Boundedness of Solutions for a Class of Sublinear Reversible Oscillators with Periodic Forcing

Abstract

We study the boundedness of all solutions for the following differential equation 𝑥^′′+𝑓(𝑥)𝑥^′+(𝐵+𝜀𝑒(𝑡))|𝑥|^𝛼−1𝑥=𝑝(𝑡), where 𝑓(𝑥),𝑝(𝑡) are odd functions, 𝑒(𝑡) is an even function, 𝑒(𝑡),𝑝(𝑡) are smooth 1-periodic functions, 𝐵 is a nonzero constant, and 𝜀 is a small parameter. A sufficient and necessary condition for the boundedness of all solutions of the above equation is established. Moreover, the existence of Aubry-Mather sets is obtained as well.

1. Introduction

It is well known that the longtime behavior for periodically forced planar systems can be very intricate. For example, there are equations having unbounded solutions but with infinitely many zeros and with nearby unbounded solutions having randomly prescribed number of zeros and also periodic solutions; see [1]. In contrast to such unbounded phenomenon Littlewood [2] suggested to study the boundedness of all the solutions of the following differential equation: 𝑥̈+𝑔(𝑥)=ℎ(𝑡) (1) in the following two cases:(i)superlinear case: 𝑔(𝑥)/𝑥→+∞ as 𝑥→±∞;(ii)sublinear case: sgn(𝑥)⋅𝑔(𝑥)→+∞ and 𝑔(𝑥)/𝑥→0 as 𝑥→±∞. Later, one calls this subject as Littlewood boundedness problem.

The first result in superlinear case is obtained by Morris [3], who showed that all solutions of 𝑥̈+2𝑥³=𝑒(𝑡) (2) are bounded, where 𝑒(𝑡)∈𝐶⁰. Later, a series results in superlinear case were obtained by several authors, see [4–13] and references therein. However, in general, it is harder to study the Lagrange stability of sublinear systems since smoothness of sublinear term is insufficient. There are only a few works in sublinear case so far. In 1999, Küpper and You [14] proved the first result in the study of the equation 𝑥̈+|𝑥|^𝛼−1𝑥=𝑝(𝑡), (3) where 0<𝛼<1 and 𝑝(𝑡)∈𝐶^∞(𝕋). Later, Liu [15] proved the same result for more general equation 𝑥̈+𝑔(𝑥)=𝑒(𝑡), (4) where 𝑔(𝑥)∈𝐶⁶ satisfying the sublinear condition (ii) and some inequalities, and 𝑒(𝑡)∈𝐶⁵(𝕋). In 2004, Ortega and Verzini [16] studied the boundedness of (4) in a special case with the variational method. In 2009, Wang [17] gave a sufficient and necessary condition for the boundedness of all solutions for sublinear equation 𝑥̈+𝑒(𝑡)|𝑥|^𝛼−1𝑥=𝑝(𝑡), (5) where 𝑒(𝑡),𝑝(𝑡)∈𝐶⁵(𝕋).

As is widely known, there is a deep similarity between reversible and Hamiltonian dynamics. Many fundamental results of the Hamiltonian systems possess reversible counterparts. On boundedness problem for sublinear reversible systems, the first results were obtained by Li [18], later, Yang [19], in the study of a sublinear reversible systems 𝑥̈+𝑓(𝑥)𝑥̇+|𝑥|^𝛼−1𝑥=𝑒(𝑡). (6) Recently, Wang [20] gave a sufficient and necessary condition for the boundedness of all solutions of the differential equation 𝑥̈+𝑓(𝑥)𝑔(𝑥̇)+𝛾|𝑥|^𝛼−1𝑥=𝑝(𝑡) (7) with 0<𝛼<1, 𝛾≠0.

By the discussions about the sublinear Hamiltonian equation (1.3) in [17] motivations, we will study the boundedness of all solutions for a sublinear reversible system like 𝑥̈+𝑓(𝑥)𝑥̇+(𝐵+𝜀𝑒(𝑡))|𝑥|^𝛼−1𝑥=𝑝(𝑡), (8) where 𝐵≠0 and 0<𝛼<1. Furthermore, we also show that (8) has solutions of Mather type. The results obtained in [18–20] can be regarded as corollary of result of this paper.

Remark 1. Using the method of this paper we also can consider the more general equation 𝑥̈+𝑓(𝑥)𝑔(𝑥̇)+(𝐵+𝜀𝑒(𝑡))|𝑥|^𝛼−1𝑥=𝑝(𝑡) (9) provided of adding suitable conditions for 𝑔(𝑥). For convenience, we only consider the case 𝑔(𝑥)≡𝑥.

Remark 2. Adding the perturbation term 𝜀𝑒(𝑡)|𝑥|^𝛼−1𝑥 will lead to a new difficulty for estimating |𝑆(𝜃𝑇₀)|^𝛼−1𝐶(𝜃𝑇₀) appeared in (86). Fortunately, we can easily verify that ∫¹₀|𝑆(𝜃𝑇₀)|^𝛼−1𝐶(𝜃𝑇₀)𝑑𝜃 is bounded by a constant (see in the proof of Lemma 12).

Throughout this paper, we denote two universal positive constants without regarding their values by 𝑐<1 and 𝐶≥1, and suppose that the following conditions hold:(A1)𝑓(𝑥)∈𝐶⁴(ℝ), 𝑝(𝑡)∈𝐶³(𝕋) and 𝑒(𝑡)∈𝐶³(𝕋), 𝑓(𝑥) and 𝑝(𝑡) are odd, 𝑒(𝑡) is even, and 𝑒(𝑡), 𝑝(𝑡) are both 1-periodic functions, 𝕋=ℝ/ℤ;(A2) there is some positive constant 𝜇 such that the inequalities 𝑥^𝑖+1𝑓^(𝑖)(𝑥)≤𝐶|𝑥|^{𝛼/2−𝛽} (10) are satisfied for 0≤𝑖≤4 and all |𝑥|≥𝜇, where 0<𝛽<𝛼/2.

We decompose 𝑒(𝑡) as 𝑒(𝑡)=𝑒+𝑒(̂𝑡), where 𝑒 is the average of 𝑒(𝑡) and 𝑒(̂𝑡) has zero mean value. That is 𝑒=∫¹₀𝑒(𝑠)𝑑𝑠 and ∫¹₀𝑒(̂𝑠)𝑑𝑠=0. If we write that 𝐴=𝐵+𝜀𝑒, then it is easy to see that 𝐴 and 𝐵 have the same sign when 0<𝜀<𝜀^∗ with 0<𝜀^∗<|𝐵/𝑒|.

Now we state the main results of this paper.

Theorem 3. Assume that 𝐵≠0 and (A1)-(A2) hold. Then there exists an 0<𝜀^∗∗<𝜀^∗ such that for any 0<𝜀<𝜀^∗∗, every solution of (8) is bounded if and only if 𝐵>0.

Theorem 4. Under the conditions of Theorem 3, there is an 𝜀₀>0 such that, for any 𝜔∈(𝑛,𝑛+𝜀₀), (8) has a solution (𝑥_𝜔(𝑡),𝑥^′_𝜔(𝑡)) of Mather type with rotation number 𝜔. More precisely:(i)if 𝜔=𝑝/𝑞 is rational, the solutions (𝑥_𝜔(𝑡+𝑖),𝑥^′_𝜔(𝑡+𝑖)), 1≤𝑖≤𝑞−1, are periodic solutions of period q; moreover, in this case lim_𝜔→𝑛min_𝑡∈ℝ𝑥_𝜔(𝑡)+𝑥^′_𝜔(𝑡)=+∞; (11)(ii)if 𝜔 is irrational, the solution (𝑥_𝜔(𝑡),𝑥^′_𝜔(𝑡)) is either a usual quasi-periodic solution or a generalized one.
We recall that a solution is called generalized quasi-periodic if the closed set 𝑥(𝑖),𝑥^′(𝑖),𝑖∈ℤ (12) is a Denjoys minimal set.

2. Reversible Systems and Action-Angle Variables

In this section, we will assume that 𝐵>0 and 𝐴>0. Firstly, we consider (8) which is equivalent to the following system: ̇(𝑡),̇|𝑥|𝑥=𝑧+𝑃𝑧=−𝐴^𝛼−1𝑥−𝜀𝑒̂(𝑡)|𝑥|^𝛼−1𝑥−𝑓(𝑥)(𝑧+𝑃(𝑡)), (13) where 𝑃(𝑡)=∫^𝑡₀𝑝(𝑠)𝑑𝑠. Then we can obtain that (13) is reversible with respect to the transformation (𝑥,𝑧)↦(−𝑥,𝑧) by (A1).

Lemma 5. There exists a 𝐺-invariant diffeomorphism (𝑥,𝑦)→(𝑥,𝑧) such that (13) is transformed into the following system: 𝑥̇=𝑦+𝜀𝐸(𝑡)|𝑥|^𝛼−1(𝑡),̇|𝑥|𝑥+𝑃𝑦=−𝐴^𝛼−1𝑥−𝛼𝜀𝐸(𝑡)|𝑥|^𝛼−1+𝑓(𝑥)𝑦+𝜀𝐸(𝑡)|𝑥|^𝛼−1𝑥+𝑃(𝑡), (14) where 𝐸(𝑡)=−∫^𝑡₀𝑒(̂𝑠)𝑑𝑠.

Proof. Introduce a transformation Ψ: 𝑥=𝑥,𝑧=𝑦+𝑈(𝑥,𝑡), (15) where 𝑈(𝑥,𝑡) will be determined later. Under this transformation, the system (13) is transformed into a new system as follows: ̇()(𝑡),̇|𝑥|𝑥=𝑦+𝑈𝑥,𝑡+𝑃𝑦=−𝐴^𝛼−1𝑥−𝜀𝑒̂(𝑡)|𝑥|^𝛼−1𝑥−𝑓(𝑥)+𝜕𝑈(𝑥,𝑡)()(𝑡−()𝜕𝑥𝑦+𝑈𝑥,𝑡+𝑃)𝜕𝑈𝑥,𝑡.𝜕𝑡 (16) Now, we define the function 𝑈(𝑥,𝑡) by −𝜀𝑒̂(𝑡)|𝑥|^𝛼−1()𝑥−𝜕𝑈𝑥,𝑡𝜕𝑡=0. (17) Since ∫¹₀𝑒(̂𝑡)𝑑𝑡=0, so we can obtain 𝑈(𝑥,𝑡)=𝜀𝐸(𝑡)|𝑥|^𝛼−1𝑥. Then the new system can be expressed as in (14) by direct computation.
It is easy to know that 𝑈(−𝑥,−𝑡)=𝑈(𝑥,𝑡) by (A1), then we can obtain that the transformation Ψ is a 𝐺-invariant diffeomorphism.

Let us consider the auxiliary system ̇̇|𝑥|𝑥=𝑦,𝑦=−𝐴^𝛼−1𝑥, (18) which is a time-independent Hamiltonian system with Hamiltonian 𝐻₀=𝑦𝑥,𝑦²2+𝐴|𝑥|𝛼+1^𝛼+1. (19)

It is easy to see that 𝐻₀(𝑥,𝑦)>0, (𝑥,𝑦)∈𝑅²⧵{0}, 𝐻₀(0,0)=0. Note that each level line 𝐻₀(𝑥,𝑦)=ℎ>0 is a close orbit of system (18), hence, all the solutions of (18) are periodic with period tending to zero as ℎ tends to infinity.

Assume that (𝑆(𝑡),𝐶(𝑡)) is the solution of (18) with initial conditions (𝑆(0),𝐶(0))=(0,1), and let 𝑇₀>0 be the minimal period. We can find that 𝑆(𝑡) and 𝐶(𝑡) satisfy(i)𝑆(𝑡)∈𝐶²(𝑅), 𝐶(𝑡)∈𝐶¹(𝑅);(ii)(𝑆(−𝑡),𝐶(−𝑡))=(−𝑆(𝑡),𝐶(𝑡)), (𝑆(𝑡+𝑇₀),𝐶(𝑡+𝑇₀))=(𝑆(𝑡),𝐶(𝑡));(iii)𝑆(̇𝑡)=𝐶(𝑡), 𝐶̇(𝑡)=−𝐴|𝑆(𝑡)|^𝛼−1𝑆(𝑡);(iv)(1/2)𝐶²(𝑡)+(𝐴/(𝛼+1))|𝑆(𝑡)|^𝛼+1=1/2; (v)𝐶(𝑇₀𝑡)=0⇔𝑡(mod(1/4))=0; (vi)(𝑆(𝑇₀(1/2−𝑡)), 𝐶(𝑇₀(1/2−𝑡)))=(𝑆(𝑇₀𝑡),−C(𝑇₀𝑡));(vii)𝑆(𝑇₀𝑡)=0⇔𝑡(mod(1/2))=0.

Then we introduce the transformation Φ∶ℝ⁺×𝕋⟶ℝ²⧵{0},𝜌,𝜑⟼𝑥,𝑦 (20) which is 𝑥=𝜌^𝑏𝑆𝜑𝑇₀,𝑦=𝜌^1−𝑏𝐶𝜑𝑇₀, (21) where 𝑏=2/(3+𝛼). It is easy to see that 1/2<𝑏<2/3 by 0<𝛼<1. Since (𝑆(−𝑡),𝐶(−𝑡))=(−𝑆(𝑡),𝐶(𝑡)), this transformation is invariant with respect to the involutions (𝜌,𝜑)↦(𝜌,−𝜑) and (𝑥,𝑦)↦(−𝑥,𝑦), and we can find that the mapping Φ is a generalized canonical transformation by (iv). In fact, 𝜕𝑥,𝑦𝜕=𝜌,𝜑𝐴𝑏𝑇₀𝑆𝜑𝑇₀^𝛼+1+(1−𝑏)𝑇₀𝐶²𝜑𝑇₀=(1−𝑏)𝑇₀−𝛼+12𝑏𝑇₀𝐶²𝜑𝑇₀+(1−𝑏)𝑇₀𝐶²𝜑𝑇₀=(1−𝑏)𝑇₀,𝜌̇𝜑̇=−𝑑𝑦_𝜑𝑑𝑥_𝜑𝑑𝑦_𝜌−𝑑𝑥_𝜌𝑥̇𝑦̇, (22) where 𝑑=((1−𝑏)𝑇₀)⁻¹.

Under the transformation Φ, the system (18) is transformed into the simpler form 𝜌̇=−𝜕ℎ₀̇𝜕𝜑=0,𝜑=𝜕ℎ₀=1𝜕𝜌𝑇₀⋅𝜌^1−2𝑏, (23) where ℎ₀(𝜌)=((2−2𝑏)𝑇₀)⁻¹⋅𝜌^2(1−𝑏).

The original system (13) is transformed into the system 𝑑𝜌𝑑𝑡=𝑙₁𝜌,𝜑+𝑙₂𝜌,𝜑,𝑡+𝜀𝑙₃𝜌,𝜑,𝑡+𝛼𝑇₀𝑆𝜑𝑇₀^𝛼−1𝐶𝜑𝑇₀𝜀𝑙₄𝜌,𝜑,𝑡,𝑑𝜑𝑑𝑡=ℎ^′₀𝜌+ℎ₁𝜌,𝜑+ℎ₂𝜌,𝜑,𝑡+𝜀ℎ₃𝜌,𝜑,𝑡, (24) where 𝑙₁𝜌,𝜑=−𝑑𝑇₀𝜌𝑓𝜌^𝑏𝑆𝜑𝑇₀𝐶²𝜑𝑇₀=∶−𝑑𝑥_𝜑𝑓(𝑥)𝑙𝑦,₂𝜌,𝜑,𝑡=−𝑑𝑇₀𝜀𝜌^2−2𝑏𝑓𝜌^𝑏𝑆𝜑𝑇₀×𝑆𝜑𝑇₀^𝛼−1𝑆𝜑𝑇₀𝐶𝜑𝑇₀𝐸(𝑡)+𝐴𝑑𝑇₀𝜌^1−𝑏𝑆𝜑𝑇₀^𝛼−1𝑆𝜑𝑇₀𝑃(𝑡)−𝑑𝑇₀𝜌^𝑏𝑓𝜌^𝑏𝑆𝜑𝑇₀𝐶𝜑𝑇₀𝑃(𝑡)=∶−𝑑𝜀𝑥_𝜑|𝑥|^𝛼−1𝑥𝑓(𝑥)𝐸(𝑡)−𝑑𝑦_𝜑𝑃(𝑡)−𝑑𝑥_𝜑𝑓(𝑥)𝑃(𝑡),𝑙₃𝜌,𝜑,𝑡=𝐴𝑑𝑇₀𝜌^3−4𝑏𝑆𝜑𝑇₀^2𝛼𝐸(𝑡)=∶−𝑑𝑦_𝜑|𝑥|^𝛼−1(𝑡),𝑙𝑥𝐸₄𝜌,𝜑,𝑡=−𝑑𝜌^3−4𝑏𝐶𝜑𝑇₀𝐸(𝑡)−𝑑𝜀𝜌^4−6𝑏𝑆𝜑𝑇₀^𝛼−1𝑆𝜑𝑇₀𝐸²(𝑡)−𝑑𝜌^2−3𝑏𝑃(𝑡)𝐸(𝑡),ℎ₁𝜌,𝜑=𝑑𝑏𝑓𝜌^𝑏𝑆𝜑𝑇₀𝐶𝜑𝑇₀𝑆𝜑𝑇₀=∶𝑑𝑥_𝜌𝑓(𝑥)ℎ𝑦,₂𝜌,𝜑,𝑡=𝑑𝑏𝜀𝜌^1−2𝑏𝑓𝜌^𝑏𝑆𝜑𝑇₀𝑆𝜑𝑇₀^𝛼+1𝐸(𝑡)+𝛼𝑑𝑏𝜀²𝜌^3−6𝑏𝑆𝜑𝑇₀^2𝛼𝐸²(𝑡)+𝑑(1−𝑏)𝜌^−𝑏𝐶𝜑𝑇₀𝑃(𝑡)+𝑑𝑏𝜌^𝑏−1𝑓𝜌^𝑏𝑆𝜑𝑇₀𝑆𝜑𝑇₀𝑃(𝑡)+𝛼𝑑𝑏𝜀𝜌^1−3𝑏𝑆𝜑𝑇₀^𝛼−1𝑆𝜑𝑇₀𝐸(𝑡)𝑃(𝑡)=∶𝑑𝜀𝑥_𝜌|𝑥|^𝛼−1𝑥𝑓(𝑥)𝐸(𝑡)+𝛼𝑑𝜀²𝑥_𝜌|𝑥|^2𝛼−2𝑥𝐸²(𝑡)+𝑑𝑦_𝜌𝑃(𝑡)+𝑑𝑥_𝜌𝑓(𝑥)𝑃(𝑡)+𝛼𝑑𝜀𝑥_𝜌|𝑥|^𝛼−1𝐸(𝑡)𝑃(𝑡),ℎ₃()𝜌𝜌,𝜑,𝑡=𝑑1−𝑏+𝛼𝑏^2−4𝑏𝑆𝜑𝑇₀^𝛼−1𝑆𝜑𝑇₀𝐶𝜑𝑇₀𝐸(𝑡)=∶𝑑𝑦_𝜌|𝑥|^𝛼−1𝑥𝐸(𝑡)+𝛼𝑑𝑥_𝜌|𝑥|^𝛼−1𝑦𝐸(𝑡). (25)

Let 𝐿₂𝜌,𝜑,𝑡=𝑙₂𝜌,𝜑,𝑡+𝜀𝑙₃𝜌,𝜑,𝑡+𝛼𝑇₀𝑆𝜑𝑇₀^𝛼−1𝐶𝜑𝑇₀𝜀𝑙₄,𝐻𝜌,𝜑,𝑡₂𝜌,𝜑,𝑡=ℎ₂𝜌,𝜑,𝑡+𝜀ℎ₃𝜌,𝜑,𝑡. (26) Clearly, 𝑥 is odd in 𝜑 and 𝑦 is even in 𝜑 by the definitions of 𝑆(𝑡) and 𝐶(𝑡). Thus, by the evenness of 𝑃(𝑡) and the oddness of 𝑓(𝑥) and 𝐸(𝑡) we have 𝑙₁𝜌,−𝜑=−𝑙₁𝜌,𝜑,𝐿₂𝜌,−𝜑,−𝑡=−𝐿₂,ℎ𝜌,𝜑,𝑡₁𝜌,−𝜑=ℎ₁𝜌,𝜑,𝐻₂𝜌,−𝜑,−𝑡=𝐻₂𝜌,𝜑,𝑡. (27) This implies that system (24) is reversible with respect to the involutions (𝜌,𝜑)↦(𝜌,−𝜑).

Lemma 6. For 0≤𝑘+𝑚≤4, the following inequalities hold:(1)|(𝜕^𝑘/𝜕𝜌^𝑘)𝑙₁(𝜌,𝜑)|≤𝐶𝜌^{−𝑘+2−𝛾−(5/2)𝑏}, (2)|(𝜕^𝑘+𝑚/𝜕𝜌^𝑘𝜕𝑡^𝑚)𝑙₂(𝜌,𝜑,𝑡)|≤𝐶𝜌^−𝑘+𝑎, (3)|(𝜕^𝑘+𝑚/𝜕𝜌^𝑘𝜕𝑡^𝑚)𝑙₃(𝜌,𝜑,𝑡)|≤𝐶𝜌^{−𝑘+3−4𝑏}, (4)|(𝜕^𝑘+𝑚/𝜕𝜌^𝑘𝜕𝑡^𝑚)𝑙₄(𝜌,𝜑,𝑡)|≤𝐶𝜌^{−𝑘+3−4𝑏}, (5)|(𝜕^𝑘/𝜕𝜌^𝑘)ℎ₁(𝜌,𝜑)|≤𝐶𝜌^{−𝑘+1−𝛾−(5/2)𝑏}, (6)|(𝜕^𝑘+𝑚/𝜕𝜌^𝑘𝜕𝑡^𝑚)ℎ₂(𝜌,𝜑,𝑡)|≤𝐶𝜌^−𝑘+𝜏, (7)|(𝜕^𝑘+𝑚/𝜕𝜌^𝑘𝜕𝑡^𝑚)ℎ₃(𝜌,𝜑,𝑡)|≤𝐶𝜌^{−𝑘+2−4𝑏},
where 𝛾=𝛽𝑏, 𝑎=max(3−(9/2)𝑏−𝛾,1−𝑏), and 𝜏=max(3−6𝑏,−𝑏).

Proof. (1) It is easy to know that (𝜕^𝑘/𝜕𝜌^𝑘)𝑙₁(𝜌,𝜑) is a sum of terms of the form 𝑑𝜕^𝑖1𝑥_𝜑𝜕𝜌^𝑖1⋅𝜕^𝑖2𝑓(𝑥)𝜕𝜌^𝑖2⋅𝜕^𝑖3𝑦𝜕𝜌^𝑖3,𝑖₁+𝑖₂+𝑖₃=𝑘, (28) where 0≤𝑖₁,𝑖₂,𝑖₃≤𝑘. Meanwhile, 𝜕^𝑖2𝑓(𝑥)/𝜕𝜌^𝑖2 is a sum terms of the form 𝑓^(𝑠)(𝑥)⋅𝜕^𝑙1𝑥𝜕𝜌^𝑙1𝜕^𝑙2𝑥𝜕𝜌^𝑙2⋯𝜕^𝑙𝑠𝑥𝜕𝜌^𝑙𝑠,0≤𝑠≤𝑖₂,𝑙₁+⋯+𝑙_𝑠=𝑖₂. (29) Hence, we obtain 𝜕^𝑘𝜕𝜌^𝑘𝑙₁𝜌,𝜑≤𝐶𝜌^−𝑖1𝑥⋅𝜌^−𝑖2𝑓(𝑥)⋅𝜌^−𝑖3𝑦≤𝐶𝜌^−𝑘⋅𝑥⋅𝑓(𝑥)⋅𝑦≤𝐶𝜌^−𝑘|𝑥|^{𝛼/2−𝛽}𝑦≤𝐶𝜌^{−𝑘+2−𝛾−(5/2)𝑏} (30) by the assumptions on 𝑓(𝑥) and the definitions of 𝑥(𝜌,𝜑) and 𝑦(𝜌,𝜑).
(2) From the expression of 𝑙₂(𝜌,𝜑,𝑡), we have 𝜕^𝑘+𝑚−𝑑𝜀𝑥_𝜑|𝑥|^𝛼−1𝑥𝑓(𝑥)𝐸(𝑡)𝜕𝜌^𝑘𝜕𝑡^𝑚𝜕≤𝐶^𝑘−𝑑𝜀𝑥_𝜑|𝑥|^𝛼−1𝑥𝑓(𝑥)𝜕𝜌^𝑘𝐸^(𝑚)(𝑡𝜌)≤𝐶𝜀^−𝑖1𝑥⋅𝜌^−𝑖2𝑓(𝑥)⋅𝜌^{𝛼𝑏−𝑖3}≤𝐶𝜀𝜌^{−𝑘+𝛼𝑏}𝑥⋅𝑓(𝑥)≤𝐶𝜀𝜌^{−𝑘+3−𝛾−9𝑏/2},𝜕^𝑘+𝑚−𝑑𝑦_𝜑𝑃(𝑡)𝜕𝜌^𝑘𝜕𝑡^𝑚𝜕≤𝐶^𝑘𝑑𝑦_𝜑𝜕𝜌^𝑘𝑑^𝑚𝑃(𝑡)𝑑𝑡^𝑚≤𝐶𝜌^{−𝑘+1−𝑏},𝜕^𝑘+𝑚−𝑑𝑥_𝜑𝑓(𝑥)𝑃(𝑡)𝜕𝜌^𝑘𝜕𝑡^𝑚𝜕≤𝐶^𝑘−𝑑𝑥_𝜑𝑓(𝑥)𝜕𝜌^𝑘𝑑^𝑚𝑃(𝑡)𝑑𝑡^𝑚≤𝐶𝜌^{−𝑘+1−𝛾−3𝑏/2}. (31)
We can find that 𝜕^𝑘+𝑚𝜕𝜌^𝑘𝜕𝑡^𝑚𝑙₂𝜌,𝜑,𝑡≤𝐶𝜌^−𝑘+𝑎, (32) where 𝑎=max(3−9𝑏/2−𝛾,1−𝑏).
(3) From the expression of 𝑙₃(𝜌,𝜑,𝑡), we have 𝜕^𝑘+𝑚𝑑𝑦_𝜑|𝑥|^𝛼−1𝑥𝐸(𝑡)𝜕𝜌^𝑘𝜕𝑡^𝑚𝜕≤𝐶^𝑘𝑑𝑦_𝜑|𝑥|^𝛼−1𝑥𝜕𝜌^𝑘𝑑^𝑚(𝐸(𝑡))𝑑𝑡^𝑚≤𝐶𝜌^{−𝑘+3−4𝑏}. (33)
(4) From the expression of 𝑙₄(𝜌,𝜑,𝑡), we can obtain that 𝜕^𝑘+𝑚𝑙₄𝜌,𝜑,𝑡𝜕𝜌^𝑘𝜕𝑡^𝑚𝜕≤𝐶^𝑘−𝑑𝜌^3−4𝑏𝐶𝜑𝑇₀𝜕𝜌^𝑘𝑑^𝑚(𝐸(𝑡))𝑑𝑡^𝑚𝜕+𝐶^𝑘−𝑑𝜌^2−3𝑏𝜕𝜌^𝑘𝑑^𝑚(𝑃(𝑡)𝐸(𝑡))𝑑𝑡^𝑚𝜕+𝐶^𝑘−𝑑𝜀𝜌^4−6𝑏𝑆𝜑𝑇₀^𝛼−1𝑆𝜑𝑇₀𝜕𝜌^𝑘𝑑^𝑚𝐸²(𝑡)𝑑𝑡^𝑚≤𝐶𝜌^{−𝑘+3−4𝑏}. (34)
(5) From the definition of ℎ₁(𝜌,𝜑), we have 𝜕^𝑘𝜕𝜌^𝑘ℎ₁𝜌𝜌,𝜑≤𝐶^−𝑖1−1𝑥⋅𝜌^−𝑖2𝑓(𝑥)⋅𝜌^−𝑖3𝑦≤𝐶𝜌^−𝑘⋅𝑥⋅𝑓(𝑥)⋅𝑦≤𝐶𝜌^−𝑘−1|𝑥|^{𝛼/2−𝛽}𝑦≤𝐶𝜌^{−𝑘+1−𝛾−(5/2)𝑏}. (35)
(6) From the definition of ℎ₂(𝜌,𝜑,𝑡), we can obtain 𝜕^𝑘+𝑚𝑑𝜀𝑥_𝜌|𝑥|^𝛼−1𝑥𝑓(𝑥)𝐸(𝑡)𝜕𝜌^𝑘𝜕𝑡^𝑚𝜕≤𝐶^𝑘𝑑𝑥_𝜌|𝑥|^𝛼−1𝑥𝑓(𝑥)𝜕𝜌^𝑘𝐸^(𝑚)(𝑡𝜌)≤𝐶𝜀^−𝑖1−1𝑥⋅𝜌^−𝑖2𝑓(𝑥)⋅𝜌^{𝛼𝑏−𝑖3}≤𝐶𝜀𝜌^{−𝑘+𝛼𝑏−1}𝑥⋅𝑓(𝑥)≤𝐶𝜀𝜌^{−𝑘+2−𝛾−9𝑏/2},𝜕^𝑘+𝑚𝛼𝑑𝜀²𝑥_𝜌|𝑥|^2𝛼−2𝑥𝐸²(𝑡)𝜕𝜌^𝑘𝜕𝑡^𝑚𝜕≤𝐶^𝑘𝛼𝑑𝜀²𝑥_𝜌|𝑥|^2𝛼−2𝑥𝜕𝜌^𝑘𝑑^𝑚𝐸²(𝑡)𝑑𝑡^𝑚≤𝐶𝜀𝜌^{−𝑘+3−6𝑏},𝜕^𝑘+𝑚−𝑑𝑦_𝜌𝑃(𝑡)𝜕𝜌^𝑘𝜕𝑡^𝑚𝜕≤𝐶^𝑘𝑑𝑦_𝜌𝜕𝜌^𝑘𝑑^𝑚𝑃(𝑡)𝑑𝑡^𝑚≤𝐶𝜌^{−𝑘−𝑏},𝜕^𝑘+𝑚𝑑𝑥_𝜌𝑓(𝑥)𝑃(𝑡)𝜕𝜌^𝑘𝜕𝑡^𝑚𝜕≤𝐶^𝑘−𝑑𝑥_𝜌𝑓(𝑥)𝜕𝜌^𝑘𝑑^𝑚𝑃(𝑡)𝑑𝑡^𝑚≤𝐶𝜌^{−𝑘−𝛾−3𝑏/2},𝜕^𝑘+𝑚𝛼𝑑𝜀𝑥_𝜌|𝑥|^𝛼−1𝐸(𝑡)𝑃(𝑡)𝜕𝜌^𝑘𝜕𝑡^𝑚𝜕≤𝐶^𝑘𝛼𝑑𝜀²𝑥_𝜌|𝑥|^2𝛼−2𝑥𝜕𝜌^𝑘𝑑^𝑚(𝐸(𝑡)𝑃(𝑡))𝑑𝑡^𝑚≤𝐶𝜀𝜌^{−𝑘+1−3𝑏}. (36) Hence, we can know that 𝜕^𝑘+𝑚𝜕𝜌^𝑘𝜕𝑡^𝑚ℎ₂𝜌,𝜑,𝑡≤𝐶𝜌^−𝑘+𝜏, (37) where 𝜏=max(3−6𝑏,−𝑏).
(7) From the expression of ℎ₃(𝜌,𝜑,𝑡), we have𝜕^𝑘+𝑚𝜕𝜌^𝑘𝜕𝑡^𝑚ℎ₃≤𝜕𝜌,𝜑,𝑡_𝑘+𝑚𝑑(1−𝑏+𝛼𝑏)𝜌^2−4𝑏𝑆𝜑𝑇₀^𝛼−1𝑆𝜑𝑇₀𝐶𝜑𝑇₀𝐸(𝑡)𝜕𝜌^𝑘𝜕𝑡^𝑚𝜕≤𝐶^𝑘𝑑(1−𝑏+𝛼𝑏)𝜌^2−4𝑏𝑆𝜑𝑇₀^𝛼−1𝑆𝜑𝑇₀𝐶𝜑𝑇₀𝜕𝜌^𝑘×𝑑^𝑚(𝐸(𝑡))𝑑𝑡^𝑚≤𝐶𝜌^{−𝑘+2−4𝑏}. (38)

For 𝜆₀>0, we define the domain 𝒜_𝜆0=𝜆,𝜑,𝑡∶𝜆≥𝜆₀,𝜑,𝑡∈𝕋². (39)

Lemma 7. There exists a 𝐺-invariant diffeomorphism Ψ₁: 𝜌=𝐼+𝑈₁(𝐼,𝜃),𝜑=𝜃 (40) such that 𝒜_𝐼⁺⊂Ψ₁(𝒜_𝐼0)⊂𝒜_𝐼− for some 𝐼₋<𝐼₀<𝐼₊. Under this transformation, (24) is transformed into the system 𝑑𝐼=𝑙̃𝑑𝑡¹(𝐼,𝜃)+𝑙₂̃(𝐼,𝜃,𝑡)+𝜀𝑙₃̃(𝐼,𝜃,𝑡)+𝛼𝑇₀𝑆𝜃𝑇₀^𝛼−1𝐶𝜃𝑇₀𝜀𝑙₄̃(𝐼,𝜃,𝑡),𝑑𝜃𝑑𝑡=ℎ^′₀(𝐼)+ℎ₁(𝐼,𝜃)+ℎ₂(𝐼,𝜃,𝑡)+𝜀ℎ₃(𝐼,𝜃,𝑡), (41) where 𝑙₁̃(𝐼,𝜃)=𝜕𝑉₁𝜌,𝜑𝜕𝜌⋅𝑙₁𝜌,𝜑+𝜕𝑉₁𝜌,𝜑𝜕𝜑⋅ℎ₁,𝑙̃𝜌,𝜑₂(𝐼,𝜃,𝑡)=𝑙₂+𝜌,𝜑,𝑡𝜕𝑉₁𝜌,𝜑⋅𝑙𝜕𝜌²𝜌,𝜑,𝑡+𝜀𝑙₃+𝜌,𝜑,𝑡𝜕𝑉₁𝜌,𝜑⋅ℎ𝜕𝜑²𝜌,𝜑,𝑡+𝜀ℎ₃𝑙𝜌,𝜑,𝑡+𝜀₃𝜌,𝜑,𝑡−𝑙₃(),𝑙̃𝐼,𝜃,𝑡₃(𝐼,𝜃,𝑡)=𝑙₃(),𝑙̃𝐼,𝜃,𝑡₄(𝐼,𝜃,𝑡)=𝑙₄𝜌,𝜑,𝑡+𝜕𝑉₁𝜌,𝜑𝜕𝜌⋅𝑙₄,ℎ𝜌,𝜑,𝑡₁(𝐼,𝜃)=ℎ^′₀𝜌−ℎ^′₀(𝐼)+ℎ₁,ℎ𝜌,𝜑₂(𝐼,𝜃,𝑡)=ℎ₂𝜌,𝜑,𝑡+𝜀ℎ₃𝜌,𝜑,𝑡−ℎ₃(),ℎ𝐼,𝜃,𝑡₃(𝐼,𝜃,𝑡)=ℎ₃(𝐼,𝜃,𝑡), (42) with 𝑉₁𝜌,𝜑=−^𝜑₀𝑙₁𝜌,𝑠ℎ^′₀𝜌𝑑𝑠. (43)

Proof. Define a transformation Φ₁ by Φ₁∶𝐼=𝜌+𝑉₁𝜌,𝜑,𝜃=𝜑. (44) By 𝑙₁𝜌,−𝜑=−𝑙₁,𝜕𝜌,𝜑_𝑘𝜕𝜌^𝑘𝑙₁𝜌,𝜑≤𝐶𝜌^{−𝑘+2−𝛾−(5/2)𝑏},0≤𝑘≤4, (45) we get 𝑉₁𝜌,−𝜑=𝑉₁,𝜕𝜌,𝜑_𝑘𝜕𝜌^𝑘𝑉₁𝜌,𝜑≤𝐶𝜌^{−𝑘+1−𝛾−𝑏/2}. (46)(47) Let Ψ₁=Φ₁⁻¹∶𝜌=𝐼+𝑈₁(𝐼,𝜃), 𝜑=𝜃. The system (24) is transformed into (41).

Lemma 8. For 𝐼 large enough, the following conclusions hold:(i)|𝜕^𝑘𝑈₁(𝐼,𝜃)/𝜕𝐼^𝑘|≤𝐶𝐼^{−𝑘+1−𝛾−𝑏/2}, (ii)𝑈₁(𝐼,−𝜃)=𝑈₁(𝐼,𝜃).

Proof. In view of 𝐼=𝜌+𝑉₁𝜌,𝜑,𝜌=𝐼+𝑈₁(𝐼,𝜃), (48) we obtain 𝑈₁(𝐼,𝜃)=−𝑉₁𝐼+𝑈₁(𝐼,𝜃),𝜃. (49) By |(𝜕^𝑘/𝜕𝜌^𝑘)𝑉₁(𝜌,𝜑)|≤𝐶𝜌^{−𝑘+1−𝛾−𝑏/2}, we have |(𝜕/𝜕𝜌)𝑉₁(𝜌,𝜑)|≤𝐶𝜌^{−𝛾−𝑏/2}≤1/2 for 𝜌 large enough. Hence, 𝑈₁ is uniquely determined by the contraction mapping principle. Moreover, 𝑈₁(⋅,𝜃)∈𝐶^∞(𝒜_𝐼0), for some 𝐼₀>0, as a consequence of the implicit function theorem and 𝐼^{−(1−𝛾−𝑏/2)}𝑈₁(𝐼,𝜃)≤𝐶. (50)
Above all, if 𝑘=1, from (47) and (49), we get 𝜕𝑈₁=𝜕𝐼𝜕𝑉₁/𝜕𝜌1+𝜕𝑉₁≤/𝜕𝜌^∞_𝑛=0𝐶𝜌^{−1+1−𝛾−𝑏/2𝑛+1}≤𝐶⋅𝜌^{−1+1−𝛾−𝑏/2}=𝐶⋅𝐼^{−1+1−𝛾−𝑏/2}𝑈1+¹𝐼^{−1+1−𝛾−𝑏/2}≤𝐶⋅𝐼^{−1+1−𝛾−𝑏/2}. (51)
We note that 𝜕^𝑘𝑈₁(𝐼,𝜃)𝜕𝐼^𝑘=𝜕^𝑘𝑉₁𝐼+𝑈₁(𝐼,𝜃),𝜃𝜕𝐼^𝑘, (52) and the right side hand is sum of the term 𝜕^𝑠𝑉₁𝜕𝜌^𝑠⋅𝜕^𝑘1𝐼+𝑈₁𝜕𝐼^𝑘1⋯𝜕^𝑘𝑠𝐼+𝑈₁𝜕𝐼^𝑘𝑠, (53) where 1≤𝑠≤𝑘, 𝑘₁+⋯+𝑘_𝑠=𝑘, 𝑘_𝑖≥1(for1≤𝑖≤𝑠). The highest order term in 𝑈₁ is the one with 𝑠=1, namely, (𝜕𝑉₁/𝜕𝜌)⋅(𝜕^𝑘(𝐼+𝑈₁)/𝜕𝐼^𝑘). We move the part (𝜕𝑉₁/𝜕𝜌)⋅(𝜕^𝑘𝑈₁/𝜕𝐼^𝑘) to the left hand side of (52). Since |(𝜕/𝜕𝜌)𝑉₁(𝜌,𝜑)|≤1/2 for 𝜌 large enough, this also provides immediately a bound on 𝜕^𝑘𝑈₁(𝐼,𝜃)/𝜕𝐼^𝑘. The rest part |(𝜕𝑉₁/𝜕𝜌)⋅(𝜕^𝑘𝐼/𝜕𝐼^𝑘)|≤𝐶𝐼^{−𝑘+1−𝛾−𝑏/2}.
Now, we proceed inductively by assuming that for 𝑗≤𝑘−1 the estimates 𝜕^𝑗𝑈₁(𝐼,𝜃)𝜕𝐼^𝑗≤𝐶𝐼^{−𝑗+1−𝛾−𝑏/2} (54) hold and we wish to conclude that the same estimate holds for 𝑗=𝑘.
Indeed, if 𝑠≥2, we have 𝜕^𝑠𝑉₁𝜕𝜌^𝑠⋅𝜕^𝑘1𝐼+𝑈₁𝜕𝐼^𝑘1⋯𝜕^𝑘𝑠𝐼+𝑈₁𝜕𝐼^𝑘𝑠≤𝐶⋅𝐼+𝑈₁^{−𝑠+1−𝛾−𝑏/2}⋅𝐼^−𝑘1+1⋯𝐼^{−𝑘𝑠+1}≤𝐶⋅𝐼^{−𝑘+1−𝛾−𝑏/2} (55) by 𝜕^𝑗𝐼+𝑈₁(𝐼,𝜃)𝜕𝐼^𝑗≤𝐶𝐼^−𝑗+1,1≤𝑗≤𝑘−1. (56) This proves (i) of Lemma 8.
Now we check (ii). In fact, since 𝑈₁(𝐼,𝜃)=−𝑉₁𝐼+𝑈₁(),𝑈𝐼,𝜃,𝜃₁(𝐼,−𝜃)=−𝑉₁𝐼+𝑈₁(𝐼,−𝜃),𝜃, (57) we have 𝑈₁(𝐼,𝜃)−𝑈₁(𝐼,−𝜃)≤sup_𝐼≥𝐼0𝜕𝑉₁𝑈𝜕𝜌¹(𝐼,𝜃)−𝑈₁(𝐼,−𝜃). (58) From (47), we have |(𝜕/𝜕𝜌)𝑉₁(𝜌,𝜑)|≤1/2 for 𝐼≥𝐼₀ sufficiently large and therefore we obtain 𝑈₁(𝐼,𝜃)=𝑈₁(𝐼,−𝜃).

By the estimates in Lemma 6, we can prove the following inequalities.

Lemma 9. For 0≤𝑘+𝑚≤4, the following inequalities hold:(1)|(𝜕^𝑘/𝜕𝐼^𝑘)𝑙₁̃(𝐼,𝜃)|≤𝐶𝐼^{−𝑘+2−2𝛾−3𝑏}, (2)|(𝜕^𝑘+𝑚/𝜕𝐼^𝑘𝜕𝑡^𝑚)𝑙₂̃(𝐼,𝜃,𝑡)|≤𝐶𝐼^−𝑘+𝑎, (3)|(𝜕^𝑘+𝑚/𝜕𝐼^𝑘𝜕𝑡^𝑚)𝑙₃̃(𝐼,𝜃,𝑡)|≤𝐶𝐼^{−𝑘+3−4𝑏}, (4)|(𝜕^𝑘+𝑚/𝜕𝐼^𝑘𝜕𝑡^𝑚)𝑙₄̃(𝐼,𝜃,𝑡)|≤𝐶𝐼^{−𝑘+3−4𝑏}, (5)|(𝜕^𝑘/𝜕𝐼^𝑘)ℎ₁(𝐼,𝜃)|≤𝐶𝐼^{−𝑘+1−𝛾−(5/2)𝑏}, (6)|(𝜕^𝑘+𝑚/𝜕𝐼^𝑘𝜕𝑡^𝑚)ℎ₂(𝐼,𝜃,𝑡)|≤𝐶𝐼^−𝑘+𝜏, (7)|(𝜕^𝑘+𝑚/𝜕𝐼^𝑘𝜕𝑡^𝑚)ℎ₃(𝐼,𝜃,𝑡)|≤𝐶𝐼^{−𝑘+2−4𝑏}.

Proof. (1) From the estimates (1) and (5) of Lemmas 6 and 8, it follows that𝜕^𝑘𝜕𝐼^𝑘𝑙₁̃(≤𝜕𝐼,𝜃)_𝑘𝜕𝐼^𝑘𝜕𝑉₁𝜌,𝜑𝜕𝜌⋅𝑙₁+𝜕𝜌,𝜑^𝑘𝜕𝐼^𝑘𝜕𝑉₁𝜌,𝜑𝜕𝜑⋅ℎ₁𝜌,𝜑≤𝐶_{𝑖1+𝑖2=𝑘}𝜕^𝑖1+1𝑉₁𝜌,𝜑𝜕𝐼^𝑖1𝜕𝜕𝜌^𝑖2𝑙₁𝜕𝐼^𝑖2+𝐶_{𝑖1+𝑖2+𝑖3=𝑘}𝜕^𝑖1𝜌^2𝑏−1𝜕𝐼^𝑖1𝜕^𝑖2𝑙₁𝜕𝐼^𝑖2𝜕^𝑖3ℎ₁𝜕𝐼^𝑖3≤𝐶_{𝑖1+𝑖2=𝑘𝜏1+⋯+𝜏𝑠=𝑖1}𝜕^𝑠+1𝑉₁𝜌,𝜑𝜕𝜌^𝑠+1𝜕^𝜏1𝐼+𝑈₁𝜕𝐼^𝜏1⋯𝜕^𝜏𝑠𝐼+𝑈₁𝜕𝐼^𝜏𝑠×𝜕^𝑖2𝑙₁𝜕𝐼^𝑖2+𝐶_{𝑖1+𝑖2+𝑖3=𝑘𝜏1+⋯+𝜏𝑠=𝑖1}𝜕^𝑠𝜌^2𝑏−1𝜕𝜌^𝑠𝜕^𝜏1𝐼+𝑈₁𝜕𝐼^𝜏1⋯𝜕^𝜏𝑠𝐼+𝑈₁𝜕𝐼^𝜏𝑠×𝜕^𝑖2𝑙₁𝜕𝐼^𝑖2𝜕^𝑖3ℎ₁𝜕𝐼^𝑖3≤𝐶𝜌^{−𝑘+2−2𝛾−3𝑏}≤𝐶𝐼^{−𝑘+2−2𝛾−3𝑏}. (59) (2) Since 𝑙₂̃(𝐼,𝜃,𝑡)=𝑙₂+𝜌,𝜑,𝑡𝜕𝑉₁𝜌,𝜑⋅𝑙𝜕𝜌²𝜌,𝜑,𝑡+𝜀𝑙₃+𝜌,𝜑,𝑡𝜕𝑉₁𝜌,𝜑⋅ℎ𝜕𝜑²𝜌,𝜑,𝑡+𝜀ℎ₃𝑙𝜌,𝜑,𝑡+𝜀₃𝜌,𝜑,𝑡−𝑙₃(𝐼,𝜃,𝑡), (60) we can prove that 𝜕^𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚𝜕𝑉₁𝜌,𝜑⋅𝑙𝜕𝜌²𝜌,𝜑,𝑡+𝜀𝑙₃𝜌,𝜑,𝑡≤𝐶𝐼^−𝑘+𝑎,𝜕^𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚𝜕𝑉₁𝜌,𝜑⋅ℎ𝜕𝜑²𝜌,𝜑,𝑡+𝜀ℎ₃𝜌,𝜑,𝑡≤𝐶𝐼^−𝑘+𝑎,𝜕^𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚𝑙₂𝜌,𝜑,𝑡≤𝐼^−𝑘+𝑎. (61) Their proofs are similar to the proofs in (1).
Next, we check the last part of   𝑙₂̃(𝐼,𝜃,𝑡). We get 𝜕^𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚𝑙₃𝜌,𝜑,𝑡−𝑙₃(=𝜕𝐼,𝜃,𝑡)_𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚10𝜕𝑙₃𝜕𝜌𝐼+𝑠𝑈₁(𝐼,𝜃),𝜃,𝑡⋅𝑈₁()≤𝐼,𝜃𝑑𝑠_{10𝑖1+𝑖2=𝑘}𝜕^𝑖1+𝑚𝜕𝐼^𝑖1𝜕𝑡^𝑚𝜕𝑙₃𝐼+𝑠𝑈₁,𝜃,𝑡𝜕𝜕𝜌^𝑖2𝑈₁𝜕𝐼^𝑖2𝑑𝑠≤𝐶¹_{0𝑖1+𝑖2=𝑘}𝐼+𝑈₁^{−𝑖1+2−4𝑏}𝐼^{−𝑖2+1−𝛾−𝑏/2}𝑑𝑠≤𝐶𝐼^{−𝑘+3−𝛾−(9/2)𝑏}≤𝐶𝐼^−𝑘+𝑎, (62) by the estimate in Lemma 6 and the definition of 𝑎.
(3) It is clearly by (3) in Lemma 6.
(4) It is clearly by (4) in Lemmas 6 and 8.
(5) We have that ℎ₁(𝐼,𝜃)=ℎ^′₀𝜌−ℎ^′₀(𝐼)+ℎ₁,𝜕𝜌,𝜑_𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚ℎ^′₀𝜌−ℎ^′₀(𝐼)≤𝜕^𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚10𝑑²ℎ₀𝐼+𝑠𝑈₁𝑑𝜌²𝑈₁(𝐼,𝜃)𝑑𝑠≤𝐶𝐼^{−𝑘−2𝑏+1−𝛾−𝑏/2}≤𝐶𝐼^{−𝑘+1−𝛾−5𝑏/2}. (63) From the last inequalities and (5) in Lemma 6, we obtain 𝜕^𝑘𝜕𝐼^𝑘ℎ₁(𝐼,𝜃)≤𝐶𝐼^{−𝑘+1−𝛾−(5/2)𝑏}. (64) (6) Since ℎ₂(𝐼,𝜃,𝑡)=ℎ₂𝜌,𝜑,𝑡+𝜀ℎ₃𝜌,𝜑,𝑡−ℎ₃(),𝜕𝐼,𝜃,𝑡_𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚ℎ₂𝜌,𝜑,𝑡≤𝐶𝐼^−𝑘+𝜏, (65) we just have to prove that 𝜕^𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚ℎ₃𝜌,𝜑,𝑡−ℎ₃(𝐼,𝜃,𝑡)≤𝐶𝐼^−𝑘+𝜏. (66) In fact, 𝜕^𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚ℎ₃𝜌,𝜑,𝑡−ℎ₃(=𝜕𝐼,𝜃,𝑡)_𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚10𝜕ℎ₃𝜕𝜌𝐼+𝑠𝑈₁(𝐼,𝜃),𝜃,𝑡⋅𝑈₁()≤𝐼,𝜃𝑑𝑠_{10𝑖1+𝑖2=𝑘}𝜕^𝑖1+𝑚𝜕𝐼^𝑖1𝜕𝑡^𝑚𝜕ℎ₃𝐼+𝑠𝑈₁,𝜃,𝑡𝜕𝜕𝜌^𝑖2𝑈₁𝜕𝐼^𝑖2𝑑𝑠≤𝐶¹_{0𝑖1+𝑖2=𝑘}𝐼+𝑈₁^{−𝑖1+1−4𝑏}𝐼^{−𝑖2+1−𝛾−𝑏/2}𝑑𝑠≤𝐶𝐼^{−𝑘+2−𝛾−(9/2)𝑏}≤𝐶𝐼^−𝑘+𝜏, (67) so we have proved (6).
(7) We have 𝜕^𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚ℎ₃(𝐼,𝜃,𝑡)≤𝐶𝐼^{−𝑘+2−4𝑏}, (68) by (7) in Lemma 6.

3. The Proof of Boundedness

In this section, all the solutions of (8) which are bounded will be proved via the KAM theory for reversible systems developed by Sevryuk [21] or Moser [22, 23] if 𝐵>0.

We define the functions 𝜂₀, 𝜂₁, 𝜂₂, 𝜂₃, 𝜉₁, 𝜉₂, and 𝜉₃ as 𝜂₀(𝐼)=1ℎ^′₀(𝐼),𝜂₁()ℎ𝐼,𝜃=−¹(𝐼,𝜃)ℎ^′₀(𝐼)ℎ^′₀(𝐼)+ℎ₁(),𝜂𝐼,𝜃₂()ℎ𝐼,𝜃,𝑡=−₂()×ℎ𝐼,𝜃,𝑡_′0(𝐼)+ℎ₁()×ℎ𝐼,𝜃_′0(𝐼)+ℎ₁(𝐼,𝜃)+ℎ₂(𝐼,𝜃,𝑡)+𝜀ℎ₃(𝐼,𝜃,𝑡)⁻¹,𝜂₃()ℎ𝐼,𝜃,𝑡=−₃()×ℎ𝐼,𝜃,𝑡_′0(𝐼)+ℎ₁()×ℎ𝐼,𝜃_′0(𝐼)+ℎ₁(𝐼,𝜃)+ℎ₂(𝐼,𝜃,𝑡)+𝜀ℎ₃(𝐼,𝜃,𝑡)⁻¹,𝜉₁(𝐼,𝜃,𝑡)=𝑙₁̃(𝐼,𝜃)+𝑙₂̃()⋅𝜂𝐼,𝜃,𝑡₀(𝐼)+𝜂₁(𝐼,𝜃)+𝜂₂(𝐼,𝜃,𝑡)+𝜀𝜂₃(),𝜉𝐼,𝜃,𝑡₂(𝐼,𝜃,𝑡)=𝑙₃̃()⋅𝜂𝐼,𝜃,𝑡₀(𝐼)+𝜂₁(𝐼,𝜃)+𝜂₂(𝐼,𝜃,𝑡)+𝜀𝜂₃(),𝜉𝐼,𝜃,𝑡₃(𝐼,𝜃,𝑡)=𝑙₄̃()⋅𝜂𝐼,𝜃,𝑡₀(𝐼)+𝜂₁(𝐼,𝜃)+𝜂₂(𝐼,𝜃,𝑡)+𝜀𝜂₃(𝐼,𝜃,𝑡). (69) Then system (41) is equivalent to the following system: 𝑑𝑡𝑑𝜃=𝜂₀(𝐼)+𝜂₁(𝐼,𝜃)+𝜂₂(𝐼,𝜃,𝑡)+𝜀𝜂₃(𝐼,𝜃,𝑡),𝑑𝐼𝑑𝜃=𝜉₁(𝐼,𝜃,𝑡)+𝜀𝜉₂(𝐼,𝜃,𝑡)+𝛼𝑇₀𝑆𝜃𝑇₀^𝛼−1𝐶𝜃𝑇₀𝜀𝜉₃(𝐼,𝜃,𝑡). (70)

In addition, one can verify that system (70) is reversible with respect to involution 𝐺∶(𝑡,𝐼)↦(−𝑡,𝐼).

Then some estimates on the functions 𝜂_𝑖(𝑖=0,1,2,3) and 𝜉_𝑖(𝑖=1,2,3) are given.

Lemma 10. The following inequalities hold:(1)𝑐𝐼^2𝑏−1≤|𝜂₀(𝐼)|≤𝐶𝐼^2𝑏−1, (2)|(𝜕^𝑘/𝜕𝐼^𝑘)𝜂₁(𝐼,𝜃)|≤𝐶𝐼^{−𝑘−1−𝛾+3𝑏/2}, (3)|(𝜕^𝑘+𝑚/𝜕𝐼^𝑘𝜕𝑡^𝑚)𝜂₂(𝐼,𝜃,𝑡)|≤𝐶𝐼^{−𝑘+𝜏+4𝑏−2}, (4)|(𝜕^𝑘+𝑚/𝜕𝐼^𝑘𝜕𝑡^𝑚)𝜂₃(𝐼,𝜃,𝑡)|≤𝐶𝐼^−𝑘, (5)|(𝜕^𝑘+𝑚/𝜕𝐼^𝑘𝜕𝑡^𝑚)𝜉₁(𝐼,𝜃,𝑡)|≤𝐶𝐼^{−𝑘+𝑎+2𝑏−1}, (6)|(𝜕^𝑘+𝑚/𝜕𝐼^𝑘𝜕𝑡^𝑚)𝜉₂(𝐼,𝜃,𝑡)|≤𝐶𝐼^{−𝑘+2−2𝑏}, (7)|(𝜕^𝑘+𝑚/𝜕𝐼^𝑘𝜕𝑡^𝑚)𝜉₃(𝐼,𝜃,𝑡)|≤𝐶𝐼^{−𝑘+2−2𝑏}, for 0≤𝑘+𝑚≤4.

Proof. (1) It is clear.
(2) Note that 1−2𝑏>1−𝛾−2𝑏/5, and ℎ₁(𝐼,𝜃)≤𝐶𝐼^{1−𝛾−(5/2)𝑏}, (71) it follows that ℎ^′₀(𝐼)+ℎ₁(𝐼,𝜃)≥ℎ^′₀(𝐼)−ℎ₁(≥ℎ𝐼,𝜃)_′0(𝐼)−ℎ₁(≥1𝐼,𝜃)𝑇₀𝐼^1−2𝑏−𝐶𝐼^{1−𝛾−(5/2)𝑏}≥𝑐𝐼^1−2𝑏 (72) as 𝐼≫1.
Moreover, we also have 𝜕^𝑙𝜕𝐼^𝑙ℎ^′₀(𝐼)+ℎ₁(≤𝜕𝐼,𝜃)^𝑙𝜕𝐼^𝑙ℎ^′₀(𝐼+𝜕)^𝑙𝜕𝐼^𝑙ℎ₁(𝐼,𝜃)≤𝐶𝐼^{−𝑙+1−2𝑏}+𝐶𝐼^{−𝑙+1−𝛾−(2/5)𝑏}≤𝐶𝐼^{−𝑙+1−2𝑏}. (73) So 𝜕^𝑖𝜕𝐼^𝑖1ℎ^′₀(𝐼)+ℎ₁(𝐼,𝜃)≤𝐶_{𝑙1+⋯+𝑙𝑠=𝑖}(−1)^𝑠𝑠!ℎ^′₀(𝐼)+ℎ₁(𝐼,𝜃)^𝑠+1×𝜕^𝑙1𝜕𝐼^𝑙1ℎ^′₀(𝐼)+ℎ₁()⋯𝜕𝐼,𝜃_𝑙𝑠𝜕𝐼^𝑙𝑠ℎ^′₀(𝐼)+ℎ₁(𝐼,𝜃)≤𝐶_{𝑙1+⋯+𝑙𝑠=𝑖}𝐼^{(2𝑏−1)(𝑠+1)}⋅𝐼^{−𝑖+(1−2𝑏)𝑠}≤𝐶𝐼^{−𝑖+2𝑏−1}. (74) From (72) and (74), it is easy to see that 𝜕^𝑘𝜕𝐼^𝑘𝜂₁(=𝜕𝐼,𝜃)_𝑘𝜕𝐼^𝑘ℎ₁(𝐼,𝜃)ℎ^′₀(𝐼)ℎ^′₀(𝐼)+ℎ₁(𝐼,𝜃)≤𝐶_{𝑖1+𝑖2+𝑖3=𝑘}𝜕^𝑖1𝜕𝐼^𝑖1ℎ₁(⋅𝜕𝐼,𝜃)_𝑖2𝜕𝐼^𝑖21ℎ^′₀(𝐼)⋅𝜕^𝑖3𝜕𝐼^𝑖31ℎ^′₀(𝐼)+ℎ₁(𝐼,𝜃)≤𝐶_{𝑖1+𝑖2+𝑖3=𝑘}𝐼^{−𝑖1+1−𝛾−(2/5)𝑏}𝐼^{−𝑖2−1+2𝑏}𝐼^{−𝑖3−1+2𝑏}≤𝐶𝐼^{−𝑘−1−𝛾+(3/2)𝑏}. (75) (3) We have 𝜕^𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚ℎ₂(𝐼,𝜃,𝑡)≤𝐶𝐼^−𝑘+𝜏,𝜕^𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚ℎ₃(𝐼,𝜃,𝑡)≤𝐶𝐼^{−𝑘+2−4𝑏}. (76) By (72), 1−2𝑏>𝜏(𝜏=max(3−6𝑏,−𝑏)) and 1−2𝑏>2−4𝑏, we have ℎ^′₀(𝐼)+ℎ₁(𝐼,𝜃)+ℎ₂(𝐼,𝜃,𝑡)+𝜀ℎ₃(≥ℎ𝐼,𝜃,𝑡)_′0(𝐼)+ℎ₁(𝐼,𝜃)−ℎ₂(𝐼,𝜃,𝑡)+𝜀ℎ₃(≥ℎ𝐼,𝜃,𝑡)_′0(𝐼)+ℎ₁(𝐼,𝜃)−ℎ₂(𝐼,𝜃,𝑡)−𝜀ℎ₃(𝐼,𝜃,𝑡)≥𝑐𝐼^1−2𝑏−𝐶𝐼^𝜏−𝐶𝜀𝐼^2−4𝑏≥𝑐𝐼^1−2𝑏, (77) for 𝐼≫1.
Let ℎ^′₀(𝐼)+ℎ₁(𝐼,𝜃)+ℎ₂(𝐼,𝜃,𝑡)+𝜀ℎ₃(𝐼,𝜃,𝑡)=𝐻(𝐼,𝜃,𝑡). We find that 𝜕^𝑙𝜕𝑡^𝑙1ℎ^′₀(𝐼)+ℎ₁(𝐼,𝜃)+ℎ₂(𝐼,𝜃,𝑡)+𝜀ℎ₃(𝐼,𝜃,𝑡)^𝑠+1≤𝐶_{𝑖1+⋯+𝑖𝑟=𝑙}(−1)^𝑟𝑟!(𝐻(𝐼,𝜃,𝑡))^𝑠+1+𝑟𝜕^𝑖1𝜕𝑡^𝑖1(𝐻(⋯𝜕𝐼,𝜃,𝑡))_𝑖𝑟𝜕𝑡^𝑖𝑟(𝐻(𝐼,𝜃,𝑡))≤𝐶_{𝑖1+⋯+𝑖𝑟=𝑙}𝐼^{(2𝑏−1)(𝑠+1+𝑟)}⋅𝜀^𝑟𝐼^{(2−4𝑏)𝑟}≤𝐶𝐼^{(2𝑏−1)(𝑠+1)}, (78) so 𝜕^𝑘+𝑙𝜕𝐼^𝑘𝜕𝑡^𝑙1ℎ^′₀(𝐼)+ℎ₁(𝐼,𝜃)+ℎ₂(𝐼,𝜃,𝑡)+𝜀ℎ₃()𝜕𝐼,𝜃,𝑡≤𝐶^𝑙𝜕𝑡^𝑙_{𝑖1+⋯+𝑖𝑠=𝑘}(−1)^𝑠𝑠!(𝐻(𝐼,𝜃,𝑡))^𝑠+1⋅𝜕^𝑖1𝜕𝐼^𝑖1(𝐻(⋯𝜕𝐼,𝜃,𝑡))_𝑖𝑠𝜕𝐼^𝑖𝑠(𝐻(𝐼,𝜃,𝑡))≤𝐶_{𝑖1+⋯+𝑖𝑠=𝑘𝑗0+𝑗1+⋯+𝑗𝑠=𝑙}𝜕^𝑗0𝜕𝑡^𝑗0(−1)^𝑠𝑠!(𝐻(𝐼,𝜃,𝑡))^𝑠+1×𝜕^𝑖1+𝑗1𝜕𝐼^𝑖1𝜕𝑡^𝑗1(𝐻(⋯𝜕𝐼,𝜃,𝑡))_{𝑖𝑠+𝑗𝑠}𝜕𝐼^𝑖𝑠𝜕𝑡^𝑗𝑠(𝐻(𝐼,𝜃,𝑡))≤𝐶_{𝑖1+⋯+𝑖𝑠=𝑘}𝐼^{(2𝑏−1)(𝑠+1)}⋅𝐼^{−(𝑖1+⋯+𝑖𝑠)−(1−2𝑏)𝑠}≤𝐶𝐼^{−𝑘+(2𝑏−1)}. (79)
When 𝑚=0, the proof of (3) is similar to the proof of (2).
When 𝑚>0, then 𝜕^𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚𝜂₂(=𝜕𝐼,𝜃,𝑡)_𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚ℎ₂(𝐼,𝜃,𝑡)ℎ^′₀(𝐼)+ℎ₁(𝐼,𝜃)ℎ^′₀(𝐼)+ℎ₁+ℎ₂+𝜀ℎ₃𝜕≤𝐶^𝑚𝜕𝑡^𝑚_{𝑖1+𝑖2+𝑖3=𝑘}𝜕^𝑖1𝜕𝐼^𝑖1ℎ₂⋅𝜕^𝑖2𝜕𝐼^𝑖21ℎ^′₀(𝐼)+ℎ₁⋅𝜕^𝑖3𝜕𝐼^𝑖31ℎ^′₀(𝐼)+ℎ₁+ℎ₂+𝜀ℎ₃≤𝐶_{𝑖1+𝑖2+𝑖3=𝑘𝑙1+𝑙2=𝑚}𝜕^𝑖1+𝑙1ℎ₂(𝐼,𝜃,𝑡)𝜕𝐼^𝑖1𝜕𝑡^𝑙1⋅𝜕^𝑖2𝜕𝐼^𝑖21ℎ^′₀(𝐼)+ℎ₁()⋅𝜕𝐼,𝜃_𝑖3+𝑙2𝜕𝐼^𝑖3𝜕𝑡^𝑙21ℎ^′₀(𝐼)+ℎ₁+ℎ₂+𝜀ℎ₃≤𝐶_{𝑖1+𝑖2+𝑖3=𝑘}𝐼^{−𝑖1+𝜏}𝐼^{−𝑖2+2𝑏−1}𝐼^{−𝑖3+2𝑏−1}≤𝐶𝐼^{−𝑘+𝜏+4𝑏−2}. (80)
(4) The proof of (4) is similar to the proof of (3).
(5) Let 𝜂₀(𝐼)+𝜂₁(𝐼,𝜃)+𝜂₂(𝐼,𝜃,𝑡)+𝜀𝜂₃(𝐼,𝜃,𝑡)=𝜂(𝐼,𝜃,𝑡). By using the estimates on the functions 𝑙_𝑖̃(𝑖=1,2) and 𝜂_𝑗(𝑗=0,1,2,3), it follows that𝜕^𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚𝜉₁(𝜕𝐼,𝜃,𝑡)≤𝐶^𝑚𝜕𝑡^𝑚_{𝑘1+𝑘2=𝑘}𝜕^𝑘1𝑙₁̃(𝐼,𝜃)+𝑙₂̃(𝐼,𝜃,𝑡)𝜕𝐼^𝑘1⋅𝜕^𝑘2𝜂(𝐼,𝜃,𝑡)𝜕𝐼^𝑘2≤𝐶_{𝑘1+𝑘2=𝑘}𝜕^𝑘1𝑙₁̃(𝐼,𝜃)𝜕𝐼^𝑘1⋅𝜕^𝑘2+𝑚𝜂₂(𝐼,𝜃,𝑡)+𝜀𝜂₃(𝐼,𝜃,𝑡)𝜕𝐼^𝑘2𝜕𝑡^𝑚+𝐶_{𝑙𝑘+𝑘2=𝑘𝑚1+𝑚2=𝑚}𝜕^𝑘1+𝑚1𝑙₂̃(𝐼,𝜃,𝑡)𝜕𝐼^𝑘1𝜕𝑡^𝑚1⋅𝜕^𝑘2+𝑚2𝜂(𝐼,𝜃,𝑡)𝜕𝐼^𝑘2𝜕𝑡^𝑚2≤𝐶𝐼^{−𝑘1+2−𝛾−(5/2)𝑏}⋅𝐼^{−𝑘2+𝜏+4𝑏−2}+𝜀𝐼^−𝑘2+𝐶𝐼^{−𝑘1+𝑎}⋅𝐼^{−𝑘2+2𝑏−1}≤𝐶𝐼^{−𝑘+𝑎+2𝑏−1}, (81)when 𝑚≠0.
When 𝑚=0, then 𝜕^𝑘𝜕𝐼^𝑘𝜉₁(𝐼,𝜃,𝑡)≤𝐶_{𝑘1+𝑘2=𝑘}𝜕^𝑘1𝑙₁̃(𝐼,𝜃)+𝑙₂̃(𝐼,𝜃,𝑡)𝜕𝐼^𝑘1⋅𝜕^𝑘2𝜂(𝐼,𝜃,𝑡)𝜕𝐼^𝑘2≤𝐶𝐼^{−𝑘1+2−𝛾−(5/2)𝑏}+𝐼^{−𝑘1+𝑎}⋅𝐼^{−𝑘2+2𝑏−1}≤𝐶𝐼^{−𝑘+𝑎+2𝑏−1}, (82) by 𝑎>2−𝛾−5𝑏/2.
(6) By using the estimates on the functions 𝑙₃̃ and 𝜂_𝑖(𝑖=0,1,2,3), it follows that 𝜕^𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚𝜉₂(𝜕𝐼,𝜃,𝑡)≤𝐶^𝑚𝜕𝑡^𝑚_{𝑘1+𝑘2=𝑘}𝜕^𝑘1𝑙₃̃(𝐼,𝜃,𝑡)𝜕𝐼^𝑘1⋅𝜕^𝑘2𝜂(𝐼,𝜃,𝑡)𝜕𝐼^𝑘2≤𝐶_{𝑙𝑘+𝑘2=𝑘𝑚1+𝑚2=𝑚}𝜕^𝑘1+𝑚1𝑙₃̃(𝐼,𝜃,𝑡)𝜕𝐼^𝑘1𝜕𝑡^𝑚1⋅𝜕^𝑘2+𝑚2𝜂(𝐼,𝜃,𝑡)𝜕𝐼^𝑘2𝜕𝑡^𝑚2≤𝐶𝐼^{−𝑘1+3−4𝑏}⋅𝐼^{−𝑘2+2𝑏−1}≤𝐶𝐼^{−𝑘+2−2𝑏}. (83)
(7) By using the estimates on the functions 𝑙₄̃ and 𝜂_𝑖(𝑖=0,1,2,3), it follows that 𝜕^𝑘+𝑚𝜕𝐼^𝑘𝜕𝑡^𝑚𝜉₃(𝜕𝐼,𝜃,𝑡)≤𝐶^𝑚𝜕𝑡^𝑚_{𝑘1+𝑘2=𝑘}𝜕^𝑘1𝑙₄̃(𝐼,𝜃,𝑡)𝜕𝐼^𝑘1⋅𝜕^𝑘2𝜂(𝐼,𝜃,𝑡)𝜕𝐼^𝑘2≤𝐶_{𝑙𝑘+𝑘2=𝑘𝑚1+𝑚2=𝑚}𝜕^𝑘1+𝑚1𝑙₄̃(𝐼,𝜃,𝑡)𝜕𝐼^𝑘1𝜕𝑡^𝑚1⋅𝜕^𝑘2+𝑚2𝜂(𝐼,𝜃,𝑡)𝜕𝐼^𝑘2𝜕𝑡^𝑚2≤𝐶𝐼^{−𝑘1+3−4𝑏}⋅𝐼^{−𝑘2+2𝑏−1}≤𝐶𝐼^{−𝑘+2−2𝑏}. (84)

Let 𝑡=𝑡, 𝜃=𝜃, 𝑟=𝜂₀(𝐼) and 𝐹₀(𝑟,𝜃)=𝜂₁(𝐼(𝑟)),𝐹,𝜃₁(𝑟,𝜃,𝑡)=𝜂₂(𝐼(𝑟),𝜃,𝑡)+𝜀𝜂₃(𝐼(𝑟)),𝐹,𝜃,𝑡₂(𝑟,𝜃,𝑡)=𝜂^′₀(𝐼(𝑟))⋅𝜉₁(𝐼(𝑟),𝜃,𝑡)+𝜀𝜉₂(𝐼(𝑟)),𝐹,𝜃,𝑡₃(𝑟,𝜃,𝑡)=𝜀𝜂^′₀(𝐼(𝑟))⋅𝜉₃(𝐼(𝑟),𝜃,𝑡), (85) where 𝐼(𝑟) is the inverse function of 𝑟=𝜂₀(𝐼).

Then system (70) is transformed into the following form: 𝑑𝑡𝑑𝜃=𝑟+𝐹₀(𝑟,𝜃)+𝐹₁(𝑟,𝜃,𝑡),𝑑𝑟𝑑𝜃=𝐹₂(𝑟,𝜃,𝑡)+𝛼𝑇₀𝑆𝜃𝑇₀^𝛼−1𝐶𝜃𝑇₀⋅𝐹₃(𝑟,𝜃,𝑡). (86) Moreover, one can verify that system (86) is reversible with respect to involution 𝐺∶(𝑡,𝑟)↦(−𝑡,𝑟).

It is easy to see that 𝐼≫1 if and only if 𝑟≫1, and the solutions of system (86) do exist on 0≤𝜃≤1 when 𝑟(0)=𝑟≫1.

By using the estimates on 𝜂_𝑖 and 𝜉_𝑖(𝑖=1,2,3) in Lemma 10, the following inequalities can be proved.

Lemma 11. For 0≤𝑘+𝑚≤4 and 𝑟≫1, the following inequalities hold:(1)|(𝜕^𝑘/𝜕𝑟^𝑘)𝐹₀(𝑟,𝜃)|≤𝐶𝑟^{−𝑘−(1+𝛾−3𝑏/2)/(2𝑏−1)}, (2)|(𝜕^𝑘+𝑚/𝜕𝑟^𝑘𝜕𝑡^𝑚)𝐹₁(𝑟,𝜃,𝑡)|≤𝐶(𝑟^{−𝑘+(𝜏+4𝑏−2)/(2𝑏−1)}+𝜀𝑟^−𝑘), (3)|(𝜕^𝑘+𝑚/𝜕𝑟^𝑘𝜕𝑡^𝑚)𝐹₂(𝑟,𝜃,𝑡)|≤𝐶(𝑟^{−𝑘+(𝑎+4𝑏−3)/(2𝑏−1)}+𝜀𝑟^−𝑘), (4)|(𝜕^𝑘+𝑚/𝜕𝑟^𝑘𝜕𝑡^𝑚)𝐹₃(𝑟,𝜃,𝑡)|≤𝐶𝜀𝑟^−𝑘.

Proof. Above all, we know that 𝑟=𝜂₀(𝐼)=𝑇₀𝐼^2𝑏−1, so we can get 𝐼=((1/𝑇₀)𝑟)^{1/(2𝑏−1)}. Then we have 𝑑^𝑗𝐼𝑑𝑟^𝑗≤𝐶𝑟^{−𝑗+1/(2𝑏−1)},𝑑^𝑗𝜂^′₀(𝐼(𝑟))𝑑𝑟^𝑗𝑑≤𝐶^𝑗𝑟^{1−1/(2𝑏−1)}𝑑𝑟^𝑗≤𝐶𝑟^{−𝑗+(2𝑏−2)/(2𝑏−1)}. (87)
(1) We have that 𝜕^𝑘𝐹₀(𝑟,𝜃)𝜕𝑟^𝑘≤_{𝑘1+⋯+𝑘𝑠=𝑘}𝜕^𝑠𝜂₁(𝐼,𝜃)𝜕𝐼^𝑠⋅𝑑^𝑘1𝐼𝑑𝑟^𝑘1⋯𝑑^𝑘𝑠𝐼𝑑𝑟^𝑘𝑠≤𝐶𝐼^{−𝑠−1−𝛾+(3/2)𝑏}𝑟^{−𝑘+(1/(2𝑏−1))𝑠}≤𝐶𝑟^{−𝑠(1/(2𝑏−1))−(1+𝛾−(3/2)𝑏)/(2𝑏−1)}𝑟^{−𝑘+(1/(2𝑏−1))𝑠}≤𝐶𝑟^{−𝑘−(1+𝛾−(3/2)𝑏)/(2𝑏−1)}. (88)
(2) We have that 𝜕^𝑘+𝑚𝐹₁(𝑟,𝜃,𝑡)𝜕𝑟^𝑘𝜕𝑡^𝑚≤𝐶_{𝑖1+⋯+𝑖𝑠=𝑘}𝜕^𝑠+𝑚𝜂₂(𝐼,𝜃,𝑡)𝜕𝐼^𝑠𝜕𝑡^𝑚𝑑^𝑖1𝐼𝑑𝑟^𝑖1⋯𝑑^𝑖𝑠𝐼𝑑𝑟^𝑖𝑠+𝐶𝜀_{𝑗1+⋯+𝑗𝜈=𝑘}𝜕^𝜈+𝑚𝜂₃(𝐼,𝜃,𝑡)𝜕𝐼^𝜈𝜕𝑡^𝑚𝑑^𝑗1𝐼𝑑𝑟^𝑗1⋯𝑑^𝑗𝜈𝐼𝑑𝑟^𝑗𝜈≤𝐶𝑟^{−𝑘+((𝜏+4𝑏−2)/(2𝑏−1))}+𝜀𝑟^−𝑘. (89)
(3) We have that𝜕^𝑘+𝑚𝐹₂(𝑟,𝜃,𝑡)𝜕𝑟^𝑘𝜕𝑡^𝑚≤_{𝑘1+𝑘2=𝑘}𝑑^𝑘1𝜂^′₀(𝐼(𝑟))𝑑𝑟^𝑘1⋅𝜕^𝑘2+𝑚𝜉₁(𝐼(𝑟),𝜃,𝑡)𝜕𝑟^𝑘2𝜕𝑡^𝑚𝜕+𝜀^𝑘2+𝑚𝜉₂(𝐼(𝑟),𝜃,𝑡)𝜕𝑟^𝑘2𝜕𝑡^𝑚≤𝐶_{𝑘1+𝑘2=𝑘}𝑟^{−𝑘1+(2𝑏−2)/(2𝑏−1)}×_{𝑖1+⋯+𝑖𝑠=𝑘2}𝜕^𝑠+𝑚𝜉₁(𝐼,𝜃,𝑡)𝜕𝐼^𝑠𝜕𝑡^𝑚𝑑^𝑖1𝐼𝑑𝑟^𝑖1⋯𝑑^𝑖𝑠𝐼𝑑𝑟^𝑖𝑠+𝜀_{𝑗1+⋯+𝑗𝜈=𝑘2}𝜕^𝜈+𝑚𝜉₂(𝐼,𝜃,𝑡)𝜕𝐼^𝜈𝜕𝑡^𝑚𝑑^𝑗1𝐼𝑑𝑟^𝑗1⋯𝑑^𝑗𝜈𝐼𝑑𝑟^𝑗𝜈≤𝐶𝑟^{−𝑘1+(2𝑏−2)/(2𝑏−1)}𝑟^{(1/(2𝑏−1))(−𝑠+𝑎+2𝑏−1)}𝑟^{−𝑘1+(1/(2𝑏−1))𝑠}+𝐶𝜀𝑟^{−𝑘1+(2𝑏−2)/(2𝑏−1)}𝑟^{(1/(2𝑏−1))(−𝜈+2−2𝑏)}𝑟^{−𝑘1+(1/(2𝑏−1))𝜈}≤𝐶𝑟^{−𝑘+(𝑎+4𝑏−3)/(2𝑏−1)}+𝐶𝜀𝑟^−𝑘. (90) (4) We have that𝜕^𝑘+𝑚𝐹₃(𝑟,𝜃,𝑡)𝜕𝑟^𝑘𝜕𝑡^𝑚≤𝜀_{𝑘1+𝑘2=𝑘}𝑑^𝑘1𝜂^′₀(𝐼(𝑟))𝑑𝑟^𝑘1⋅𝜕^𝑘2+𝑚𝜉₃(𝐼(𝑟),𝜃,𝑡)𝜕𝑟^𝑘2𝜕𝑡^𝑚≤𝐶𝜀_{𝑘1+𝑘2=𝑘}𝑟^{−𝑘1+(2𝑏−2)/(2𝑏−1)}×_{𝑖1+⋯+𝑖𝑠=𝑘2}𝜕^𝑠+𝑚𝜉₃(𝐼,𝜃,𝑡)𝜕𝐼^𝑠𝜕𝑡^𝑚𝑑^𝑖1𝐼𝑑𝑟^𝑖1⋯𝑑^𝑖𝑠𝐼𝑑𝑟^𝑖𝑠≤𝐶𝜀𝑟^−𝑘. (91)

Lemma 12. The time 1 map Φ¹ of the flow Φ^𝜃 of the system (86) is of the form Φ¹∶𝑟₁=𝑟+𝑄₂(𝑟,𝑡),𝑡₁=𝑡+𝜔(𝑟)+𝑄₁(𝑟,t), (92) where 𝜔(𝑟)=𝑟+∫¹₀𝐹₀(𝑟,𝜃)𝑑𝜃. And there exists a 𝜇₀>0 such that, for 0≤𝑘+𝑚≤4, sufficiently large 𝑟 and sufficiently small 𝜀, 𝜕^𝑘+𝑚𝜕𝑟^𝑘𝜕𝑡^𝑚𝑄_𝑖(𝑟,𝑡)≤𝐶𝑟^−𝜇0+𝜀,𝑖=1,2 (93) hold. Moreover, the map Φ¹ is reversible with respect to the involution 𝐺∶(𝑡,𝑟)↦(−𝑡,𝑟).

Proof. Since ¹₀𝛼𝑇₀𝑆𝜃𝑇₀^𝛼−1𝐶𝜃𝑇₀𝑑𝜃=lim_{𝜖→0+𝜖}^1/4𝛼𝑇₀𝑆𝜃𝑇₀^𝛼−1𝐶𝜃𝑇₀𝑑𝜃−lim_𝜖→0+^1/2−𝜖_1/4𝛼𝑇₀𝑆𝜃𝑇₀^𝛼−1𝐶𝜃𝑇₀𝑑𝜃−lim_𝜖→0+^3/4_1/2+𝜖𝛼𝑇₀𝑆𝜃𝑇₀^𝛼−1𝐶𝜃𝑇₀𝑑𝜃+lim_𝜖→0+^1−𝜖_3/4𝛼𝑇₀𝑆𝜃𝑇₀^𝛼−1𝐶𝜃𝑇₀=𝑆𝑇𝑑𝜃₀4^𝛼−1𝑆𝑇₀4−lim_𝜖→0+𝑆𝜖𝑇₀^𝛼−1𝑆𝜖𝑇₀−lim_𝜖→0+𝑆12−𝜖𝑇₀^𝛼−1𝑆12−𝜖𝑇₀−𝑆𝑇₀4^𝛼−1𝑆𝑇₀4−𝑆3𝑇₀4^𝛼−1𝑆3𝑇₀4−lim_𝜖→0+𝑆12+𝜖𝑇₀^𝛼−1𝑆12+𝜖𝑇₀+lim_𝜖→0+𝑆(1−𝜖)𝑇₀^𝛼−1𝑆(1−𝜖)𝑇₀−𝑆3𝑇₀4^𝛼−1𝑆3𝑇₀4𝑆𝑇=4⁰4^𝛼−1𝑆𝑇₀4=4, (94) then we get ∫¹₀𝛼𝑇₀|𝑆(𝜃𝑇₀)|^𝛼−1|𝐶(𝜃𝑇₀)|𝑑𝜃 is bounded.
Let 𝛼𝑇₀|𝑆(𝜐𝑇₀)|^𝛼−1𝐶(𝜐𝑇₀)=𝑆₁(𝜐). Set (𝑟(𝜃),𝑡(𝜃))=Φ^𝜃(𝑟,𝑡) with Φ⁰=𝑖𝑑 for the flow: 𝑡(𝜃)=𝑡+𝑟𝜃+𝐷₁(𝑟,𝑡,𝜃),𝑟(𝜃)=𝑟+𝐷₂(𝑟,𝑡,𝜃). (95) Since Φ^𝜃=Φ⁰+^𝜃₀𝑋⋅Φ^𝜐𝑑𝜐, (96) where 𝑋 denotes the vector field of the system (86), we have 𝑡(𝜃)+=𝑡_𝜃0𝑟(𝜐)+𝐹₀(𝑟(𝜐),𝜐)+𝐹₁(𝑟(𝜐)(𝜐+,𝜐,𝑡))𝑑𝜐=𝑡+𝑟𝜃_𝜃0𝐷₂(𝑟,𝑡,𝜐)+𝐹₀𝑟+𝐷₂,𝜐+𝐹₁𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁𝑑𝜐=𝑡+𝑟𝜃+𝐷₁(),𝑟(𝜃)+𝑟,𝑡,𝜃=𝑟_𝜃0𝐹₂(𝑟(𝜐),𝜐,𝑡(𝜐))+𝑆₁(𝜐)𝐹₃(𝑟(𝜐)(𝜐+,𝜐,𝑡))𝑑𝜐=𝑟_𝜃0𝐹₂𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁+𝑆₁(𝜐)𝐹₃𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁𝑑𝜐=𝑟+𝐷₂(𝑟,𝑡,𝜃), (97) which is equivalent to the following equations for 𝐷₁ and 𝐷₂: 𝐷₁(𝑟,𝑡,𝜃)=^𝜃₀𝐷₂(𝑟,𝑡,𝜐)+𝐹₀𝑟+𝐷₂,𝜐+𝐹₁𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁𝐷𝑑𝜐,₂(𝑟,𝑡,𝜃)=^𝜃₀𝐹₂𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁+𝑆₁(𝜐)𝐹₃𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁𝑑𝜐. (98)
Let 𝐷(𝑟,𝑡,𝜃)=(𝐷₁(𝑟,𝑡,𝜃),𝐷₂(𝑟,𝑡,𝜃)), |𝐷₁(𝑟,𝑡,𝜃)|=sup₊^{(𝑟,𝑡,𝜃)∈}_{(𝑅×𝕋×(0,1])}|𝐷₁(𝑟,𝑡,𝜃)|. Define ‖𝐷‖=∶|𝐷₁|/3+2|𝐷₁|/3, and 𝑇(𝐷)=∶(𝑇₁(𝐷),𝑇₂(𝐷)), where 𝑇₁(𝐷)=^𝜃₀𝐷₂(𝑟,𝑡,𝜐)+𝐹₀𝑟+𝐷₂,𝜐+𝐹₁𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁𝑇𝑑𝜐,₂(𝐷)=^𝜃₀𝐹₂𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁+𝑆₁(𝜐)𝐹₃𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁𝑑𝜐. (99) Next, we will prove that 𝑇 is a contraction map. From the definition of 𝑇(𝐷), we have 𝑇₁𝐷−𝑇₁𝐷=^𝜃₀𝐷₂−𝐷₂+𝐹₀𝑟+𝐷₂,𝜐−𝐹₀𝑟+𝐷₂+𝐹₁𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁−𝐹₁𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁≤𝐷𝑑𝜐₂−𝐷₂+¹₀𝜕𝐹₀𝑟+𝑠𝐷₂−𝐷₂,𝜐⋅𝐷𝜕𝑟²−𝐷₂+𝑑𝑠₁₀𝜕𝐹₁𝑟+𝑠𝐷₂−𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝐷𝜕𝑟₂−𝐷₂+𝑑𝑠₁₀𝜕𝐹₁𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝑠𝐷₁−𝐷₁⋅𝐷𝜕𝑡₁−𝐷₁≤6𝑑𝑠5𝐷₂−𝐷₂+14𝐷₁−𝐷₁,𝑇₂𝐷−𝑇₂𝐷=^𝜃₀𝐹₂𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁−𝐹₂𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁+𝑆₁(𝜐)𝐹₃𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁−𝑆₁(𝜐)𝐹₃𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁≤𝑑𝜐₁₀𝜕𝐹₂𝑟+𝑠𝐷₂−𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝐷𝜕𝑟₂−𝐷₂𝑑𝑠+¹₀𝜕𝐹₂𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝑠𝐷₁−𝐷₁⋅𝐷𝜕𝑡₁−𝐷₁+𝑑𝑠₁₀𝑆₁(𝜐⋅)𝑑𝜐₁₀𝜕𝐹₃𝑟+𝑠𝐷₂−𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝐷𝜕𝑟₂−𝐷₂+𝑑𝑠₁₀𝑆₁(𝜐⋅)𝑑𝜐₁₀𝜕𝐹₃𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝑠𝐷₁−𝐷₁⋅𝐷𝜕𝑡₁−𝐷₁≤3𝑑𝑠𝐷20²−𝐷₂+18𝐷₁−𝐷₁, (100) by Lemma 11 and the boundedness of ∫¹₀|𝑆₁(𝜐)|𝑑𝜐. Then we have 𝑇(𝐷)𝐷=1−𝑇3𝑇₁(𝐷)−𝑇₁𝐷+23𝑇₂(𝐷)−𝑇₂𝐷≤13×65𝐷₂−𝐷₂+14𝐷₁−𝐷₁+23×3𝐷20²−𝐷₂+18𝐷₁−𝐷₁=16𝐷₁−𝐷₁+12𝐷₂−𝐷₂≤34×13𝐷₁−𝐷₁+23𝐷₂−𝐷₂≤34𝐷−𝐷, (101) by the definition of the norm ‖⋅‖.
Using the contraction principle, one verifies easily that for 𝑟≥𝑟₀, (98) has a unique solution in the space {|𝐷₁|≤1,|𝐷₂|≤1}. Moreover, 𝐷₁ and 𝐷₂ are smooth.
Next, we will estimate 𝑄₁(𝑟,𝑡) and 𝑄₂(𝑟,𝑡) as follows: 𝑄₁(𝑟,𝑡)=𝐷₁(𝑟,𝑡,1)−¹₀𝐹₀(𝑟,𝜐)𝑑𝜐=¹₀𝐷₂()+𝑟,𝑡,𝜐₁₀𝜕𝐹₀𝑟+𝑠𝐷₂,𝜐𝜕𝑟⋅𝐷₂𝑑𝑠+𝐹₁𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁𝑄𝑑𝜐,₂(𝑟,𝑡)=𝐷₂()=𝑟,𝑡,1₁₀𝐹₂𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁+𝑆₁(𝜐)𝐹₃𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁𝑑𝜐. (102) In order to prove (93), we just need to prove that 𝜕^𝑘+𝑚𝜕𝑟^𝑘𝜕𝑡^𝑚𝐷_𝑖(𝑟,𝑡,𝜃)≤𝐶𝑟^−𝜇0+𝐶𝜀,𝑖=1,2 (103) hold for 𝑘+𝑚≤4.
(1) When 𝑘+𝑚=0, 𝐷₂(≤𝑟,𝑡,𝜃)_𝜃0𝐹₂𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁+𝑆₁(𝜐)𝐹₃𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁𝑟𝑑𝜐≤𝐶^−𝜇0+𝜀+¹₀𝑆₁(𝜐)𝑑𝜐⋅(𝐶𝜀)≤𝐶𝑟^−𝜇0,𝐷+𝜀₁(≤𝐷𝑟,𝑡,𝜃)₂(+𝑟,𝑡,𝜃)_𝜃0𝐹₀𝑟+𝐷₂,𝜐+𝐹₁𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁≤𝐷𝑑𝜐,₂(𝑟,𝑡,𝜐)+^𝜃₀𝐶𝑟^−𝜇0+𝐶𝑟^−𝜇0≤𝐷+𝐶𝜀𝑑𝜐₂(𝑟,𝑡,𝜃)+𝐶𝑟^−𝜇0+𝜀≤𝐶𝑟^−𝜇0+𝜀, (104) where 𝜇₀=min((1+𝛾−3𝑏/2)/(2𝑏−1),(2−4𝑏−𝜏)/(2𝑏−1),(3−4𝑏−𝑎)/(2𝑏−1)).
(2) When 𝑚=0 and 𝑘≠0, we check the case when 𝑘=1 firstly 𝜕𝐷₂(𝑟,𝑡,𝜃)≤𝜕𝑟_𝜃0𝜕𝐹₂𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝑟1+𝜕𝐷₂(𝑟,𝑡,𝜐)𝜕𝑟𝑑𝜐+^𝜃₀𝜕𝐹₂𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝑡1+𝜕𝐷₁(𝑟,𝑡,𝜐)+𝜕𝑟𝑑𝜐₁₀𝑆₁(𝜐)𝑑𝜐⋅^𝜃₀𝜕𝐹₃𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝑟1+𝜕𝐷₂(𝑟,𝑡,𝜐)+𝜕𝑟𝑑𝜐₁₀𝑆₁(𝜐)𝑑𝜐⋅^𝜃₀𝜕𝐹₃𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝑡1+𝜕𝐷₁(𝑟,𝑡,𝜐)𝜕𝑟𝑑𝜐≤𝐶𝑟⁻¹𝑟^−𝜇0+𝜀⋅1+𝜕𝐷₂(𝑟,𝑡,𝜐)𝑟𝜕𝑟+𝐶^−𝜇0+𝜀⋅1+𝜕𝐷₁(𝑟,𝑡,𝜐),𝜕𝑟𝜕𝐷₁(𝑟,𝑡,𝜃)≤𝜕𝑟𝜕𝐷₂(𝑟,𝑡,𝜃)+𝜕𝑟_𝜃0𝜕𝐹₀𝑟+𝐷₂,𝜐⋅𝜕𝑟1+𝜕𝐷₂(𝑟,𝑡,𝜐)+𝜕𝑟𝑑𝜐_𝜃0𝜕𝐹₁𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝑟1+𝜕𝐷₂(𝑟,𝑡,𝜐)+𝜕𝑟𝑑𝜐_𝜃0𝜕𝐹₁𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝑡1+𝜕𝐷₁(𝑟,𝑡,𝜐)≤𝜕𝑟𝑑𝜐𝜕𝐷₂(𝑟,𝑡,𝜃)𝜕𝑟+𝐶𝑟⁻¹𝑟^−𝜇0+𝜀⋅1+𝜕𝐷₂(𝑟,𝑡,𝜐)𝑟𝜕𝑟+𝐶^−𝜇0+𝜀⋅1+𝜕𝐷₁(𝑟,𝑡,𝜐).𝜕𝑟 (105) Hence, 𝜕𝐷₁(𝑟,𝑡,𝜃)𝑟𝜕𝑟≤𝐶^−𝜇0+𝜀,𝜕𝐷₂(𝑟,𝑡,𝜃)𝑟𝜕𝑟≤𝐶^−𝜇0+𝜀. (106) Now, we proceed inductively by assuming that for 𝑗<𝑘−1 the estimates 𝜕^𝑗𝐷₁(𝑟,𝑡,𝜃)𝜕𝑟^𝑗≤𝐶𝑟^−𝜇0,𝜕+𝜀_𝑗𝐷₂(𝑟,𝑡,𝜃)𝜕𝑟^𝑗≤𝐶𝑟^−𝜇0+𝜀, (107) hold and we wish to conclude that the same estimate holds for 𝑗=𝑘𝜕^𝑘𝐷₂(𝑟,𝑡,𝜃)𝜕𝑟^𝑘≤^𝜃₀𝜕𝐹₂𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝜕𝑟_𝑘𝐷₂(𝑟,𝑡,𝜃)𝜕𝑟^𝑘+𝑑𝜐_𝜃0𝜕𝐹₂𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝜕𝑡_𝑘𝐷₁(𝑟,𝑡,𝜐)𝜕𝑟^𝑘𝑟𝑑𝜐+𝐶^−𝜇0+𝜀⋅_{𝑘1+𝑘2=𝑘𝑖1𝑠1𝑗1𝜈2+⋯+𝑖=𝑘+⋯+𝑗=𝑘}𝜕^𝑖1𝑟+𝐷₂𝜕𝑟^𝑖1⋯𝜕^𝑖𝑠𝑟+𝐷₂𝜕𝑟^𝑖𝑠×𝜕^𝑗1𝑟+𝐷₁𝜕𝑟^𝑗1⋯𝜕^𝑗𝜈𝑟+𝐷₁𝜕𝑟^𝑗𝜈+¹₀𝑆₁(𝜐⋅)𝑑𝜐_𝜃0𝜕𝐹₃𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝜕𝑟^𝑘𝐷₂(𝑟,𝑡,𝜐)𝜕𝑟^𝑘+𝑑𝜐₁₀𝑆₁(𝜐⋅)𝑑𝜐_𝜃0𝜕𝐹₃𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝜕𝑡^𝑘𝐷₁(𝑟,𝑡,𝜐)𝜕𝑟^𝑘𝑟𝑑𝜐+𝐶^−𝜇0⋅+𝜀_{𝑘1+𝑘2=𝑘𝑖1𝑠1𝑗1𝜈2+⋯+𝑖=𝑘+⋯+𝑗=𝑘}𝜕^𝑖1𝑟+𝐷₂𝜕𝑟^𝑖1⋯𝜕^𝑖𝑠𝑟+𝐷₂𝜕𝑟^𝑖𝑠×𝜕^𝑗1𝑟+𝐷₁𝜕𝑟^𝑗1⋯𝜕^𝑗𝜈𝑟+𝐷₁𝜕𝑟^𝑗𝜈≤𝐶𝑟⁻¹𝑟^−𝜇0⋅𝜕+𝜀^𝑘𝐷₂(𝑟,𝑡,𝜐)𝜕𝑟^𝑘+𝐶𝑟^−𝜇0⋅𝜕+𝜀^𝑘𝐷₁(𝑟,𝑡,𝜐)𝜕𝑟^𝑘+𝐶𝑟^−𝜇0,𝜕+𝜀_𝑘𝐷₁(𝑟,𝑡,𝜃)𝜕𝑟^𝑘≤𝜕^𝑘𝐷₂(𝑟,𝑡,𝜃)𝜕𝑟^𝑘+^𝜃₀𝜕𝐹₀𝑟+𝐷₂,𝜐⋅𝜕𝜕𝑟^𝑘𝐷₂(𝑟,𝑡,𝜐)𝜕𝑟^𝑘𝑟𝑑𝜐+𝐶^−𝜇0+𝜀⋅_{𝑖1+⋯+𝑖𝑠=𝑘}𝜕^𝑖1𝑟+𝐷₂𝜕𝑟^𝑖1⋯𝜕^𝑖𝑠𝑟+𝐷₂𝜕𝑟^𝑖𝑠+^𝜃₀𝜕𝐹₁𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝜕𝑟^𝑘𝐷₂(𝑟,𝑡,𝜐)𝜕𝑟^𝑘+𝑑𝜐_𝜃0𝜕𝐹₁𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝜕𝑡^𝑘𝐷₁(𝑟,𝑡,𝜐)𝜕𝑟^𝑘𝑟𝑑𝜐+𝐶^−𝜇0+𝜀⋅_{𝑘1+𝑘2=𝑘𝑖1𝑠1𝑗1𝜈2+⋯+𝑖=𝑘+⋯+𝑗=𝑘}𝜕^𝑖1𝑟+𝐷₂𝜕𝑟^𝑖1⋯𝜕^𝑖𝑠𝑟+𝐷₂𝜕𝑟^𝑖𝑠×𝜕^𝑗1𝑟+𝐷₁𝜕𝑟^𝑗1⋯𝜕^𝑗𝜈𝑟+𝐷₁𝜕𝑟^𝑗𝜈≤𝜕^𝑘𝐷₂(𝑟,𝑡,𝜃)𝜕𝑟^𝑘+𝐶𝑟⁻¹𝑟^−𝜇0⋅𝜕+𝜀^𝑘𝐷₂(𝑟,𝑡,𝜐)𝜕𝑟^𝑘+𝐶𝑟^−𝜇0⋅𝜕+𝜀^𝑘𝐷₁(𝑟,𝑡,𝜐)𝜕𝑟^𝑘+𝐶𝑟^−𝜇0+𝜀, (108) where 𝑠+𝜈≤2. Hence, 𝜕^𝑘𝐷₁(𝑟,𝑡,𝜃)𝜕𝑟^𝑘≤𝐶𝑟^−𝜇0,𝜕+𝜀_𝑘𝐷₂(𝑟,𝑡,𝜃)𝜕𝑟^𝑘≤𝐶𝑟^−𝜇0+𝜀. (109)
(3) We can prove that 𝜕^𝑚𝐷₁(𝑟,𝑡,𝜃)𝜕𝑡^𝑚≤𝐶𝑟^−𝜇0,𝜕+𝜀_𝑚𝐷₂(𝑟,𝑡,𝜃)𝜕𝑡^𝑚≤𝐶𝑟^−𝜇0+𝜀 (110) similarly to (2) when 𝑚≠0.
(4) we have that 𝜕²𝐷₂(𝑟,𝑡,𝜃)≤𝜕𝑟𝜕𝑡_𝜃0𝜕𝐹₂𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝜕𝑟²𝐷₂(𝑟,𝑡,𝜃)+𝜕𝑟𝜕𝑡𝑑𝜐_𝜃0𝜕𝐹₂𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝜕𝑡²𝐷₁(𝑟,𝑡,𝜃)𝑟𝜕𝑟𝜕𝑡𝑑𝜐+𝐶^−𝜇0⋅+𝜀𝜕𝐷₂(𝑟,𝑡,𝜃)+𝜕𝑟𝜕𝐷₁(𝑟,𝑡,𝜃)+𝜕𝑡𝜕𝐷₁(𝑟,𝑡,𝜃)+𝜕𝑟𝜕𝐷₂(𝑟,𝑡,𝜃)𝜕𝑡+1+¹₀𝑆₁(𝜐⋅)𝑑𝜐_𝜃0𝜕𝐹₃𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝜕𝑟²𝐷₂(𝑟,𝑡,𝜃)+𝜕𝑟𝜕𝑡𝑑𝜐₁₀𝑆₁(𝜐⋅)𝑑𝜐_𝜃0𝜕𝐹₃𝑟+𝐷₂,𝜐,𝑡+𝑟𝜐+𝐷₁⋅𝜕𝜕𝑡²𝐷₁(𝑟,𝑡,𝜃)𝑟𝜕𝑟𝜕𝑡𝑑𝜐≤𝐶^−𝜇0⋅𝜕+𝜀²𝐷₂(𝑟,𝑡,𝜃)𝑟𝜕𝑟𝜕𝑡+𝐶^−𝜇0⋅𝜕+𝜀²𝐷₁(𝑟,𝑡,𝜃)𝑟𝜕𝑟𝜕𝑡+𝐶^−𝜇0,𝜕+𝜀₂𝐷₁(𝑟,𝑡,𝜃)≤𝜕𝜕𝑟𝜕𝑡₂𝐷₂(𝑟,𝑡,𝜃)𝑟𝜕𝑟𝜕𝑡+𝐶^−𝜇0⋅𝜕+𝜀²𝐷₂(𝑟,𝑡,𝜃)𝑟𝜕𝑟𝜕𝑡+𝐶^−𝜇0⋅𝜕+𝜀²𝐷₁(𝑟,𝑡,𝜃)𝑟𝜕𝑟𝜕𝑡+𝐶^−𝜇0+𝜀. (111) Hence, 𝜕²𝐷₁(𝑟,𝑡,𝜃)𝑟𝜕𝑟𝜕𝑡≤𝐶^−𝜇0,𝜕+𝜀₂𝐷₂(𝑟,𝑡,𝜃)𝑟𝜕𝑟𝜕𝑡≤𝐶^−𝜇0+𝜀. (112)
(5) We can prove (103) similarly to (4) for the left 𝑘+𝑚≤4.

Proof of Boundedness. From Theorem 1.1 in [21] we can see Φ¹ possesses a sequence of invariant circles tending to infinity. So, in the original system (13), there exists a corresponding sequence of invariant tori in phase space (𝑥,𝑥̇,𝑡)∈ℝ²×𝕋. Then any solution of system (13) is bounded because it must stay within one of those tori.

4. The Proof of Unboundedness

In this section, we will prove that all solutions of (8) are unbounded if 𝐵<0. In this case, 𝐴<0.

Consider (8) which is equivalent to the following system: ̇̇|𝑥|𝑥=𝑦,𝑦=−𝐴^𝛼−1𝑥−𝑓(𝑥)𝑦−𝜀𝑒̂(𝑡)|𝑥|^𝛼−1𝑥+𝑝(𝑡). (113) Replacing (18) by an “auxiliary” system ̇̇|𝑥|𝑥=𝑦,𝑦=𝐴^𝛼−1𝑥. (114)

Under the transformation (21), the system (113) is transformed into the form 𝑑𝜌1𝑑𝑡=−𝜌2−2𝑏^2(1−𝑏)𝑔^′𝜑+ℎ₁𝜌,𝜑,𝑡,𝑑𝜑𝑑𝑡=𝜌^1−2𝑏𝑔𝜑+𝑔₁𝜌,𝜑,𝑡, (115) where 𝑔𝜑=(1−𝑏)𝑑+2𝑏𝑑𝐴𝑆𝜑𝑇₀^𝛼+1,ℎ₁𝜌,𝜑,𝑡=−𝑇₀𝑑𝜌𝑓𝜌^𝑏𝑆𝜑𝑇₀𝑆^′𝜑𝑇₀𝐶𝜑𝑇₀−𝑇₀𝑑𝜀𝜌^2−2𝑏𝑆𝜑𝑇₀^𝛼−1𝑆𝜑𝑇₀𝑆^′𝜑𝑇₀𝑒̂(𝑡)+𝑇₀𝑑𝜌^𝑏𝑆^′𝜑𝑇₀𝑝(𝑡),𝑔₁𝜌,𝜑,𝑡=𝑏𝑑𝑓𝜌^𝑏𝑆𝜑𝑇₀𝑆𝜑𝑇₀𝐶𝜑𝑇₀+𝑏𝑑𝜀𝜌^1−2𝑏𝑆𝜑𝑇₀^𝛼+1𝑒̂(𝑡)−𝑏𝑑𝜌^𝑏−1𝑆𝜑𝑇₀𝑝(𝑡). (116)

Thus, the system (115) can be written in the form 𝑑𝜌1𝑑𝑡=−2()𝑔1−𝑏^′𝜑𝜌^2(1−𝑏)+𝑂𝜀𝜌^2(1−𝑏),𝑑𝜑𝑑𝑡=𝜌^1−2𝑏𝑔𝜑+𝑂𝜀𝜌^1−2𝑏. (117) From the equality 12𝐶²(𝑡)+−𝐴|𝑆(𝑡𝛼+1)|^𝛼+1=12,∀𝑡∈ℝ, (118) it follows that 0≤𝑆𝜑𝑇₀^𝛼+1≤−𝛼+1.2𝐴 (119) Hence, the function 𝑔(𝜑) is 𝐶¹, 1-periodic and change the sign. Since |𝑆(𝑇₀−𝜑𝑇₀)|=|𝑆(𝜑𝑇₀)| for any 𝜑∈[0,1], there exists 𝜑₁∈(0,1/2) such that 𝑆𝑇₀−𝜑₁𝑇₀^𝛼+1=𝑆𝜑₁𝑇₀^𝛼+1=−𝛼+1.4𝐴 (120) That is, 𝑔(𝜑₁)=𝑔(1−𝜑₁)=0. In view of 𝑆𝑇₀−𝜑𝑇₀=−𝑆𝜑𝑇₀,𝐶𝑇₀−𝜑𝑇₀=𝐶𝜑𝑇₀, (121) we find 𝑔^′𝜑₁⋅𝑔^′1−𝜑₁=−(𝛼+1)²2𝑏𝑑𝐴𝑇₀²𝑆𝜑₁𝑇₀^2(𝛼−1)𝑆²𝜑₁𝑇₀𝐶²𝜑₁𝑇₀<0. (122) Hence, we obtain that 𝑔^′(𝜑₁) or 𝑔^′(1−𝜑₁) is negative. This proves that there exists a 𝜑^∗ such that 𝑔(𝜑^∗)=0 and 𝑔^′(𝜑^∗)<0. Therefore, there are 𝜐>0 and 𝛿₀>0 such that 𝑔^′(𝜑)<−𝛿₀ for 𝜑∈[𝜑^∗−𝜐,𝜑^∗+𝜐] and 𝑔(𝜑)>0 for 𝜑∈(𝜑^∗−𝜐,𝜑^∗), 𝑔(𝜑)<0 for 𝜑∈(𝜑^∗,𝜑^∗+𝜐). Let 𝒦_𝐽,𝜐=𝜌,𝜑∈ℝ⁺×𝕋∶𝜌>𝐽,𝜑∈𝜑^∗−𝜐,𝜑^∗+𝜐. (123) Then, if 𝐽 is sufficiently large, on the set 𝒦_𝐽,𝜐, we have −12()𝑔1−𝑏^′𝜑𝜌^2(1−𝑏)+𝑂𝜀𝜌^2(1−𝑏)>𝛿₀2⋅𝜌^2(1−𝑏),𝜌^1−2𝑏𝑔𝜑+𝑂𝜀𝜌^1−2𝑏𝜑>0,for𝜌≥𝐽,𝜑∈^∗−𝜐,𝜑^∗−𝜐2,𝜌^1−2𝑏𝑔𝜑+𝑂𝜀𝜌^1−2𝑏𝜑<0,for𝜌≥𝐽,𝜑∈^∗+𝜐2,𝜑^∗+𝜐. (124)(125)

From (117) and (124) we obtain, for 𝑡≥0, 𝜌𝑡,𝜌₀,𝜑₀=𝜌₀+^𝑡₀−12()𝑔1−𝑏^′𝜑𝜌^2(1−𝑏)+𝑂𝜀𝜌^2(1−𝑏)𝑑𝑡>𝜌₀+^𝑡₀𝛿₀2⋅𝜌^2(1−𝑏)𝑑𝑡≥𝜌₀>𝐽. (126) Moreover, for 𝜌(𝑡,𝜌₀,𝜑₀)>𝐽 and 𝜑(𝑡,𝜌₀,𝜑₀)∈[𝜑^∗−𝜐,𝜑^∗−𝜐/2]∪[𝜑^∗+𝜐/2,𝜑^∗+𝜐], we have 𝜌^1−2𝑏𝑔𝜑+𝑂𝜀𝜌^1−2𝑏=𝜌^1−2𝑏𝑔^′𝜑𝜑−𝜑^∗+𝑂𝜀𝜌^1−2𝑏𝛿<−⁰2𝜑−𝜑^∗𝜌^1−2𝑏. (127) From (126) and (127), it follows that any solution (𝜌(𝑡,𝜌₀,𝜑₀),𝜑(𝑡,𝜌₀,𝜑₀)) of (115) with the initial condition (𝜌(0,𝜌₀,𝜑₀),𝜑(0,𝜌₀,𝜑₀))=(𝜌₀,𝜑₀)∈𝒦_𝐽,𝜐 always stays in 𝒦_𝐽,𝜐 and satisfies 𝜌(𝑡,𝜌₀,𝜑₀)>𝛿𝑡+𝜌(0) with 𝛿=𝛿₀^3−2𝑏/2, for all 𝑡≥0. The proof of Theorem 3 is completed.

5. The Proof of Theorem 4

In this section, we will prove Theorem 4 by using the abstract result on the existence of quasi-periodic solutions proved in [24] in the context Aubry-Mather theory for reversible systems. We only need to show that the Poincaré map (92) has the monotone property; that is, 𝜕𝑡₁()𝜕𝑟𝑟,𝑡>0. (128)

We can get that 𝜕𝐹₀(𝑟,𝜃)𝜕𝑟≤𝐶𝑟^{−1−(1+𝛾−3𝑏/2)/(2𝑏−1)} (129) by Lemma 11, and 𝜕𝑄₂(𝑟,𝑡)𝜕𝑟≤𝑟^−𝜇0+𝜀 (130) by Lemma 12. Then we have 𝜕𝑡₁()𝜕𝑟𝑟,𝑡=1+¹₀𝜕𝐹₀𝜕𝑟𝑑𝜃+𝜕𝑄₂𝜕𝑟⟶𝑐₀,as𝑟⟶+∞, (131) where 𝑐₀≥1−𝜀. Therefore, we have 𝜕𝑡₁()𝜕𝑟𝑟,𝑡>0 (132) as 𝑟≫1 and 𝜀≪1. This proves the validity of (128).

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant nos. 11171185, 10871117) and SDNSF (Grant no. ZR2010AM013).