𝑥𝑥≤>00,,(2.6) satisfies (1.2) with 𝐿=1, (1.3), (1.4) with 𝐵=−(2𝑘)^−1/3 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 𝐽⊂ℝ. We write 𝐴𝐶(𝐽) for the set of functions absolutely continuous on 𝐽 and 𝐴𝐶_loc(𝐽) for the set of functions belonging to 𝐴𝐶(𝐼) for each compact interval 𝐼⊂𝐽. Choose 𝑡₀>0 and a function matrix 𝐴(𝑡)=(𝑎_𝑖,𝑗(𝑡))_{𝑖,𝑗≤2} which is defined on (𝑡₀,∞). Denote by 𝜆(𝑡) and 𝜇(𝑡) eigenvalues of 𝐴(𝑡),𝑡∈(𝑡₀,∞). Further, suppose 𝜆=lim_𝑡→∞𝜆(𝑡),𝜇=lim_𝑡→∞𝜇(𝑡)(2.7) be different eigenvalues of the matrix 𝐴=lim_𝑡→∞𝐴(𝑡), and let 𝐥 and 𝐦 be eigenvectors of 𝐴 corresponding to 𝜆 and 𝜇, respectively.

Theorem 2.5 (see [20]). Assume that 𝑎_𝑖,𝑗∈𝐴𝐶_loc𝑡₀,∞,||||^∞𝑡₀𝑎^𝑖,𝑗(𝑡)d𝑡||||<∞,𝑖,𝑗=1,2,(2.8) and that there exists 𝑐₀>0 such that ^𝑡𝑠Re(𝜆(𝜏)−𝜇(𝜏))d𝜏≤𝑐₀,𝑡₀≤𝑠<𝑡,(2.9) or ^∞𝑡₀Re(𝜆(𝜏)−𝜇(𝜏))d𝜏=∞,^𝑡𝑠Re(𝜆(𝜏)−𝜇(𝜏))d𝜏≥−𝑐₀,𝑡₀≤𝑠<𝑡.(2.10) Then the differential system 𝐱^(𝑡)=𝐴(𝑡)𝐱(𝑡)(2.11) has a fundamental system of solutions 𝑥(𝑡),𝑦(𝑡) such that lim_𝑡→∞𝐱(𝑡)e^{−∫𝑡𝑡0𝜆(𝜏)d𝜏}=𝐥,lim_𝑡→∞𝐲(𝑡)e^{−∫𝑡𝑡0𝜇(𝜏)d𝜏}=𝐦.(2.12)

3. Asymptotic Formula

In order to derive an asymptotic formula for a damped oscillatory solution 𝑢 of problem (1.1), (2.1), we need a little stronger assumption than (2.3). In particular, the function 𝑓(𝑥)/𝑥 should have a negative derivative at 𝑥=0.

Theorem 3.1. Assume that (1.2)–(1.6), and (2.2) hold. Assume, moreover, that there exist 𝜂>0 and 𝑐>0 such that 𝑓(𝑥)𝑥∈𝐴𝐶[−𝜂,𝜂],lim^𝑥→0𝑓(𝑥)𝑥=−𝑐.(3.1) Then for each 𝐵∈[𝐵,𝐿) problem (1.1), (2.1) has a unique solution 𝑢. If 𝐵≠0, then the solution 𝑢 is damped and oscillatory with decreasing amplitudes such that limsup_𝑡→∞√𝑝(𝑡)||𝑢(𝑡)||<∞.(3.2)

Proof. We have the following steps:Step 1 (construction of an auxiliary linear differential system). Choose 𝐵∈[𝐵,𝐿),𝐵≠0. By Theorem 2.2, problem (1.1), (2.1) has a unique oscillatory solution 𝑢 with decreasing amplitudes and satisfying (2.4). Having this solution 𝑢, define a linear differential equation 𝑣^+𝑝^(𝑡)𝑝(𝑡)𝑣^=𝑓(𝑢(𝑡))𝑢(𝑡)𝑣,(3.3) and the corresponding linear differential system 𝑥^1=𝑥₂,𝑥^2=𝑓(𝑢(𝑡))𝑢(𝑡)𝑥¹−𝑝^(𝑡)𝑝(𝑡)𝑥².(3.4) Denote 𝐴(𝑡)=𝑎_𝑖,𝑗(𝑡)_{𝑖,𝑗≤2}=⎛⎜⎜⎝𝑓(𝑢0(𝑡))1𝑢(𝑡)−𝑝^(𝑡)𝑝(𝑡)⎞⎟⎟⎠,𝐴=−0𝑐01.(3.5) By (1.6), (2.4), and (3.1), 𝐴=lim_𝑡→∞𝐴(𝑡).(3.6) Eigenvalues of 𝐴 are numbers 𝜆=𝑖√𝑐 and 𝜇=−𝑖√𝑐, and eigenvectors of 𝐴 are 𝐥=(1,𝑖√𝑐) and 𝐦=(1,−𝑖√𝑐), respectively. Denote 𝐷(𝑡)=𝑝^(𝑡)2𝑝(𝑡)²+𝑓(𝑢(𝑡))𝑢(𝑡),𝑡∈(0,∞).(3.7) Then eigenvalues of 𝐴(𝑡) have the form 𝜆(𝑡)=−𝑝^(𝑡)2𝑝(𝑡)+√𝐷(𝑡),𝜇(𝑡)=−𝑝^(𝑡)2𝑝(𝑡)−√𝐷(𝑡),𝑡∈(0,∞).(3.8) We see that lim_𝑡→∞𝜆(𝑡)=𝜆,lim_𝑡→∞𝜇(𝑡)=𝜇.(3.9)Step 2 (verification of the assumptions of Theorem 2.5). Due to (1.6), (2.4), and (3.1), we can find 𝑡₀>0 such that 𝑢𝑡₀≠0,||𝑢(𝑡)||≤𝜂,𝐷(𝑡)<0,𝑡∈𝑡₀,∞.(3.10) Therefore, by (3.1), 𝑎₂₁(𝑡)=𝑓(𝑢(𝑡))𝑢(𝑡)∈𝐴𝐶^loc𝑡₀,∞,(3.11) and so ||||^∞𝑡₀𝑓𝑢(𝑢(𝑡()𝑡))^d𝑡||||=||||lim_𝑡→∞𝑓(𝑢(𝑡))𝑢(𝑡)−𝑓𝑢𝑡⁰𝑢𝑡₀||||=||||−𝑐−𝑓𝑢𝑡⁰𝑢𝑡₀||||<∞.(3.12) Further, by (2.2), 𝑎₂₂(𝑡)=−𝑝^(𝑡)/𝑝(𝑡)∈𝐶¹(𝑡₀,∞). Hence, due to (1.6), ||||^∞𝑡₀𝑝^(𝑡)𝑝(𝑡)d𝑡||||=||||lim^𝑡→∞𝑝^(𝑡)𝑝(𝑡)−𝑝^𝑡₀𝑝𝑡₀||||=𝑝^𝑡₀𝑝𝑡₀<∞.(3.13) Since 𝑎₁₁(𝑡)≡0 and 𝑎₁₂(𝑡)≡1, we see that (2.8) is satisfied. Using (3.8) we get Re(𝜆(𝑡)−𝜇(𝑡))≡0. This yields ^𝑡𝑠Re(𝜆(𝜏)−𝜇(𝜏))d𝜏=0<𝑐₀,𝑡₀≤𝑠<𝑡,(3.14) for any positive constant 𝑐₀. Consequently (2.9) is valid.Step 3 (application of Theorem 2.5). By Theorem 2.5 there exists a fundamental system 𝐱(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡)),𝐲(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡)) of solutions of (3.4) such that (2.12) is valid. Hence lim_𝑡→∞𝑥₁(𝑡)e^{−∫𝑡𝑡0𝜆(𝜏)d𝜏}=1,lim_𝑡→∞𝑦₁(𝑡)e^{−∫𝑡𝑡0𝜇(𝜏)d𝜏}=1.(3.15) Using (3.8) and (3.10), we get exp−^𝑡𝑡₀𝜆(𝜏)d𝜏=exp^𝑡𝑡₀𝑝^(𝜏)2𝑝(𝜏)−√=exp1𝐷(𝜏)d𝜏2ln𝑝𝑝(𝑡𝑡₀)exp−𝑖^𝑡𝑡₀||𝐷(𝜏)||d𝜏,(3.16) and, hence, |||e_−∫^𝑡𝑡0_{𝜆(𝜏)d𝜏}|||=𝑝(𝑡)𝑝𝑡₀,𝑡∈𝑡⁰,∞.(3.17) Similarly |||e_−∫^𝑡𝑡0_{𝜇(𝜏)d𝜏}|||=𝑝(𝑡)𝑝𝑡₀,𝑡∈𝑡⁰,∞.(3.18) Therefore, (3.15) implies 1=lim_𝑡→∞|||𝑥₁(𝑡)e^{−∫𝑡𝑡0𝜆(𝜏)d𝜏}|||=lim_𝑡→∞||𝑥₁(𝑡)||𝑝(𝑡)1=lim_𝑡→∞𝑝𝑡₀,|||𝑦₁(𝑡)e^{−∫𝑡𝑡0𝜇(𝜏)d𝜏}|||=lim_𝑡→∞||𝑦₁(𝑡)||𝑝(𝑡)𝑝𝑡₀.(3.19)Step 4 (asymptotic formula). In Step 1, we have assumed that 𝑢 is a solution of (1.1), which means that 𝑢^(𝑡)+𝑝^(𝑡)𝑝(𝑡)𝑢^(𝑡)=𝑓(𝑢(𝑡)),for𝑡∈(0,∞).(3.20) Consequently 𝑢^(𝑡)+𝑝^(𝑡)𝑝(𝑡)𝑢^(𝑡)=𝑓(𝑢(𝑡))𝑢(𝑡)𝑢(𝑡),for𝑡∈(0,∞),(3.21) and, hence, 𝑢 is also a solution of (3.3). This yields that there are 𝑐₁,𝑐₂∈ℝ such that 𝑢(𝑡)=𝑐₁𝑥₁(𝑡)+𝑐₂𝑦₁(𝑡),𝑡∈(0,∞). Therefore, limsup_𝑡→∞√𝑝(𝑡)||𝑢(𝑡)||≤||𝑐₁||+||𝑐₂||𝑝𝑡₀<∞.(3.22)

Remark 3.2. Due to (2.2) and (3.2), we have for a solution 𝑢 of Theorem 3.1𝑢(𝑡)=𝑂𝑡^−𝑘0/2,for𝑡⟶∞.(3.23)

Example 3.3. Let 𝑘∈(1,∞). (i)The functions 𝑓(𝑥)=𝑥(𝑥−1) and 𝑓(𝑥)=𝑥(𝑥−1)(𝑥+2) satisfy all assumptions of Theorem 3.1. (ii) The functions 𝑓(𝑥)=𝑥^2𝑘−1(𝑥−1) and 𝑓(𝑥)=𝑥^2𝑘−1(𝑥−1)(𝑥+2)satisfy (1.2)–(1.4) but not (3.1) (the second condition).

Example 3.4. Consider the initial problem 𝑡²𝑢^^=𝑡²𝑢(𝑢−5)(𝑢+10),𝑢(0)=−3,𝑢^(0)=0.(3.24) Here 𝐿₀=−10,𝐿=5 and we can check that 𝐵<−3. Further, all assumptions of Theorems 2.2 and 3.1 are fulfilled. Therefore, by Theorem 2.2, there exists a unique solution 𝑢 of problem (3.24) which is damped and oscillatory and converges to 0. By Theorem 3.1, we have limsup_𝑡→∞𝑡||𝑢(𝑡)||<∞,thatis,𝑢(𝑡)=𝑂1𝑡,for𝑡⟶∞.(3.25) The behaviour of the solution 𝑢(𝑡) and of the function 𝑡𝑢(𝑡) 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 𝑢 is a homoclinic solution of (1.1), then lim_𝑡→∞𝑢(𝑡)=𝐿. 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.