A High Order Iterative Scheme for a Nonlinear Kirchhoff Wave Equation in the Unit Membrane

Abstract

A high-order iterative scheme is established in order to get a convergent sequence at a rate of order 𝑁 (𝑁≥1) to a local unique weak solution of a nonlinear Kirchhoff wave equation in the unit membrane. This extends a recent result in (EJDE, 2005, No. 138) where a recurrent sequence converges at a rate of order 2.

1. Introduction

In this paper we consider the initial and boundary value problem𝑢_𝑡𝑡−𝐵‖‖𝑢_𝑟‖‖²⁰𝑢^𝑟𝑟+1||||li𝑟m𝑢^𝑟_𝑟→0+=𝑓(𝑟,𝑡,𝑢),0<𝑟<1,0<𝑡<𝑇,𝑢√𝑟𝑢_𝑟(𝑟,𝑡)||||<∞,_𝑟𝑢(𝑟,0)=(𝑢1₀̃,𝑡)+ℎ𝑢(1,𝑡)=0,(𝑟),𝑢_𝑡(𝑟,0)=𝑢₁̃(𝑟),(1.1) where 𝐵,𝑓,𝑢₀̃,𝑢₁̃ are given functions satisfying conditions specified later, ‖𝑢_𝑟‖²⁰=∫¹⁰𝑟|𝑢_𝑟(𝑟,𝑡)|²𝑑𝑟, and ℎ>0 is a given constant.

Equation (1.1)₁ herein is the bidimensional nonlinear wave equation describing nonlinear vibrations of the unit membrane Ω₁={(𝑥,𝑦)∶𝑥²+𝑦²<1}. In the vibration process, the area of the unit membrane and the tension at various points change in time. The condition on the boundary 𝜕Ω₁ describes elastic constraints, where the constant ℎ₁ has a mechanical signification. The boundary condition |lim_𝑟→0+√𝑟𝑢_𝑟(𝑟,𝑡)|<∞ is satisfied automatically if 𝑢 is a classical solution of the problem (1.1), for example, with 𝑢∈C¹([0,1]×(0,𝑇))∩𝐶²((0,1)×(0,𝑇)). This condition is also used in connection with Sobolev spaces with weight 𝑟 (see [1–3]).

Equation (1.1)₁ is related to the Kirchhoff equation𝜌ℎ𝑢_𝑡𝑡=𝑃₀+𝐸ℎ2𝐿^𝐿0||||𝜕𝑢𝜕𝑦(𝑦,𝑡)||||²𝑑𝑦𝑢_𝑥𝑥(1.2) presented by Kirchhoff in 1876 (see [4]). This equation is an extension of the classical D’Alembert wave equation which considers the effects of the changes in the length of the string during the vibrations. The parameters in (1.2) have the following meanings: 𝑢 is the lateral deflection, 𝐿 is the length of the string, ℎ is the area of the cross-section, 𝐸 is the Young modulus of the material, 𝜌 is the mass density, and 𝑃₀ is the initial tension.

The Kirchhoff wave equation of the form (1.1)₁ received much attention. Many interesting results about the existence, stability, regularity in time variable, asymptotic behavior, and asymptotic expansion of solutions can be found, for example, in [2, 3, 5–14] and references therein.

In [2], in a special case, sufficient conditions were established for a quadratic convergence to the solution of (1.1) with 𝑓(𝑟,𝑡,𝑢)=𝑓(𝑟,𝑢) and 𝐵(‖𝑢_𝑟‖²⁰)=𝑏₀+‖𝑢_𝑟‖²⁰,𝑏₀>0. Based on the ideas about recurrence relations for a third-order method for solving the nonlinear operator equation 𝐹(𝑢)=0 in [15], we extend the above result by the construction of a high-order iterative scheme for (1.1)₁, where 𝑓 and 𝐵 are more generalized.

In this paper, we associate with (1.1)₁ a recurrent sequence {𝑢_𝑚} defined by𝜕²𝑢_𝑚𝜕𝑡²−𝐵‖‖𝑢_𝑚𝑟‖‖²⁰𝜕²𝑢_𝑚𝜕𝑟²+1𝑟𝜕𝜕𝑢𝑟^𝑚=^𝑁−1_𝑖=01𝑖!𝜕^𝑖𝑓𝜕𝑢^𝑖𝑟,𝑡,𝑢_𝑚−1𝑢_𝑚−𝑢_𝑚−1^𝑖,(1.3)0<𝑟<1,0<𝑡<𝑇, with 𝑢_𝑚 satisfying (1.1)_2-3. The first term 𝑢₀ is chosen as 𝑢₀≡0. If𝐵∈𝐶¹(ℝ₊) and 𝑓∈𝐶^𝑁([0,1]×ℝ₊×ℝ), we prove that the sequence {𝑢_𝑚} converges at a rate of order 𝑁 to a unique weak solution of the problem (1.1). This result is a relative generalization of [2, 3, 8, 9, 14, 16].

2. Preliminary Results, Notations, Function Spaces

Put Ω=(0,1). We omit the definitions of the usual function spaces 𝐶^𝑚(Ω),𝐿^𝑝(Ω),𝐻^𝑚(Ω), and 𝑊^𝑚,𝑝(Ω). For any function 𝑣∈𝐶⁰(Ω) we define ‖𝑣‖₀ as ‖𝑣‖₀=(∫¹⁰𝑟𝑣²(𝑟)𝑑𝑟)^1/2 and define the space 𝑉₀ as completion of the space 𝐶⁰(Ω) with respect to the norm ‖⋅‖₀. Similarly, for any function 𝑣∈𝐶¹(Ω) we define ‖𝑣‖₁ as ‖𝑣‖₁=(‖𝑣‖²⁰+‖𝑣_𝑟‖²⁰)^1/2 and define the space 𝑉₁ as completion of the space 𝐶¹(Ω) with respect to the norm ‖⋅‖₁. Note that the norms ‖⋅‖₀ and ‖⋅‖₁ can be defined, respectively, from the inner products ⟨𝑢,𝑣⟩=¹⁰𝑟𝑢(𝑟)𝑣(𝑟)𝑑𝑟,⟨𝑢,𝑣⟩+⟨𝑢_𝑟,𝑣_𝑟⟩.(2.1) Identifying 𝑉₀ with its dual 𝑉^0 we obtain the dense and continuous embedding 𝑉₁↪𝑉₀≡𝑉^0↪𝑉^1. The inner product notation will be reutilized to denote the duality pairing between 𝑉₁ and 𝑉^1.

We then have the following lemmas, the proofs of which can be found in [1].

Lemma 2.1. There exist two constants 𝐾₁>0 and 𝐾₂>0 such that, for all 𝑣∈𝐶¹(Ω), we have(i)‖𝑣_𝑟‖²⁰+𝑣²(1)≥‖𝑣‖²⁰, (ii)|𝑣(1)|≤𝐾₁‖𝑣‖₁, (iii)√𝑟|𝑣(𝑟)|≤𝐾₂‖𝑣‖₁,forall𝑟∈Ω.

Lemma 2.2. The embedding 𝑉₁↪𝑉₀ is compact.

Remark 2.3. In Lemma 2.1, the two constants 𝐾₁ and 𝐾₂ can be given explicitly as 𝐾₁=1+√2 and 𝐾₂=1+√5. We also note that lim_𝑟→0+√𝑟𝑣(𝑟)=0 for all 𝑣∈𝑉₁ (see [17, page 128/Lemma 5.40]). On the other hand, by 𝐻¹(𝜀,1)↪𝐶⁰([𝜀,1]),0<𝜀<1 and √𝜀‖𝑣‖_{𝐻1(𝜀,1)}≤‖𝑣‖₁ for all 𝑣∈𝑉₁, it follows that 𝑣|_[𝜀,1]∈𝐶⁰([𝜀,1]). From both relations we deduce that √𝑟𝑣∈𝐶⁰(Ω) for all 𝑣∈𝑉₁.
Now, let the bilinear form 𝑎(⋅,⋅) be defined by𝑎(𝑢,𝑣)=ℎ𝑢(1)𝑣(1)+¹⁰𝑟𝑢_𝑟(𝑟)𝑣_𝑟(𝑟)𝑑𝑟,𝑢,𝑣∈𝑉₁,(2.2) where ℎ is a positive constant. Then, there exists a unique bounded linear operator 𝐴∶𝑉₁→𝑉^1 such that 𝑎(𝑢,𝑣)=⟨𝐴𝑢,𝑣⟩ for all 𝑢,𝑣∈𝑉₁. We then have the following lemma.

Lemma 2.4. The symmetric bilinear form 𝑎(⋅,⋅) defined by (2.2) is continuous on 𝑉₁×𝑉₁ and coercive on 𝑉₁, that is,(i)|𝑎(𝑢,𝑣)|≤𝐶₁‖𝑢‖₁‖𝑣‖₁, (ii)𝑎(𝑣,𝑣)≥𝐶₀‖𝑣‖²¹, for all 𝑢,𝑣∈𝑉₁, where 𝐶₀=(1/2)min{1,ℎ} and 𝐶₁=1+(1+√2)ℎ.

The proof of Lemma 2.4 is straightforward and we omit it.

Lemma 2.5. There exists an orthonormal Hilbert basis {𝑤_𝑗} of the space 𝑉₀ consisting of eigenfunctions 𝑤_𝑗 corresponding to eigenvalues 𝜆_𝑗 such that(i)0<𝜆₁≤𝜆₂≤⋯≤𝜆_𝑗↑+∞as𝑗→∞, (ii)𝑎(𝑤_𝑗,𝑣)=𝜆_𝑗⟨𝑤_𝑗,𝑣⟩forall𝑣∈𝑉₁and𝑗∈ℕ. Note that it follows from (ii) that {𝑤_𝑗/√𝜆_𝑗} is automatically an orthonormal set in 𝑉₁ with respect to 𝑎(⋅,⋅) as inner product. The eigensolutions 𝑤_𝑗 are indeed eigensolutions for the boundary value problem 𝐴𝑤_𝑗≡−1𝑟𝑑𝑑𝑟𝑟𝑑𝑑𝑤𝑟^𝑗=𝜆^𝑗||||lim_𝑟→0+𝑤_𝑗,inΩ,√𝑟𝑑𝑤^𝑗𝑑r(𝑟)||||<+∞,𝑑𝑑𝑤𝑟^𝑗(1)+ℎ𝑤^𝑗(1)=0.(2.3)

The proof of Lemma 2.5 can be found in ([18, page 87, Theorem 7.7]) with 𝑉=𝑉₁,𝐻=𝑉₀ and 𝑎(⋅,⋅) as defined by (2.2).

For any function 𝑣∈𝐶²(Ω) we define ‖𝑣‖₂ as ‖𝑣‖₂=‖𝑣‖²⁰+‖‖𝑣_𝑟‖‖²⁰+‖𝐴𝑣‖²⁰^1/2(2.4) and define the space 𝑉₂ as completion of 𝐶²(Ω) with respect to the norm ‖⋅‖₂. Note that 𝑉₂ is also a Hilbert space with respect to the scalar product ⟨𝑢,𝑣⟩+⟨𝑢_𝑟,𝑣_𝑟⟩+⟨𝐴𝑢,𝐴𝑣⟩(2.5) and that 𝑉₂ can be defined also as 𝑉₂={𝑣∈𝑉₁∶𝐴𝑣∈𝑉₀}.

We then have the following two lemmas the proof of which can be found in [1].

Lemma 2.6. The embedding 𝑉₂↪𝑉₁ is compact.

Lemma 2.7. For all 𝑣∈𝑉₂ we have (i)‖‖𝑣_𝑟‖‖_𝐿∞(Ω)≤1√2‖𝐴𝑣‖⁰(ii)‖‖𝑣_𝑟𝑟,‖‖₀≤32‖𝐴𝑣‖⁰(iii)‖𝑣‖_2𝐿∞(Ω),≤2‖𝑣‖₀+1√2‖𝐴𝑣‖⁰‖𝑣‖⁰.(2.6)

For a Banach space 𝑋, we denote by ‖⋅‖_𝑋 its norm, by 𝑋^ its dual space, and by 𝐿^𝑝(0,𝑇;𝑋),1≤𝑝≤∞ the Banach space of all real measurable functions 𝑢∶(0,𝑇)→𝑋 such that ‖𝑢‖‖𝑢‖_𝐿^𝑝_{(0,𝑇;𝑋)}=^𝑇0_{𝐿∞(0,𝑇;𝑋)}‖𝑢(𝑡)‖^𝑝𝑋𝑑𝑡^1/𝑝<∞,for1≤𝑝<∞,=esssup_0<𝑡<𝑇‖𝑢(𝑡)‖_𝑋for𝑝=∞.(2.7)

Let 𝑢(𝑡),𝑢^(𝑡)=𝑢_𝑡(𝑡)=𝑢̇(𝑡),𝑢^(𝑡)=𝑢_𝑡𝑡(𝑡)=𝑢̈(𝑡),𝑢_𝑟(𝑡)=∇𝑢(𝑡),𝑢_𝑟𝑟(𝑡)(2.8) denote 𝑢(𝑟,𝑡),𝜕𝑢𝜕𝑡(𝑟,𝑡),𝜕²𝑢𝜕𝑡²(𝑟,𝑡),𝜕𝑢𝜕𝑟(𝑟,𝑡),𝜕²𝑢𝜕𝑟²(𝑟,𝑡),(2.9) respectively.

With 𝑓∈𝐶^𝑘(Ω×ℝ₊×ℝ),𝑓=𝑓(𝑟,𝑡,𝑢), we put 𝐷₁𝑓=𝜕𝑓/𝜕𝑟,𝐷₂𝑓=𝜕𝑓/𝜕𝑡,𝐷₃𝑓=𝜕𝑓/𝜕𝑢, and 𝐷^𝛾𝑓=𝐷^𝛾1₁𝐷^𝛾2₂𝐷^𝛾3₃𝑓,𝛾=(𝛾₁,𝛾₂,𝛾₃)∈ℤ³⁺,|𝛾|=𝛾₁+𝛾₂+𝛾₃=𝑘.

3. The Hight Order Iterative Schemes

Fix 𝑇^∗>0, we make the following assumptions: (𝐻₁)𝑢₀̃∈𝑉₂and𝑢₁̃∈𝑉₁; (𝐻₂)𝐵∈𝐶¹(ℝ₊)andthereexistconstants𝑏_∗>0,𝛼>1,𝑑₀,𝑑₁>0suchthat(i)𝑏_∗≤𝐵(𝜂)≤𝑑₀(1+𝜂^𝛼),forall𝜂≥0, (ii)|𝐵^(𝜂)|≤𝑑₁(1+𝜂^𝛼−1),forall𝜂≥0; (𝐻₃)𝑓∈𝐶^𝑁(Ω×[0,𝑇^∗]×ℝ)andsatisfiesthefollowingcondition∶forall𝑀>0, 𝐾^(1)∗𝑖(𝑀,𝑓)=sup_{(𝑟,𝑡,𝑢)∈𝐴𝑀}||||√𝑟^−𝑖𝜕^𝑖𝑓𝜕𝑢^𝑖𝐾(𝑟,𝑡,𝑢)||||<+∞,𝑖=0,1,…,𝑁−1,_(2)∗𝑖(𝑀,𝑓)=sup_{(𝑟,𝑡,𝑢)∈𝐴𝑀}||||√𝑟^−𝑖𝜕^𝑖𝑓𝜕𝑟𝜕𝑢^𝑖−1(𝑟,𝑡,𝑢)||||<+∞,𝑖=1,…,𝑁−1,(3.1)

where 𝐴_𝑀={(𝑟,𝑡,𝑢)∈[0,1]×[0,𝑇^∗]×ℝ∶|𝑢|≤𝑀2+1/√2}. We put 𝐾_∗𝑖(𝑀,𝑓)=⎧⎪⎨⎪⎩𝐾^(1)∗0max(𝑀𝐾,𝑓_(1)∗𝑖),𝑖=0,(𝑀,𝑓),𝐾^(2)∗𝑖(𝑀,𝑓),𝑖=1,…,𝑁−1.(3.2)

With 𝐵 and 𝑓 satisfying assumptions (𝐻₂) and (𝐻₃), respectively, for each 𝑀>0 given, we introduce the following constants:𝐾_𝑀(𝐵)=sup_{0≤𝜂≤𝑀2}𝐵(𝜂)+||𝐵^(𝜂)||,𝐾₀(𝑀,𝑓)=sup_{(𝑟,𝑡,𝑢)∈𝐴𝑀}||𝑓(𝑟,𝑡,𝑢)||,𝐾_𝑁(𝑀,𝑓)=_{||𝛾||≤𝑁}𝐾₀(𝑀,𝐷^𝛾𝑓).(3.3)

For each 𝑇∈(0,𝑇^∗] and 𝑀>0 we get 𝑊(𝑀,𝑇)=𝑣w∈ith𝐿‖^∞𝑣‖0,𝑇;𝑉₂_{𝐿∞(0,𝑇;𝑉2)}∶𝑣^∈𝐿^∞0,𝑇;𝑉₁,𝑣^∈𝐿²0,𝑇;𝑉₀,𝑊,‖‖𝑣^‖‖_{𝐿∞(0,𝑇;𝑉1)},‖‖𝑣^‖‖_{𝐿2(0,𝑇;𝑉0)}≤𝑀₁,(𝑀,𝑇)=𝑣∈𝑊(𝑀,𝑇)∶𝑣^∈𝐿^∞0,𝑇;𝑉₀.(3.4)

We will choose as first initial term 𝑢₀≡0, suppose that𝑢_𝑚−1∈𝑊₁(𝑀,𝑇),(3.5) and associate with the problem (1.1) the following variational problem.

Find 𝑢_𝑚∈𝑊₁(𝑀,𝑇)(𝑚≥1) so that𝑢^𝑚(𝑡),𝑣+𝑏𝑢_𝑚𝑚(𝑡)𝑎𝑢_𝑚(𝑡),𝑣=⟨𝐹_𝑚(𝑡),𝑣⟩,∀𝑣∈𝑉₁,(0)=𝑢₀̃,𝑢^𝑚(0)=𝑢₁̃,(3.6) where𝑏_𝑚(𝑡)=𝐵‖‖∇𝑢_𝑚(𝑡)‖‖²⁰,𝐹^𝑚(𝑟,𝑡)=^𝑁−1_𝑖=01𝑖!𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑢_𝑚−𝑢_𝑚−1^𝑖.(3.7)

Then, we have the following theorem.

Theorem 3.1. Let assumptions (𝐻₁)-(𝐻₃) hold. Then there exist a constant 𝑀>0 depending on 𝑇^∗,𝑢₀̃,𝑢₁̃,𝐵,ℎ and a constant 𝑇>0 depending on 𝑇^∗,𝑢₀̃,𝑢₁̃,𝐵,ℎ,𝑓 such that, for 𝑢₀≡0, there exists a recurrent sequence {𝑢_𝑚}⊂𝑊₁(𝑀,𝑇) defined by (3.6), (3.7).

Proof. The proof consists of several steps.
Step 1. The Faedo-Galerkin approximation (introduced by Lions [19]). Consider as in Lemma 2.5 the basis {𝑤_𝑗} for 𝑉₁ and put 𝑢^𝑚(𝑘)(𝑡)=^𝑘_𝑗=1𝑐^{(𝑘)𝑚𝑗}(𝑡)𝑤_𝑗,(3.8) where the coefficients 𝑐^{(𝑘)𝑚𝑗} satisfy the system of the following nonlinear differential equations: 𝑢̈^𝑚(𝑘)(𝑡),𝑤_𝑗+𝑏𝑢^𝑚(𝑘)(𝑡)𝑎𝑢^𝑚(𝑘)(𝑡),𝑤_𝑗=𝐹^𝑚(𝑘)(𝑡),𝑤_𝑗,1≤𝑗≤𝑘,_𝑚(𝑘)(0)=𝑢̃_0𝑘,𝑢̇^𝑚(𝑘)(0)=𝑢̃_1𝑘,(3.9) where 𝑢̃_0𝑘=^𝑘_𝑗=1𝑢̃_1𝑘𝛼^𝑗(𝑘)𝑤_𝑗⟶𝑢₀̃stronglyin𝑉₂,=^𝑘_𝑗=1𝑏𝛽^𝑗(𝑘)𝑤_𝑗⟶𝑢₁̃stronglyin𝑉₁.(3.10)_𝑚(𝑘)𝐹_𝑚(𝑘)(𝑡)=𝐵‖‖∇𝑢^𝑚(𝑘)(𝑡)‖‖²⁰,(𝑟,𝑡)=^𝑁−1_𝑖=01𝑖!𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^𝑚(𝑘)−𝑢_𝑚−1^𝑖=^𝑁_𝑗=0Ψ_𝑗Ψ_𝑗𝑟,𝑡,𝑢_𝑚−1𝑢^𝑚(𝑘)^𝑗,𝑟,𝑡,𝑢_𝑚−1=^𝑁−1_𝑖=𝑗1𝑗!(𝑖−𝑗)!(−1)^𝑖−𝑗𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^{𝑖−𝑗𝑚−1},0≤𝑗≤𝑁−1.(3.11)
Let us suppose that 𝑢_𝑚−1 satisfies (3.5). Then we have the following lemma.
Lemma 3.2. Let assumptions (𝐻₁)-(𝐻₃) hold. For fixed 𝑀>0 and 𝑇>0, then, the system (3.8)–(3.11) has a unique solution 𝑢^𝑚(𝑘)(𝑡) on an interval [0,𝑇^𝑚(𝑘)]⊂[0,𝑇].
Proof of Lemma 3.2. The system of (3.8)–(3.11) is rewritten in the form 𝑐̈^{(𝑘)𝑚𝑗}(𝑡)=−𝜆_𝑗𝑏𝑐^𝑚(𝑘)(𝑡)𝑐^{(𝑘)𝑚𝑗}(𝑡)+𝐹^𝑚(𝑘)(𝑡),𝑤_𝑗,1≤𝑗≤𝑘,_{(𝑘)𝑚𝑗}(0)=𝛼^𝑗(𝑘),𝑐̇^{(𝑘)𝑚𝑗}(0)=𝛽^𝑗(𝑘),(3.12) and it is equivalent to the system of integral equations 𝑐^{(𝑘)𝑚𝑗}(𝑡)=𝛼^𝑗(𝑘)+𝛽^𝑗(𝑘)𝑡−𝜆_𝑗^𝑡0𝑑𝜏^𝜏0𝑏^𝑚(𝑘)(𝑠)𝑐^{(𝑘)𝑚𝑗}(𝑠)𝑑𝑠+^𝑡0𝑑𝜏^𝜏0𝐹^𝑚(𝑘)(𝑠),𝑤_𝑗𝑑𝑠,(3.13) for 1≤𝑗≤𝑘. Omitting the index 𝑚, it is written as follows: 𝑐=ℱ[𝑐],(3.14) where ℱ[𝑐]=(ℱ₁[𝑐],…,ℱ_𝑘[𝑐]),𝑐=(𝑐₁,…,𝑐_𝑘), ℱ_𝑗[𝑐](𝑡)=𝑞_𝑗(𝑡)−𝜆_𝑗^𝑡0+𝑑𝜏^𝜏0𝑏̃(𝑠)𝑐_𝑗(𝑠)𝑑𝑠_𝑁−1_𝑖=1^𝑡0𝑞𝑑𝜏^𝜏0Ψ_𝑖⋅,𝑠,𝑢_𝑚−1𝑢^𝑖(𝑠),𝑤_𝑗𝑑𝑠,1≤𝑗≤𝑘,_𝑗𝑏̃((𝑡𝑡))==𝛼𝑏̃[𝑐^𝑗(𝑘)]+(𝑡𝛽)=^𝑗(𝑘)𝑡𝐵+‖^𝑡0∇𝑑𝑢𝜏(𝑡)^𝜏0‖₂₀Ψ₀⋅,𝑠,𝑢_𝑚−1,𝑤_𝑗𝑑𝑠,1≤𝑗≤𝑘,,𝑢(𝑡)=^𝑘_𝑗=1𝑐_𝑗(𝑡)𝑤_𝑗.(3.15)
For every 𝑇^𝑚(𝑘)∈(0,𝑇] and 𝜌>0 that will be chosen later, we put 𝑋=𝐶⁰([0,𝑇^𝑚(𝑘)];ℝ^𝑘),𝑆={𝑐∈𝑌∶‖𝑐‖_𝑋≤𝜌}, where ‖𝑐‖_𝑋=sup_{0≤𝑡≤𝑇}^𝑚(𝑘)|𝑐(𝑡)|₁,|𝑐(𝑡)|₁=∑^𝑘𝑗=1|𝑐_𝑗(𝑡)|, for each 𝑐=(𝑐₁,…,𝑐_𝑘)∈𝑋. Clearly 𝑆 is a closed nonempty subset in 𝑋, and we have the operator ℱ∶𝑋→𝑋. In what follows, we will choose 𝜌>0 and 𝑇^𝑚(𝑘)>0 such that(i)𝑆ismappedintoitselfbyℱ, (ii)ℱ∶𝑆→𝑆iscontractive.
Proof (i). First we note that, for all 𝑐=(𝑐₁,…,𝑐_𝑘)∈𝑆, 𝑢(𝑡)=^𝑘_𝑗=1‖𝑢(𝑡)‖₀𝑐_𝑗(𝑡)𝑤_𝑗,=⎷^𝑘_𝑗=1‖∇𝑢(𝑡)‖₂₀𝑐^2𝑗(𝑡)≤||𝑐(𝑡)||₁,≤𝑎(𝑢(𝑡),𝑢(𝑡))=^𝑘_𝑖_,𝑗=1𝑐_𝑖(𝑡)𝑐_𝑗(𝑡)𝑎𝑤_𝑖,𝑤_𝑗=^𝑘_𝑗=1‖𝑢(𝑡)‖₀𝜆_𝑗𝑐^2𝑗(𝑡)≤𝜆_𝑘‖𝑢(𝑡)‖²⁰,‖𝑢(𝑡)‖₁≤||𝑐(𝑡)||₁≤‖𝑐‖_𝑋≤𝜌,‖∇𝑢(𝑡)‖₀≤√𝜆_𝑘||𝑐(𝑡)||₁≤√𝜆_𝑘𝜌,≤1𝐶₀√𝑎(𝑢(𝑡),𝑢(𝑡))≤1𝐶₀√𝜆_𝑘‖𝑢(𝑡)‖₀≤𝜆_𝑘𝐶₀||𝑐(𝑡)||₁≤𝜆_𝑘𝐶₀𝜌,(3.16) so 𝑏̃(𝑡)=𝐵‖∇𝑢(𝑡)‖²⁰≤𝑑₀1+‖∇𝑢(𝑡)‖^02𝛼≤𝑑₀1+𝜆^𝑘2𝛼𝜌^2𝛼.(3.17) On the other hand, by ||𝑢_𝑚−1||≤𝑀2+1||Ψ√2≡𝜃,₀𝑟,𝑡,𝑢_𝑚−1||=|||||^𝑁−1_𝑖=01𝑖!(−1)^𝑖𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^𝑖𝑚−1|||||≤_𝑁−1_𝑖=0|||1𝑖!𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^𝑖𝑚−1≤|||𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=01𝑖!𝑀2+1√2^𝑖=𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖𝑖!,(3.18) we have ||Ψ₀𝑡,𝑢_𝑚−1,𝑤_𝑗||≤‖‖Ψ₀𝑡,𝑢_𝑚−1‖‖₀‖‖𝑤_𝑗‖‖₀=‖‖Ψ₀𝑡,𝑢_𝑚−1‖‖₀≤1√2𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖𝑖!.(3.19) By Lemma 2.1, (iii), and the assumption (𝐻₃), we deduce from (3.16) that ||Ψ=_𝑖|||||𝑠,_𝑁−1𝑢_𝑚−1𝑢^𝑖(𝑠),𝑤_𝑗||_𝑙=𝑖1𝑖!(𝑙−𝑖)!(−1)^𝑙−𝑖𝐷^𝑙3=|||||_𝑁−1𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^{𝑙−𝑖𝑚−1}𝑢^𝑖(𝑠),𝑤_𝑗|||||_𝑙=𝑖1𝑖!(𝑙−𝑖)!(−1)^𝑙−𝑖√𝑟^−𝑙𝐷^𝑙3𝑓𝑟,𝑠,𝑢_𝑚−1√𝑟^𝑙−𝑖𝑢^{𝑙−𝑖𝑚−1}√𝑟^𝑖𝑢^𝑖(𝑠),𝑤_𝑗≤_𝑁−1|||||_𝑙=𝑖1𝑖!(𝑙−𝑖)!𝐾^∗𝑙(𝑀,𝑓)𝜃^𝑙−𝑖𝐾^𝑖2‖𝑢(𝑠)‖^𝑖1√≤_𝑁−1𝑟^𝑙−𝑖,||𝑤_𝑗||_𝑙=𝑖1𝑖!(𝑙−𝑖)!𝐾^∗𝑙(𝑀,𝑓)𝜃^𝑙−𝑖𝐾^𝑖2𝜆_𝑘𝐶₀𝜌^𝑖1≤√2+𝑙−𝑖_𝑁−1_𝑙=𝑖1𝑖!(𝑙−𝑖)!√2+1𝑙−𝑖𝐾^∗𝑙(𝑀,𝑓)𝜃^𝑙−𝑖𝐾^𝑖2𝜆_𝑘𝐶₀𝜌^𝑖,1≤𝑖≤𝑁−1.(3.20)
It follows that||ℱ_𝑗[𝑐](𝑡)||≤+||𝑞1_𝑗(𝑡)||+𝜆_𝑘𝑑₀1+𝜆^𝑘2𝛼𝜌^2𝛼^𝑡0𝑑𝜏^𝜏0||𝑐_𝑗(𝑠)||𝑑𝑠2𝑡^2𝑁−1_{𝑖=1𝑁−1𝑙=𝑖}1𝑖!(𝑙−𝑖)!√2+1𝑙−𝑖𝐾^∗𝑙(𝑀,𝑓)𝜃^𝑙−𝑖𝐾^𝑖2𝜆_𝑘𝐶₀𝜌^𝑖.(3.21) Thus ||ℱ[𝑐](𝑡)||₁≤+||𝑞1(𝑡)||₁+𝜆_𝑘𝑑₀1+𝜆^𝑘2𝛼𝜌^2𝛼^𝑡0𝑑𝜏^𝜏0||𝑐(𝑠)||₁𝑑𝑠2𝑘𝑡^2𝑁−1_{𝑖=1𝑁−1𝑙=𝑖}1𝑖!(𝑙−𝑖)!√2+1𝑙−𝑖𝐾^∗𝑙(𝑀,𝑓)𝜃^𝑙−𝑖𝐾^𝑖2𝜆_𝑘𝐶₀𝜌^𝑖≤‖𝑞‖_𝑇+12𝐷_𝜌𝑇^𝑚(𝑘)²,(3.22) where ‖𝑞‖_𝑇=sup_{0≤𝑡≤𝑇}|𝑞(𝑡)|₁ and 𝐷_𝜌==𝐷𝜆_𝜌𝑘(𝑓,𝜌,𝑘,𝑀,𝑚,𝑁)𝑑₀1+𝜆^𝑘2𝛼𝜌^2𝛼𝜌+𝑘^𝑁−1_{𝑖=1𝑁−1𝑙=𝑖}1𝑖!(𝑙−𝑖)!√2+1𝑙−𝑖𝐾^∗𝑙(𝑀,𝑓)𝜃^𝑙−𝑖𝐾^𝑖2𝜆_𝑘𝐶₀𝜌^𝑖.(3.23)
Hence, we obtain‖ℱ[𝑐]‖_𝑋≤‖𝑞‖_𝑇+12𝐷_𝜌𝑇^𝑚(𝑘)²,(3.24)
choosing 𝜌>‖𝑞‖_𝑇 and𝑇^𝑚(𝑘)∈(0,𝑇], such that12𝐷_𝜌𝑇^𝑚(𝑘)²≤𝜌−‖𝑞‖_𝑇,12𝐷^𝜌𝑇^𝑚(𝑘)²<1,(3.25) where 𝐷_𝜌=≡𝑑𝐷_𝜌0(𝜌,𝑘,𝑀,𝑇,𝑚,𝑁,𝑓)+𝑘𝜆_𝑘_𝑁−11+𝜆^𝑘2𝛼𝜌^2𝛼+2𝑑₁𝜆^2𝑘𝜌²1+𝜆^𝑘𝛼−1𝜌^2𝛼−2_𝑖=1𝑖𝜆_𝑘𝐶₀𝐾₂^𝑖𝜌^{𝑖−1𝑁−1}_𝑙=𝑖1𝑖!(𝑙−𝑖)!√2+1𝑙−𝑖𝐾^∗𝑙(𝑀,𝑓)𝜃^𝑙−𝑖.(3.26) Then ‖ℱ[𝑐]‖_𝑋≤‖𝑞‖_𝑇+12𝐷_𝜌𝑇^𝑚(𝑘)²≤𝜌,∀𝑐∈𝑆,(3.27) which means that ℱ maps 𝑆 into itself.
Proof (ii). We now prove that, for all 𝑐,𝑑∈𝑆, for all 𝑡∈[0,𝑇^𝑚(𝑘)], ||ℱ[𝑐](𝑡)−ℱ[𝑑](𝑡)||₁≤12𝐷^𝜌𝑡²‖𝑐−𝑑‖_𝑋,∀𝑛∈ℕ,(3.28) where 𝐷_𝜌 is defined as (3.26).
Proof of (3.28) is as follows.
For all 𝑗=1,2,…,𝑘, for all 𝑡∈[0,𝑇^𝑚(𝑘)], we have||ℱ_𝑗[𝑐](𝑡)−ℱ_𝑗[𝑑](𝑡)||≤𝜆_𝑗^𝑡0+𝜆_𝑗𝑑𝜏^𝜏0||𝑏̃[𝑐](𝑠)𝑐_𝑗(𝑠)−𝑑_𝑗(𝑠)||𝑑𝑠^𝑡0+𝑑𝜏^𝜏0||𝑏̃[𝑐](𝑠)−𝑏̃[𝑑](𝑠)𝑑_𝑗(𝑠)||𝑑𝑠_𝑁−1_𝑖=1^𝑡0𝑑𝜏^𝜏0||Ψ_𝑖𝑠,𝑢_𝑚−1𝑢^𝑖(𝑠)−𝑣^𝑖(𝑠),𝑤_𝑗||𝑑𝑠,(3.29) where 𝑏̃[𝑐](𝑡)𝑢=(𝑡)𝐵=‖_𝑘∇𝑢(𝑡)‖²⁰,𝑏̃[𝑑](𝑡)=𝐵‖∇𝑣(𝑡)‖²⁰,_𝑗=1𝑐_𝑗(𝑡)𝑤_𝑗,𝑣(𝑡)=^𝑘_𝑗=1𝑑_𝑗(𝑡)𝑤_𝑗,(3.30) so ||ℱ[𝑐](𝑡)−ℱ[𝑑](𝑡)||₁≤𝜆_𝑘^𝑡0+𝜆_𝑘𝑑𝜏^𝜏0𝑏̃[𝑐](𝑠)||𝑐(𝑠)−𝑑(𝑠)||₁𝑑𝑠^𝑡0+𝑑𝜏^𝜏0||𝑏̃[𝑐](𝑠)−𝑏̃[𝑑](𝑠)||||𝑑(𝑠)||₁𝑑𝑠_𝑘_{𝑗=1𝑁−1𝑖=1}^𝑡0≤𝜆_𝑘𝑑𝜏^𝜏0||Ψ_𝑖𝑠,𝑢_𝑚−1𝑢^𝑖(𝑠)−𝑣^𝑖(𝑠),𝑤_𝑗||𝑑𝑠^𝑡0+𝑑𝜏^𝜏0𝑏̃[𝑐](𝑠)||𝑐(𝑠)−𝑑(𝑠)||₁𝑑𝑠+𝜆_𝑘𝜌^𝑡0𝑑𝜏^𝜏0||𝑏̃[𝑐](𝑠)−𝑏̃[𝑑](𝑠)||𝑑𝑠_𝑘_{𝑗=1𝑁−1𝑖=1}^𝑡0𝑑𝜏^𝜏0||Ψ_𝑖𝑠,𝑢_𝑚−1𝑢^𝑖(𝑠)−𝑣^𝑖(𝑠),𝑤_𝑗||𝑑𝑠≡𝐽₁+𝐽₂+𝐽₃,(3.31) in which 𝐽₁=𝜆_𝑘^𝑡0≤𝜆_𝑘𝑑𝜏^𝜏0𝑏̃[𝑐](𝑠)||𝑐(𝑠)−𝑑(𝑠)||₁𝑑𝑠≡𝜁₁𝑑₀1+𝜆^𝑘2𝛼𝜌^2𝛼^𝑡0𝑑𝜏^𝜏0||𝑐(𝑠)−𝑑(𝑠)||₁𝑑𝑠^𝑡0𝑑𝜏^𝜏0||𝑐(𝑠)−𝑑(𝑠)||₁𝑑𝑠.(3.32) In order to consider 𝐽₂, we also note that 𝑏̃[𝑐](𝑠)−𝑏̃[𝑑](𝑠)=𝐵^(𝜉)‖∇𝑢(𝑠)‖²⁰−‖∇𝑣(𝑠)‖²⁰,(3.33) where 𝜉=𝜃‖∇𝑢(𝑠)‖²⁰+(1−𝜃)‖∇𝑣(𝑠)‖²⁰,0≤𝜉≤𝜆_𝑘𝜌²,0<𝜃<1,(3.34) and 𝐵^(𝜉) satisfy the following inequality: ||𝐵^(𝜉)||≤𝑑₁1+𝜉^𝛼−1≤𝑑₁1+𝜆^𝑘𝛼−1𝜌^2𝛼−2.(3.35)
It implies that||𝑏̃[𝑐](𝑠)−𝑏̃[𝑑](𝑠)||≤=||𝑑𝐵^₁(𝜉)‖∇𝑢(𝑠)‖²⁰−‖∇𝑣(𝑠)‖²⁰||1+𝜆^𝑘𝛼−1𝜌^2𝛼−22𝜆_𝑘𝜌||𝑐(𝑠)−𝑑(𝑠)||₁=2𝜆_𝑘𝜌𝑑₁1+𝜆^𝑘𝛼−1𝜌^2𝛼−2||𝑐(𝑠)−𝑑(𝑠)||₁,(3.36) and then 𝐽₂=≤𝜆2𝜆_𝑘𝜌_2𝑘^𝑡0𝑑𝜏^𝜏0||𝑏̃[𝑐](𝑠)−𝑏̃[𝑑](𝑠)||𝑑𝑠≡𝜁₂𝜌²𝑑₁1+𝜆^𝑘𝛼−1𝜌^2𝛼−2^𝑡0𝑑𝜏^𝜏0||𝑐(𝑠)−𝑑(𝑠)||₁𝑑𝑠^𝑡0𝑑𝜏^𝜏0||𝑐(𝑠)−𝑑(𝑠)||₁𝑑𝑠.(3.37)
It remains to estimate 𝐽₃. By√𝑟𝑢(𝑠)^𝑖−√𝑟𝑣(𝑠)^𝑖=^𝑖−1_𝑗=0√𝑟𝑢(𝑠)^𝑗√𝑟𝑣(𝑠)^{𝑖−𝑗−1}√𝑟(𝑢(𝑠)−𝑣(𝑠)),(3.38) we obtain |||√𝑟𝑢(𝑠)^𝑖−√𝑟𝑣(𝑠)^𝑖|||≤𝐾_{𝑖2𝑖−1}_𝑗=0‖𝑢(𝑠)‖^𝑗1‖𝑣(𝑠)‖^{1𝑖−𝑗−1}‖𝑢(𝑠)−𝑣(𝑠)‖₁≤𝐾^{𝑖2𝑖−1}_𝑗=0‖𝑢(𝑠)‖^𝑗1‖𝑣(𝑠)‖^{1𝑖−𝑗−1}‖𝑢(𝑠)−𝑣(𝑠)‖₁≤𝐾^{𝑖2𝑖−1}_𝑗=0𝜆_𝑘𝐶₀𝜌^𝑗𝜆_𝑘𝐶₀𝜌^{𝑖−𝑗−1}𝜆_𝑘𝐶₀||𝑐(𝑠)−𝑑(𝑠)||₁=𝐾^{𝑖2𝑖−1}_𝑗=0𝜆_𝑘𝐶₀^𝑖𝜌^𝑖−1||𝑐(𝑠)−𝑑(𝑠)||₁=𝑖𝜆_𝑘𝐶₀𝐾₂^𝑖𝜌^𝑖−1||𝑐(𝑠)−𝑑(𝑠)||₁.(3.39) On the other hand, Ψ_𝑖𝑟,𝑡,𝑢_𝑚−1=^𝑁−1_𝑙=𝑖1𝑖!(𝑙−𝑖)!(−1)^𝑙−𝑖√𝐷^𝑙3𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^{𝑙−𝑖𝑚−1},𝑟^−𝑖||Ψ_𝑖𝑟,𝑡,𝑢_𝑚−1||=|||||^𝑁−1_𝑙=𝑖1𝑖!(𝑙−𝑖)!(−1)^𝑙−𝑖√𝑟^−𝑙𝐷^𝑙3𝑓𝑟,𝑡,𝑢_𝑚−1√𝑟^𝑙−𝑖𝑢^{𝑙−𝑖𝑚−1}≤|||||_𝑁−1|||||_𝑙=𝑖1𝑖!(𝑙−𝑖)!√≤_𝑁−1𝑟^−𝑙||𝐷^𝑙3𝑓𝑟,𝑡,𝑢_𝑚−1||√𝑟^𝑙−𝑖||𝑢^{𝑙−𝑖𝑚−1}|||||||_𝑙=𝑖1𝑖!(𝑙−𝑖)!𝐾^∗𝑙(𝑀,𝑓)𝜃^𝑙−𝑖√𝑟^𝑙−𝑖.(3.40)
Hence, we deduce from (3.39), (3.40) that𝐽₃=^𝑘_{𝑗=1𝑁−1𝑖=1}^𝑡0𝑑𝜏^𝜏0||||√𝑟^−𝑖Ψ_𝑖𝑠,𝑢_𝑚−1√𝑟𝑢(𝑠)^𝑖−√𝑟𝑣(𝑠)^𝑖≤,𝑤^𝑗||||𝑑𝑠_𝑘_{𝑗=1𝑁−1}_{𝑖=1𝑁−1𝑙=𝑖}1𝑖!(𝑙−𝑖)!𝐾^∗𝑙(𝑀,𝑓)𝜃^𝑙−𝑖𝑖𝜆_𝑘𝐶₀𝐾₂^𝑖𝜌^𝑖−1×^𝑡0𝑑𝜏^𝜏0||||√=𝑟^𝑙−𝑖,||𝑤_𝑗||||||||𝑐(𝑠)−𝑑(𝑠)||₁𝑑𝑠_𝑘_{𝑗=1𝑁−1}_{𝑖=1𝑁−1𝑙=𝑖}1𝑖!(𝑙−𝑖)!𝐾^∗𝑙(𝑀,𝑓)𝜃^𝑙−𝑖𝑖𝜆_𝑘𝐶₀𝐾₂^𝑖𝜌^𝑖−11×√2+𝑙−𝑖_𝑡0=𝑘_𝑁−1𝑑𝜏^𝜏0||𝑐(𝑠)−𝑑(𝑠)||₁𝑑𝑠_𝑖=1𝑖𝜆_𝑘𝐶₀𝐾₂^𝑖𝜌^{𝑖−1𝑁−1}_𝑙=𝑖1𝑖!(𝑙−𝑖)!𝐾^∗𝑙(𝑀,𝑓)𝜃^𝑙−𝑖1×√2+𝑙−𝑖_𝑡0≡𝜁₃𝑑𝜏^𝜏0||𝑐(𝑠)−𝑑(𝑠)||₁𝑑𝑠^𝑡0𝑑𝜏^𝜏0||𝑐(𝑠)−𝑑(𝑠)||₁𝑑𝑠.(3.41)
We deduce that||ℱ[𝑐](𝑡)−ℱ[𝑑](𝑡)||₁≤𝜁₁+𝜁₂+𝜁₃^𝑡0𝑑𝜏^𝜏0||𝑐(𝑠)−𝑑(𝑠)||₁𝑑𝑠≤12𝐷^𝜌𝑡²‖𝑐−𝑑‖_𝑋.(3.42) We note that 𝜁₁+𝜁₂+𝜁₃=𝐷_𝜌(𝜌,𝑘,𝑀,𝑇,𝑚,𝑁,𝑓)=𝐷_𝜌.(3.43)
It follows from (3.28) that‖ℱ[𝑐]−ℱ[𝑑]‖_𝑋≤12𝐷^𝜌𝑇^𝑚(𝑘)²‖𝑐−𝑑‖_𝑋,∀𝑐,𝑑∈𝑆.(3.44)
By (3.25), it follows that ℱ∶𝑆→𝑆 is contractive. We deduce that ℱ has a unique fixed point in 𝑆; that is, the system (3.8)–(3.11) has a unique solution 𝑢^𝑚(𝑘)(𝑡) on an interval [0,𝑇^𝑚(𝑘)]. The proof of Lemma 3.2 is complete.
The following estimates allow one to take constant 𝑇^𝑚(𝑘)=𝑇 for all 𝑚 and 𝑘.
Step 2. A priori estimates. Put 𝑆^𝑚(𝑘)(𝑡)=𝑋^𝑚(𝑘)(𝑡)+𝑌^𝑚(𝑘)(𝑡)+^𝑡0‖‖𝑢̈^𝑚(𝑘)(𝑠)‖‖²⁰𝑑𝑠,(3.45) where 𝑌𝑋^𝑚(𝑘)(𝑡)=‖‖𝑢̇^𝑚(𝑘)(𝑡)‖‖²⁰+𝑏^𝑚(𝑘)(𝑡)𝑎𝑢^𝑚(𝑘)(𝑡),𝑢^𝑚(𝑘)(𝑡)_𝑚(𝑘),(𝑡)=𝑎𝑢̇^𝑚(𝑘)(𝑡),𝑢̇^𝑚(𝑘)(𝑡)+𝑏^𝑚(𝑘)(𝑡)‖‖𝐴𝑢^𝑚(𝑘)(𝑡)‖‖²⁰,(3.46) with 𝐴 is defined by (2.2). Then it follows that 𝑆^𝑚(𝑘)(𝑡)=+𝑆2^𝑚(𝑘)(_𝑡00)+^𝑡0𝑏̇^𝑚(𝑘)(𝑠)𝑎𝑢^𝑚(𝑘)(𝑠),𝑢^𝑚(𝑘)(𝑠)+‖‖𝐴𝑢^𝑚(𝑘)(𝑠)‖‖²⁰𝑑𝑠+𝐹^𝑚(𝑘)(𝑠),𝑢̇^𝑚(𝑘)(𝑠)𝑑𝑠+2^𝑡0𝑎𝐹^𝑚(𝑘)(𝑠),𝑢̇^𝑚(𝑘)(𝑠)𝑑𝑠_𝑡0≡𝑆_𝑚(𝑘)𝐹^𝑚(𝑘)(𝑠),𝑢̈^𝑚(𝑘)(𝑠)𝑑𝑠−^𝑡0𝑏^𝑚(𝑘)(𝑠)𝐴𝑢^𝑚(𝑘)(𝑠),𝑢̈^𝑚(𝑘)(𝑠)𝑑𝑠(0)+⁵_𝑗=1𝐼_𝑗.(3.47)
We will now require the following lemma.
Lemma 3.3. We have ((ii)i)0||𝑏<̇_𝑚(𝑘)𝑏_∗≤𝑏^𝑚(𝑘)(𝑡)≤𝑑₀1+‖‖∇𝑢^𝑚(𝑘)(𝑡)‖‖^02𝛼,(𝑡)||≤2𝑑¹√𝑏_∗(iii)‖‖𝐹_𝑚(𝑘)𝑆^𝑚(𝑘)(𝑡)+𝑏^∗1−𝛼𝑆^𝑚(𝑘)(𝑡)^𝛼,(𝑡)‖‖₀≤^𝑁−1_𝑗=0𝑎̃^𝑗(0)(iv)‖‖‖𝜕𝑆^𝑚(𝑘)(𝑡)^𝑗,𝜕𝑟𝐹^𝑚(𝑘)(𝑡)‖‖‖₀≤^𝑁−1_𝑗=0𝑎̃^𝑗(1)𝑆^𝑚(𝑘)(𝑡)^𝑗,(3.48) where 𝑎̃^𝑗(0),𝑎̃^𝑗(1),𝑗=0,1,…,𝑁−1 are defined as follows: 𝑎̃^𝑗(0)=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑎̃⁰⁽⁰⁾=1√2𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖𝑎̃_𝑗(0)𝑖!,𝑗=0,=𝐾^𝑗2√𝑏_∗𝐶₀^𝑗𝑁−1_𝑖=𝑗𝜃^𝑖−𝑗𝑗!(𝑖−𝑗)!√2+1𝑖−𝑗𝐾^∗𝑖𝑎̃_𝑗(1)(𝑀,𝑓),1≤𝑗≤𝑁−1,=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩1√2+2𝑀𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖𝐾𝑖!,𝑗=0,_𝑗2√𝑏_∗𝐶₀^𝑗𝑁−1_𝑖=𝑗𝜃^𝑖−𝑗𝑗!(𝑖−𝑗)!𝑗𝐾²⁻¹+1𝐾_∗𝑖(𝑀,𝑓)√𝑖−𝑗+3+2𝑀𝐾^∗𝑖+1𝜃=𝑀(𝑀,𝑓),1≤𝑗≤𝑁−1,2+1√2.(3.49)
Proof of Lemma 3.3. Proof (i), (ii). Note that𝑆𝑆^𝑚(𝑘)(𝑡_𝑚(𝑘))≥𝑋^𝑚(𝑘)(𝑡)≥𝑏_∗𝑎𝑢^𝑚(𝑘)(𝑡),𝑢^𝑚(𝑘)(𝑡)≥𝑏_∗‖‖∇𝑢^𝑚(𝑘)(𝑡)‖‖²⁰,(𝑡)≥𝑌^𝑚(𝑘)(𝑡)≥𝑎𝑢̇^𝑚(𝑘)(𝑡),𝑢̇^𝑚(𝑘)(𝑡)≥‖‖∇𝑢̇^𝑚(𝑘)(𝑡)‖‖²⁰.(3.50) We deduce that ||𝑏𝑏̇^𝑚(𝑘)_𝑚(𝑘)(𝑡)=𝐵‖‖∇𝑢^𝑚(𝑘)(𝑡)‖‖²⁰≤𝑑⁰1+‖‖∇𝑢^𝑚(𝑘)(𝑡)‖‖^02𝛼,(𝑡)||≤=22𝑑|||𝐵^₁‖‖∇𝑢^𝑚(𝑘)(𝑡)‖‖²⁰||||||∇𝑢^𝑚(𝑘)(𝑡),∇𝑢̇^𝑚(𝑘)(𝑡)|||1+‖‖∇𝑢^𝑚(𝑘)(𝑡)‖‖^02𝛼−2‖‖∇𝑢^𝑚(𝑘)(𝑡)‖‖₀‖‖∇𝑢̇^𝑚(𝑘)(𝑡)‖‖₀≤2𝑑₁1+𝑏^∗1−𝛼𝑆^𝑚(𝑘)(𝑡)^𝛼−1√1𝑏_∗=2𝑑₁𝑆^𝑚(𝑘)(𝑡)√𝑏_∗𝑆^𝑚(𝑘)(𝑡)+𝑏^∗1−𝛼𝑆^𝑚(𝑘)(𝑡)^𝛼.(3.51)
Proof (iii). We have ‖‖𝐹^𝑚(𝑘)(𝑡)‖‖₀≤‖‖Ψ₀𝑟,𝑡,𝑢_𝑚−1‖‖₀+^𝑁−1_𝑗=1‖‖‖Ψ_𝑗𝑟,𝑡,𝑢_𝑚−1𝑢^𝑚(𝑘)^𝑗‖‖‖₀.(3.52) By (3.18)₃, we have ‖‖Ψ₀𝑟,𝑡,𝑢_𝑚−1‖‖₀≤1√2𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖𝑖!≡𝑎̃⁰⁽⁰⁾.(3.53) On the other hand, it follows from (3.49) and 𝑢_𝑚−1∈𝑊₁(𝑀,𝑇) that ‖‖‖Ψ_𝑗𝑟,𝑡,𝑢_𝑚−1𝑢^𝑚(𝑘)^𝑗‖‖‖₀≤^𝑁−1_𝑖=𝑗1𝑗!(𝑖−𝑗)!‖‖‖𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^{𝑖−𝑗𝑚−1}𝑢^𝑚(𝑘)^𝑗‖‖‖₀=^𝑁−1_𝑖=𝑗1𝑗!(𝑖−𝑗)!‖‖‖√𝑟^𝑖−𝑗√𝑟^−𝑖𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^{𝑖−𝑗𝑚−1}√𝑟^𝑗𝑢^𝑚(𝑘)^𝑗‖‖‖₀≤^𝑁−1_𝑖=𝑗1𝑗!(𝑖−𝑗)!‖‖‖√𝑟^𝑖−𝑗‖‖‖₀𝐾_∗𝑖(𝑀,𝑓)𝜃^𝑖−𝑗𝐾^𝑗2‖‖𝑢^𝑚(𝑘)(𝑡)‖‖^𝑗1=^𝑁−1_𝑖=𝑗1𝑗!(𝑖−𝑗)!√2+1𝑖−𝑗𝐾^∗𝑖(𝑀,𝑓)𝜃^𝑖−𝑗𝐾^𝑗2‖‖𝑢^𝑚(𝑘)(𝑡)‖‖^𝑗1≤𝐾^𝑗2√𝑏_∗𝐶₀^𝑗𝑁−1_𝑖=𝑗𝜃^𝑖−𝑗𝑗!(𝑖−𝑗)!√2+1𝑖−𝑗𝐾^∗𝑖(𝑀,𝑓)𝑆^𝑚(𝑘)(𝑡)^𝑗≡𝑎̃^𝑗(0)𝑆^𝑚(𝑘)(𝑡)^𝑗.(3.54)
It follows from (3.52)–(3.54) that ‖‖𝐹^𝑚(𝑘)(𝑡)‖‖₀≤^𝑁−1_𝑗=0𝑎̃^𝑗(0)𝑆^𝑚(𝑘)(𝑡)^𝑗,(3.55) where 𝑎̃^𝑗(0),0≤𝑗≤𝑁−1 are defined by (3.49)₁.
Proof (iv). We have 𝜕𝜕𝑟𝐹^𝑚(𝑘)(𝑟,𝑡)=𝜕𝜕𝑟Ψ⁰𝑟,𝑡,𝑢_𝑚−1+^𝑁−1_𝑗=1𝑗Ψ_𝑗𝑟,𝑡,𝑢_𝑚−1𝑢^𝑚(𝑘)^𝑗−1∇𝑢^𝑚(𝑘)+^𝑁−1_𝑗=1𝜕𝜕𝑟Ψ^𝑗𝑟,𝑡,𝑢_𝑚−1𝑢^𝑚(𝑘)^𝑗.(3.56)
Hence‖‖‖𝜕𝜕𝑟𝐹^𝑚(𝑘)(𝑡)‖‖‖₀≤‖‖‖𝜕𝜕𝑟Ψ⁰𝑟,𝑡,𝑢_𝑚−1‖‖‖₀+^𝑁−1_𝑗=1𝑗‖‖‖Ψ_𝑗𝑟,𝑡,𝑢_𝑚−1𝑢^𝑚(𝑘)^𝑗−1∇𝑢^𝑚(𝑘)‖‖‖₀+^𝑁−1_𝑗=1‖‖‖𝜕𝜕𝑟Ψ^𝑗𝑟,𝑡,𝑢_𝑚−1𝑢^𝑚(𝑘)^𝑗‖‖‖₀≡𝐿₁+𝐿₂+𝐿₃.(3.57) We shall estimate step by step the terms on the right-hand side of (3.57) as follows.(iv.1) Estimating 𝐿₁=‖(𝜕/𝜕𝑟)Ψ₀(𝑟,𝑡,𝑢_𝑚−1)‖₀. We have𝜕𝜕𝑟Ψ⁰𝑟,𝑡,𝑢_𝑚−1=^𝑁−1_𝑖=11𝑖!(−1)^𝑖𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑖𝑢^{𝑖−1𝑚−1}∇𝑢_𝑚−1+^𝑁−1_𝑖=01𝑖!(−1)^𝑖𝐷₁𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^𝑖𝑚−1+^𝑁−1_𝑖=01𝑖!(−1)^𝑖𝐷^3𝑖+1𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^𝑖𝑚−1∇𝑢_𝑚−1=𝑎(1)+𝑎(2)+𝑎(3).(3.58) We will estimate step by step the terms 𝑎(1),𝑎(2),𝑎(3) as follows.(iv.1.1) Estimating ‖𝑎(1)‖₀. We have||𝑎(1)||=|||||^𝑁−1_𝑖=11𝑖!(−1)^𝑖𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑖𝑢^{𝑖−1𝑚−1}∇𝑢_𝑚−1≤|||||𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=11𝑖!𝑖𝜃^𝑖−1=||∇𝑢_𝑚−1||𝐾_𝑁(𝑀,𝑓)^𝑁−2_𝑖=0𝜃^𝑖𝑖!||∇𝑢^𝑚−1≤||𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖𝑖!||∇𝑢^𝑚−1||.(3.59) Hence ‖𝑎(1)‖₀≤𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖𝑖!‖‖∇𝑢^𝑚−1‖‖₀≤𝑀𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖𝑖!.(3.60)(iv.1.2) Estimating ‖𝑎(2)‖₀. It follows from||𝑎(2)||=|||||^𝑁−1_𝑖=01𝑖!(−1)^𝑖𝐷₁𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^𝑖𝑚−1|||||≤𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖𝑖!(3.61) that ‖𝑎(2)‖₀≤1√2𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖𝑖!.(3.62)(iv.1.3) Estimating ‖𝑎(3)‖₀. Similarly, with ||𝑎(3)||=|||||^𝑁−1_𝑖=01𝑖!(−1)^𝑖𝐷^3𝑖+1𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^𝑖𝑚−1∇𝑢_𝑚−1|||||≤𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖𝑖!||∇𝑢^𝑚−1||,(3.63) we obtain‖𝑎(3)‖₀≤𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖𝑖!‖‖∇𝑢^𝑚−1‖‖₀≤𝑀𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖𝑖!.(3.64)
It follows from (3.58), (3.60), (3.62), (3.64) that 𝐿₁=‖‖‖𝜕𝜕𝑟Ψ⁰𝑟,𝑡,𝑢_𝑚−1‖‖‖₀≤‖𝑎(1)‖₀+‖𝑎(2)‖₀+‖𝑎(3)‖₀≤1√2𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖𝑖!+2𝑀𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖≤1𝑖!√2+2𝑀𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖𝑖!.(3.65)
(iv.2) Estimating 𝐿₂=∑^{𝑁−1𝑗=1}𝑗‖Ψ_𝑗(𝑟,𝑡,𝑢_𝑚−1)(𝑢^𝑚(𝑘))^𝑗−1∇𝑢^𝑚(𝑘)‖₀. By the assumption (𝐻₃), we deduce that𝐿₂=^𝑁−1_𝑗=1𝑗‖‖‖Ψ_𝑗𝑟,𝑡,𝑢_𝑚−1𝑢^𝑚(𝑘)^𝑗−1∇𝑢^𝑚(𝑘)‖‖‖₀≤^𝑁−1_𝑗=1𝑗^𝑁−1_𝑖=𝑗1𝑗!(𝑖−𝑗)!‖‖‖𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^{𝑖−𝑗𝑚−1}𝑢^𝑚(𝑘)^𝑗−1∇𝑢^𝑚(𝑘)‖‖‖₀=^𝑁−1_𝑗=1𝑗^𝑁−1_𝑖=𝑗1𝑗!(𝑖−𝑗)!‖‖‖√𝑟^−𝑖𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1√𝑟^𝑖−𝑗𝑢^{𝑖−𝑗𝑚−1}√𝑟𝑢^𝑚(𝑘)^𝑗−1√𝑟∇𝑢^𝑚(𝑘)‖‖‖₀=^𝑁−1_𝑗=1𝑗^𝑁−1_𝑖=𝑗1𝑗!(𝑖−𝑗)!𝐾^∗𝑖(𝑀,𝑓)‖‖‖√𝑟^𝑖−𝑗√𝑟∇𝑢^𝑚(𝑘)‖‖‖₀𝜃^𝑖−𝑗𝐾^2𝑗−1‖‖𝑢^𝑚(𝑘)(𝑡)‖‖^1𝑗−1≤^𝑁−1_𝑗=1𝑗^𝑁−1_𝑖=𝑗𝜃^𝑖−𝑗𝑗!(𝑖−𝑗)!𝐾^∗𝑖(𝑀,𝑓)𝐾^2𝑗−1‖‖𝑢^𝑚(𝑘)(𝑡)‖‖^𝑗1≤^𝑁−1_𝑗=1𝑗𝐾^2𝑗−1√𝑏_∗𝐶₀^𝑗𝑁−1_𝑖=𝑗𝜃^𝑖−𝑗𝑗!(𝑖−𝑗)!𝐾^∗𝑖(𝑀,𝑓)𝑆^𝑚(𝑘)(𝑡)^𝑗.(3.66)(iv.3) Estimating 𝐿₃=∑^{𝑁−1𝑗=1}‖((𝜕/𝜕𝑟)Ψ_𝑗(𝑟,𝑡,𝑢_𝑚−1))(𝑢^𝑚(𝑘))^𝑗‖₀. We have𝐿₃=^𝑁−1_𝑗=1‖‖‖𝜕𝜕𝑟Ψ^𝑗𝑟,𝑡,𝑢_𝑚−1𝑢^𝑚(𝑘)^𝑗‖‖‖₀=^𝑁−1_𝑗=1‖‖‖√𝑟^−𝑗𝜕𝜕𝑟Ψ^𝑗𝑟,𝑡,𝑢_𝑚−1√𝑟^𝑗𝑢^𝑚(𝑘)^𝑗‖‖‖₀≤^𝑁−1_𝑗=1‖‖‖√𝑟^−𝑗𝜕𝜕𝑟Ψ^𝑗𝑟,𝑡,𝑢_𝑚−1‖‖‖₀𝐾^𝑗2‖‖𝑢^𝑚(𝑘)(𝑡)‖‖^𝑗1≤^𝑁−1_𝑗=1‖‖‖√𝑟^−𝑗𝜕𝜕𝑟Ψ^𝑗𝑟,𝑡,𝑢_𝑚−1‖‖‖₀1√𝑏_∗𝐶₀^𝑗𝐾^𝑗2𝑆^𝑚(𝑘)(𝑡)^𝑗≡^𝑁−1_𝑗=1𝐿₃(𝑚,𝑗,𝑡)𝐾^𝑗2√𝑏_∗𝐶₀^𝑗𝑆^𝑚(𝑘)(𝑡)^𝑗.(3.67) Now we need an estimation of the term  𝐿₃(𝑚,𝑗,𝑡)=‖(√𝑟)^−𝑗((𝜕/𝜕𝑟)Ψ_𝑗(𝑟,𝑡,𝑢_𝑚−1))‖₀. We have 𝜕𝜕𝑟Ψ^𝑗𝑟,𝑡,𝑢_𝑚−1=^𝑁−1_𝑖=𝑗+11𝑗!(𝑖−𝑗)!(−1)^𝑖−𝑗(𝑖−𝑗)𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^{𝑖−𝑗−1𝑚−1}∇𝑢_𝑚−1+^𝑁−1_𝑖=𝑗1𝑗!(𝑖−𝑗)!(−1)^𝑖−𝑗𝐷₁𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^{𝑖−𝑗𝑚−1}+^𝑁−1_𝑖=𝑗1𝑗!(𝑖−𝑗)!(−1)^𝑖−𝑗𝐷^3𝑖+1𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^{𝑖−𝑗𝑚−1}∇𝑢_𝑚−1=𝑏(1)+𝑏(2)+𝑏(3).(3.68) It follows from (3.68) that 𝐿₃(𝑚,𝑗,𝑡)=‖‖‖√𝑟^−𝑗𝜕𝜕𝑟Ψ^𝑗𝑟,𝑡,𝑢_𝑚−1‖‖‖₀≤‖‖‖√𝑟^−𝑗𝑏(1)‖‖‖₀+‖‖‖√𝑟^−𝑗𝑏(2)‖‖‖₀+‖‖‖√𝑟^−𝑗𝑏(3)‖‖‖₀.(3.69)(iv.3.1) Estimating ‖(√𝑟)^−𝑗𝑏(1)‖₀. We have‖‖‖√𝑟^−𝑗𝑏(1)‖‖‖₀=‖‖‖‖√𝑟^{−𝑗𝑁−1}_𝑖=_𝑗+11𝑗!(𝑖−𝑗)!(−1)^𝑖−𝑗(𝑖−𝑗)𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^{𝑖−𝑗−1𝑚−1}∇𝑢_𝑚−1‖‖‖‖₀≤^𝑁−1_𝑖=_𝑗+11𝑗!(𝑖−𝑗)!(𝑖−𝑗)‖‖‖√𝑟^−𝑖𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1√𝑟^{𝑖−𝑗−1}𝑢^{𝑖−𝑗−1𝑚−1}√𝑟∇𝑢_𝑚−1‖‖‖₀≤^𝑁−1_𝑖=_𝑗+11𝑗!(𝑖−𝑗)!(𝑖−𝑗)𝐾^∗𝑖=𝑀_𝑁−2(𝑀,𝑓)𝜃^{𝑖−𝑗−1}𝑀_𝑖=𝑗𝜃^𝑖−𝑗𝑗!(𝑖−𝑗)!𝐾^∗𝑖+1(𝑀,𝑓)≤𝑀^𝑁−1_𝑖=𝑗𝜃^𝑖−𝑗𝑗!(𝑖−𝑗)!𝑀𝐾^∗𝑖+1(𝑀,𝑓).(3.70)(iv.3.2) Estimating ‖(√𝑟)^−𝑗𝑏(2)‖₀. We have‖‖‖√𝑟^−𝑗𝑏(2)‖‖‖₀=‖‖‖‖√𝑟^{−𝑗𝑁−1}_𝑖=𝑗1𝑗!(𝑖−𝑗)!(−1)^𝑖−𝑗𝐷₁𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^{𝑖−𝑗𝑚−1}‖‖‖‖₀≤^𝑁−1_𝑖=𝑗1𝑗!(𝑖−𝑗)!‖‖‖√𝑟^−𝑖−1𝐷₁𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1√𝑟^{𝑖−𝑗+1}𝑢^{𝑖−𝑗𝑚−1}‖‖‖₀≤^𝑁−1_𝑖=𝑗1𝑗!(𝑖−𝑗)!𝐾^∗𝑖+1(𝑀,𝑓)𝜃^𝑖−𝑗‖‖‖√𝑟^{𝑖−𝑗+1}‖‖‖₀=^𝑁−1_𝑖=𝑗𝜃^𝑖−𝑗𝑗!(𝑖−𝑗)!√𝑖−1𝑗+3𝐾^∗𝑖+1(𝑀,𝑓).(3.71)(iv.3.3) Estimating ‖(√𝑟)^−𝑗𝑏(3)‖₀. We have‖‖‖√𝑟^−𝑗𝑏(3)‖‖‖₀=‖‖‖‖√𝑟^{−𝑗𝑁−1}_𝑖=𝑗1𝑗!(𝑖−𝑗)!(−1)^𝑖−𝑗𝐷^3𝑖+1𝑓𝑟,𝑡,𝑢_𝑚−1𝑢^{𝑖−𝑗𝑚−1}∇𝑢_𝑚−1‖‖‖‖₀≤^𝑁−1_𝑖=𝑗1𝑗!(𝑖−𝑗)!‖‖‖√𝑟^−𝑖−1𝐷^3𝑖+1𝑓𝑟,𝑡,𝑢_𝑚−1√𝑟^𝑖−𝑗𝑢^{𝑖−𝑗𝑚−1}√𝑟∇𝑢_𝑚−1‖‖‖₀≤^𝑁−1_𝑖=𝑗1𝑗!(𝑖−𝑗)!𝐾^∗𝑖+1(𝑀,𝑓)𝜃^𝑖−𝑗𝑀=𝑀^𝑁−1_𝑖=𝑗𝜃^𝑖−𝑗𝑗!(𝑖−𝑗)!𝐾^∗𝑖+1(𝑀,𝑓).(3.72) It follows from (3.69)–(3.72) that 𝐿₃(𝑚,𝑗,𝑡)=‖‖‖√𝑟^−𝑗𝜕𝜕𝑟Ψ^𝑗𝑟,𝑡,𝑢_𝑚−1‖‖‖₀≤2𝑀^𝑁−1_𝑖=𝑗𝜃^𝑖−𝑗𝑗!(𝑖−𝑗)!𝐾^∗𝑖+1(𝑀,𝑓)+^𝑁−1_𝑖=𝑗𝜃^𝑖−𝑗𝑗!(𝑖−𝑗)!√𝑖−1𝑗+3𝐾^∗𝑖+1=(𝑀,𝑓)_𝑁−1_𝑖=𝑗𝜃^𝑖−𝑗𝑗!(𝑖−𝑗)!√𝑖−1𝑗+3+2𝑀𝐾^∗𝑖+1(𝑀,𝑓).(3.73) It follows from (3.67) – (3.73) that 𝐿₃≤^𝑁−1_𝑗=1𝐿₃(𝑚,𝑗,𝑡)𝐾^𝑗2√𝑏_∗𝐶₀^𝑗𝑆^𝑚(𝑘)(𝑡)^𝑗≤^𝑁−1_{𝑗=1𝑁−1𝑖=𝑗}𝜃^𝑖−𝑗𝑗!(𝑖−𝑗)!√𝑖−1𝑗+3+2𝑀𝐾^∗𝑖+1(𝑀,𝑓)𝐾^𝑗2√𝑏_∗𝐶₀^𝑗𝑆^𝑚(𝑘)(𝑡)^𝑗.(3.74) We deduce from (3.57), (3.65), (3.66), (3.74) that ‖‖‖𝜕𝜕𝑟𝐹^𝑚(𝑘)(𝑡)‖‖‖₀≤1√2+2𝑀𝐾_𝑁(𝑀,𝑓)^𝑁−1_𝑖=0𝜃^𝑖+𝑖!_𝑁−1_𝑗=1𝐾^𝑗2√𝑏_∗𝐶₀^𝑗𝑁−1_𝑖=𝑗𝜃^𝑖−𝑗×𝑗𝐾₂₋₁𝑗!(𝑖−𝑗)!𝐾_∗𝑖(𝑀,𝑓)+1√𝑖−𝑗+3+2𝑀𝐾^∗𝑖+1(𝑀,𝑓)𝑆^𝑚(𝑘)(𝑡)^𝑗≡^𝑁−1_𝑗=1𝑎̃^𝑗(1)𝑆^𝑚(𝑘)(𝑡)^𝑗,(3.75) where 𝑎̃^𝑗(1),0≤𝑗≤𝑁−1 are defined by (3.49)₂.
Next, we will estimate step by step all integrals 𝐼₁,…,𝐼₅.
Integral 𝐼₁=∫^𝑡0𝑏̇^𝑚(𝑘)(𝑠)[𝑎(𝑢^𝑚(𝑘)(𝑠),𝑢^𝑚(𝑘)(𝑠))+‖𝐴𝑢^𝑚(𝑘)(𝑠)‖²⁰]𝑑𝑠.
Now, using the inequalities (3.48) and𝑠^𝑞≤1+𝑠^𝑁0,∀𝑠≥0,∀𝑞,0<𝑞≤𝑁₀=max{1+𝛼,𝑁},(3.76) we estimate without difficulty the following integrals in the right-hand side of (3.47) as follows.
The integral 𝐼₁:𝐼₁=≤2^𝑡0𝑑𝑏₁̇^𝑚(𝑘)(𝑠)𝑎𝑢^𝑚(𝑘)(𝑠),𝑢^𝑚(𝑘)(𝑠)+‖‖𝐴𝑢^𝑚(𝑘)(𝑠)‖‖²⁰𝑑𝑠𝑏^∗3/2^𝑡0𝑆^𝑚(𝑘)(𝑠)²+𝑏^∗1−𝛼𝑆^𝑚(𝑘)(𝑠)^𝛼+1≤2𝑑₁𝑑𝑠𝑏^∗3/2^𝑡01+𝑆^𝑚(𝑘)(𝑠)^𝑁0+𝑏^∗1−𝛼1+𝑆^𝑚(𝑘)(𝑠)^𝑁0≤2𝑑₁𝑑𝑠𝑏^∗3/21+𝑏^∗1−𝛼^𝑡01+𝑆^𝑚(𝑘)(𝑠)^𝑁0𝑑𝑠.(3.77)
The integral 𝐼₂=2∫^𝑡0⟨𝐹^𝑚(𝑘)(𝑠),𝑢̇^𝑚(𝑘)(𝑠)⟩𝑑𝑠∶𝐼₂=≤22^𝑡0_𝑁−1𝐹^𝑚(𝑘)(𝑠),𝑢̇^𝑚(𝑘)(𝑠)𝑑𝑠≤2^𝑡0‖‖𝐹^𝑚(𝑘)(𝑠)‖‖₀‖‖𝑢̇^𝑚(𝑘)(𝑠)‖‖₀𝑑𝑠_𝑗=0𝑎̃^𝑗(0)^𝑡0𝑆^𝑚(𝑘)(𝑠)^𝑗+1𝑑𝑠≤2^𝑁−1_𝑗=0𝑎̃^𝑗(0)^𝑡01+𝑆^𝑚(𝑘)(𝑠)^𝑁0𝑑𝑠.(3.78)
The integral 𝐼₃=2∫^𝑡0𝑎(𝐹^𝑚(𝑘)(𝑠),𝑢̇^𝑚(𝑘)(𝑠))𝑑𝑠:𝐼₃=≤22^𝑡0_𝑡0𝑎𝐹^𝑚(𝑘)(𝑠),𝑢̇^𝑚(𝑘)(𝑠)𝑑𝑠≤2√𝑎𝐹^𝑚(𝑘)(𝑠),𝐹^𝑚(𝑘)(𝑠)𝑎𝑢̇^𝑚(𝑘)(𝑠),𝑢̇^𝑚(𝑘)(𝑠)𝑑𝑠𝐶₁^𝑡0‖‖𝐹^𝑚(𝑘)(𝑠)‖‖₁≤2√𝑎𝑢̇^𝑚(𝑘)(𝑠),𝑢̇^𝑚(𝑘)(𝑠)𝑑𝑠𝐶_1𝑁−1_𝑗=0𝑎̃^𝑗(1)^𝑡0≤2√𝑆^𝑚(𝑘)(𝑠)^𝑗+1𝑑𝑠𝐶_1𝑁−1_𝑗=0𝑎̃^𝑗(1)^𝑡01+𝑆^𝑚(𝑘)(𝑠)^𝑁0𝑑𝑠.(3.79)
The integral 𝐼₄=∫^𝑡0⟨𝐹^𝑚(𝑘)(𝑠),𝑢̈^𝑚(𝑘)(𝑠)⟩𝑑𝑠:𝐼₄≤=^𝑡0𝐹^𝑚(𝑘)(𝑠),𝑢̈^𝑚(𝑘)(𝑠)𝑑𝑠≤^𝑡0‖‖𝐹^𝑚(𝑘)(𝑠)‖‖₀‖‖𝑢̈^𝑚(𝑘)(𝑠)‖‖₀𝑑𝑠_𝑡0‖‖𝐹^𝑚(𝑘)(𝑠)‖‖²⁰𝑑𝑠+14^𝑡0≤‖‖𝑢̈^𝑚(𝑘)(𝑠)‖‖²⁰𝑑𝑠_𝑡0^𝑁−1_𝑗=0𝑎̃^𝑗(0)𝑆^𝑚(𝑘)(𝑠)^𝑗²𝑑𝑠+14𝑆^𝑚(𝑘)≤𝑁_𝑁−1(𝑡)_𝑗=0||𝑎̃^𝑗(0)||²^𝑡0𝑆^𝑚(𝑘)(𝑠)^𝑗𝑑𝑠+14𝑆^𝑚(𝑘)≤𝑁_𝑁−1(𝑡)_𝑗=0||𝑎̃^𝑗(0)||²^𝑡01+𝑆^𝑚(𝑘)(𝑠)^𝑁0𝑑𝑠+41𝑆^𝑚(𝑘)(𝑡).(3.80)
The integral 𝐼₅=−∫^𝑡0𝑏^𝑚(𝑘)(𝑠)⟨𝐴𝑢^𝑚(𝑘)(𝑠),𝑢̈^𝑚(𝑘)(𝑠)⟩𝑑𝑠∶𝐼₅≤=−^𝑡0𝑏^𝑚(𝑘)(𝑠)𝐴𝑢^𝑚(𝑘)(𝑠),𝑢̈^𝑚(𝑘)(𝑠)𝑑𝑠≤^𝑡0𝑏^𝑚(𝑘)(𝑠)‖‖𝐴𝑢^𝑚(𝑘)(𝑠)‖‖₀‖‖𝑢̈^𝑚(𝑘)(𝑠)‖‖₀𝑑𝑠_𝑡0𝑏^𝑚(𝑘)(𝑠)𝑏^𝑚(𝑘)(𝑠)‖‖𝐴𝑢^𝑚(𝑘)(𝑠)‖‖²⁰𝑑𝑠+14^𝑡0≤𝑑₀‖‖𝑢̈^𝑚(𝑘)(𝑠)‖‖²⁰𝑑𝑠^𝑡01+𝑏^∗−𝛼𝑆^𝑚(𝑘)(𝑠)^𝛼𝑆^𝑚(𝑘)(𝑠)𝑑𝑠+14𝑆^𝑚(𝑘)≤𝑑₀(𝑡)^𝑡0𝑆^𝑚(𝑘)(𝑠)+𝑏^∗−𝛼𝑆^𝑚(𝑘)(𝑠)^𝛼+1𝑑𝑠+14𝑆^𝑚(𝑘)≤𝑑₀(𝑡)^𝑡01+𝑆^𝑚(𝑘)(𝑠)^𝑁0+𝑏^∗−𝛼1+𝑆^𝑚(𝑘)(𝑠)^𝑁0𝑑𝑠+14𝑆^𝑚(𝑘)≤𝑑₀(𝑡)1+𝑏^∗−𝛼^𝑡01+𝑆^𝑚(𝑘)(𝑠)^𝑁0𝑑𝑠+14𝑆^𝑚(𝑘)(𝑡).(3.81)
From the convergences in (3.10), we can deduce the existence of a constant 𝑀≥2 independent of𝑘 and 𝑚 such that𝑆^𝑚(𝑘)(0)≤𝑀²4.(3.82)
Combining (3.47), (3.77)–(3.82), we then have𝑆^𝑚(𝑘)(𝑡)≤𝑀²2+𝑇𝐷⁰(𝑀)+𝐷₀(𝑀)^𝑡0𝑆^𝑚(𝑘)(𝑠)^𝑁0𝑑𝑠,(3.83) where 𝐷₀(𝑀)=2𝑑₀1+𝑏^∗−𝛼+4𝑑¹𝑏^∗3/21+𝑏^∗1−𝛼+4^𝑁−1_𝑗=0𝑎̃^𝑗(0)+√𝐶₁𝑎̃^𝑗(1)+12𝑁||𝑎̃^𝑗(0)||².(3.84)
Then, we have the following Lemma.
Lemma 3.4. There exists a constant 𝑇>0 independent of 𝑘 and 𝑚 such that 𝑆^𝑚(𝑘)(𝑡)≤𝑀²∀𝑡∈[0,𝑇],∀𝑘,𝑚.(3.85)
Proof of Lemma 3.4. Put 𝑌(𝑡)=𝑀²2+𝑇𝐷⁰(𝑀)+𝐷₀(𝑀)^𝑡0𝑆^𝑚(𝑘)(𝑠)^𝑁0𝑑𝑠,0≤𝑡≤𝑇.(3.86)
Clearly 𝑌(𝑡𝑌)>0,0≤𝑆^𝑚(𝑘)(𝑡)≤𝑌(𝑡),0≤𝑡≤𝑇,_(𝑡)𝑌≤(𝐷0)₀=(𝑀𝑀)𝑌^𝑁0₂(𝑡),0≤𝑡≤𝑇,2+𝑇𝐷⁰(𝑀).(3.87)
Put 𝑍(𝑡)=𝑌^−𝑁0+1(𝑡), after integrating of (3.87) 𝑍(𝑡)≥𝑀²2+𝑇𝐷⁰≥𝑀(𝑀)^−𝑁0+1−𝑁₀−1𝐷₀(𝑀)𝑡₂2+𝑇𝐷⁰(𝑀)^−𝑁0+1−𝑁₀−1𝑇𝐷₀(𝑀),∀𝑡∈[0,𝑇].(3.88)
Then, by lim_𝑇→0+𝑀2²+𝑇𝐷⁰(𝑀)^−𝑁0+1−𝑁₀−1𝑇𝐷₀(𝑀)=𝑀²2^−𝑁0+1>𝑀^−2𝑁0+2,(3.89) we can always choose the constant 𝑇∈(0,𝑇^∗] such that 𝑀2²+𝑇𝐷⁰(𝑀)^−𝑁0+1−𝑁₀−1𝑇𝐷₀(𝑀)≥𝑀^−2𝑁0+2.(3.90)
Finally, it follows from (3.87), (3.88) and (3.90), that 0≤𝑆^𝑚(𝑘)(𝑡)≤𝑌(𝑡)=_𝑁0−11≤√𝑍(1𝑡)_𝑁0−1𝑀²/2+𝑇𝐷₀(𝑀)^−𝑁0+1−𝑁₀−1𝑇𝐷₀(𝑀)≤𝑀²,∀𝑡∈[0,𝑇].(3.91)
The proof of Lemma 3.4 is complete.
Remark 3.5. The function 𝑌(𝑡)=[(𝑀²/2+𝑇𝐷₀(𝑀))^1−𝑁0−(𝑁₀−1)𝐷₀(𝑀)𝑡]^{1/(1−𝑁0)},0≤𝑡≤𝑇, is the maximal solution of the following Volterra integral equation with non-decreasing kernel [20]. 𝑌(𝑡)=𝑀²2+𝑇𝐷⁰(𝑀)+𝐷₀(𝑀)^𝑡0𝑌^𝑁0(𝑠)𝑑𝑠,0≤𝑡≤𝑇.(3.92) By Lemma 3.4, we can take constant 𝑇^𝑚(𝑘)=𝑇 for all 𝑘 and 𝑚. Therefore, we have 𝑢^𝑚(𝑘)∈𝑊₁(𝑀,𝑇)∀𝑚,𝑘.(3.93) From (3.92), we can extract from {𝑢^𝑚(𝑘)} a subsequence {𝑢^{(𝑘𝑖)𝑚}} such that 𝑢𝑢̇^{(𝑘𝑖)𝑚}_{(𝑘𝑖)𝑚}⟶𝑢_𝑚in𝐿^∞0,𝑇;𝑉₂weak^∗,𝑢̈_{(𝑘𝑖)𝑚}⟶𝑢^𝑚in𝐿^∞0,𝑇;𝑉₁weak^∗,⟶𝑢𝑢^𝑚in𝐿²0,𝑇;𝑉₀weak,(3.94)_𝑚∈𝑊(𝑀,𝑇).(3.95)
We can easily check from (3.9), (3.10), (3.94) that 𝑢_𝑚 satisfies (3.6), (3.7) in 𝐿²(0,𝑇), weak. On the other hand, it follows from (3.6)₁ and 𝑢_𝑚∈𝑊(𝑀,𝑇) that 𝑢^𝑚=−𝑏_𝑚(𝑡)𝐴𝑢_𝑚+𝐹_𝑚∈𝐿^∞(0,𝑇;𝑉₀), hence 𝑢_𝑚∈𝑊₁(𝑀,𝑇) and the proof of Theorem 3.1 is complete.

The following result gives a convergence at a rate of order 𝑁 of {𝑢_𝑚} to a weak solution of (1.1).

First, we note that 𝑊₁(𝑇)={𝑣∈𝐿^∞(0,𝑇;𝑉₁)∶𝑣^∈𝐿^∞(0,𝑇;𝑉₀)} is a Banach space with respect to the norm (see [19]): ‖𝑣‖_𝑊1(𝑇)=‖𝑣‖_𝐿^∞_{(0,𝑇;𝑉1)}+‖‖𝑣^‖‖_{𝐿∞(0,𝑇;𝑉0)}.(3.96)

Then, we have the following theorem.

Theorem 3.6. Let (H₁)–(H₃) hold. Then, there exist constants 𝑀>0 and 𝑇>0 such that(i)the problem (1.1) has a unique weak solution 𝑢∈𝑊₁(𝑀,𝑇);(ii)the recurrent sequence {𝑢_𝑚} defined by (3.6), (3.7) converges at a rate of order 𝑁 to the solution 𝑢 strongly in the space 𝑊₁(𝑇) in the sense ‖‖𝑢_𝑚−𝑢‖‖_𝑊1(𝑇)≤𝐶‖‖𝑢_𝑚−1−𝑢‖‖^𝑁𝑊_1(𝑇),(3.97) for all 𝑚≥1, where 𝐶 is suitable constant. Furthermore, we have the estimation ‖‖𝑢_𝑚−𝑢‖‖_𝑊1(𝑇)≤𝐶_𝑇𝛽^𝑁𝑚,(3.98) for all 𝑚≥1, where 𝐶_𝑇 and 𝛽<1 are positive constants depending only on 𝑇.

Proof. (a) Existence of the solution. First, we will prove that {𝑢_𝑚} is a Cauchy sequence in 𝑊₁(𝑇). Let𝑣_𝑚=𝑢_𝑚+1−𝑢_𝑚. Then 𝑣_𝑚 satisfies the variational problem 𝑣^𝑚=(𝑡),𝐹𝑤+𝑏_𝑚+1(𝑡)𝑎𝑣_𝑚(𝑡),𝑤+𝑏_𝑚+1(𝑡)−𝑏_𝑚(𝑡)⟨𝐴𝑢_𝑚(𝑡),𝑤⟩_𝑚+1𝑣_𝑚(𝑡)−𝐹_𝑚(𝑡),𝑤,∀𝑤∈𝑉₁,(0)=𝑣^𝑚(0)=0,(3.99) with 𝑓𝑟,𝑡,𝑢_𝑚−𝑟,𝑡,𝑢_𝑚−1=^𝑁−1_𝑖=11𝑖!𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑣^𝑖𝑚−1+1𝑁!𝐷^𝑁3𝐹_𝑚+1𝑓𝑟,𝑡,𝜆_𝑚𝑣^𝑁𝑚−1,(𝑡)−𝐹_𝑚(𝑡)=^𝑁−1_𝑖=11𝑖!𝐷^𝑖3𝑓𝑟,𝑡,𝑢_𝑚−1𝑣^𝑖𝑚+1𝑁!𝐷^𝑁3𝜆_𝑚𝑓𝑟,𝑡,𝜆_𝑚𝑣^𝑁𝑚−1,𝑏=𝑢_𝑚−1+𝜃₁𝑣_𝑚−1,0<𝜃₁<1_𝑚+1,(𝑡)−𝑏_𝑚(𝑡)=𝐵‖‖∇𝑢_𝑚+1(𝑡)‖‖²⁰−𝐵‖‖∇𝑢^𝑚(𝑡)‖‖²⁰.(3.100)
Taking 𝑤=𝑣^𝑚 in (3.99) and integrating in 𝑡 we get 𝛾_𝑚(𝑡)=−^𝑡02𝑏^𝑚+1_𝑡0(𝑠)𝑎𝑣_𝑚(𝑠),𝑣_𝑚(𝑠)𝑑𝑠+2_𝑁−1𝐵‖‖∇𝑢^𝑚+1(𝑠)‖‖²⁰−𝐵‖‖∇𝑢^𝑚(𝑠)‖‖²⁰𝐴𝑢^𝑚(𝑠),𝑣^𝑚(𝑠)𝑑𝑠_𝑖=11𝑖!^𝑡0𝐷^𝑖3𝑓𝑟,𝑠,𝑢_𝑚−1𝑣^𝑖𝑚,𝑣^𝑚(𝑠)𝑑𝑠+1𝑁!^𝑡0𝐷^𝑁3=𝐽₁𝑓𝑟,𝑠,𝜆_𝑚𝑣^𝑁𝑚−1,𝑣^𝑚(𝑠)𝑑𝑠+⋯+𝐽₄,(3.101) where 𝛾_𝑚(𝑡)=‖‖𝑣^𝑚(𝑡)‖‖²⁰+𝑏_𝑚+1(𝑡)𝑎𝑣_𝑚(𝑡),𝑣_𝑚(𝑡)≥‖‖𝑣^𝑚(𝑡)‖‖²⁰+𝑏_∗𝐶₀‖‖𝑣_𝑚(𝑡)‖‖²¹≡𝐸_𝑚(𝑡).(3.102)
We estimate without difficulty the following integrals 𝐽₁,…,𝐽₄ in the right-hand side of (3.101) as follows.
The integral 𝐽₁=∫^𝑡0𝑏^𝑚+1(𝑠)𝑎(𝑣_𝑚(𝑠),𝑣_𝑚(𝑠))𝑑𝑠𝐽₁≤≤22𝐶𝐶₁₁𝑑₁𝑀²+𝑀^2𝛼^𝑡0‖‖𝑣_𝑚(𝑠)‖‖²¹𝑑𝑠𝑑₁𝑀²+𝑀^2𝛼1𝑏_∗𝐶₀^𝑡0𝐸_𝑚(𝑠)𝑑𝑠≡𝜉₁^𝑡0𝐸_𝑚(𝑠)𝑑𝑠.(3.103)The integral 𝐽₂=−2∫^𝑡0[𝐵(‖∇𝑢_𝑚+1(𝑠)‖²⁰)−𝐵(‖∇𝑢_𝑚(𝑠)‖²⁰)]⟨𝐴𝑢_𝑚(𝑠),𝑣^𝑚(𝑠)⟩𝑑𝑠𝐽₂≤≤44𝑑𝑑₁₁𝑀²+𝑀^2𝛼^𝑡0‖‖𝑣_𝑚(𝑠)‖‖₁‖‖𝑣^𝑚(𝑠)‖‖₀𝑑𝑠𝑀²+𝑀^2𝛼12√𝑏_∗𝐶₀^𝑡0𝐸_𝑚(𝑠)𝑑𝑠≡𝜉₂^𝑡0𝐸_𝑚(𝑠)𝑑𝑠.(3.104)The integral 𝐽₃=2∑^{𝑁−1𝑖=1}(1/𝑖!)∫^𝑡0⟨𝐷^𝑖3𝑓(𝑟,𝑠,𝑢_𝑚−1)𝑣^𝑖𝑚,𝑣^𝑚(𝑠)⟩𝑑𝑠.𝐽₃=2^𝑁−1_𝑖=11𝑖!^𝑡0√≤2_𝑁−1𝑟^−𝑖𝐷^𝑖3𝑓𝑟,𝑠,𝑢_𝑚−1√𝑟^𝑖𝑣^𝑖𝑚,𝑣^𝑚(𝑠)𝑑𝑠_𝑖=11𝑖!𝐾^∗𝑖(𝑀,𝑓)𝐾^𝑖2^𝑡0≤√‖‖𝑣_𝑚(𝑠)‖‖^𝑖11,||𝑣^𝑚(𝑠)||𝑑𝑠2^𝑁−1_𝑖=11𝑖!𝐾^∗𝑖(𝑀,𝑓)𝐾^𝑖2^𝑡0≤√‖‖𝑣_𝑚(𝑠)‖‖^𝑖1‖‖𝑣^𝑚(𝑠)‖‖₀𝑑𝑠2^𝑁−1_𝑖=11𝑖!𝐾^∗𝑖(𝑀,𝑓)𝐾^𝑖2𝑀^𝑖−1^𝑡0≤√‖‖𝑣_𝑚(𝑠)‖‖₁‖‖𝑣^𝑚(𝑠)‖‖₀𝑑𝑠2^𝑁−1_𝑖=11𝑖!𝐾^∗𝑖(𝑀,𝑓)𝐾^𝑖2𝑀^𝑖−112√𝑏_∗𝐶₀^𝑡0𝐸_𝑚(𝑠)𝑑𝑠≡𝜉₃^𝑡0𝐸_𝑚(𝑠)𝑑𝑠.(3.105)The integral 𝐽₄=(1/𝑁!)∫^𝑡0⟨𝐷^𝑁3𝑓(𝑟,𝑠,𝜆_𝑚)𝑣^𝑁𝑚−1,𝑣^𝑚(𝑠)⟩𝑑𝑠𝐽₄=1𝑁!^𝑡0√≤1𝑟^−𝑁𝐷^𝑁3𝑓𝑟,𝑠,𝜆_𝑚√𝑟^𝑁𝑣^𝑁𝑚−1,𝑣^𝑚(𝑠)𝑑s𝑁!𝐾^∗𝑁(𝑀,𝑓)𝐾^𝑁2^𝑡0≤1‖‖𝑣_𝑚−1(𝑠)‖‖^𝑁11,||𝑣^𝑚(𝑠)||𝑑𝑠𝑁!𝐾^∗𝑁(𝑀,𝑓)𝐾^𝑁21√2^𝑡0≤1‖‖𝑣_𝑚−1(𝑠)‖‖^𝑁1‖‖𝑣^𝑚(𝑠)‖‖₀𝑑𝑠𝑁!𝐾^∗𝑁(𝑀,𝑓)𝐾^𝑁212√22‖‖𝑣^𝑚−1‖‖^𝑁𝑊_1(𝑇)^𝑡0≤1‖‖𝑣^𝑚(𝑠)‖‖₀𝑑𝑠𝑁!𝐾^∗𝑁(𝑀,𝑓)𝐾^𝑁212√2𝑇‖‖𝑣^𝑚−1=𝜉₄‖‖^𝑊2𝑁_1(𝑇)+^𝑡0‖‖𝑣^𝑚(𝑠)‖‖²⁰𝑑𝑠𝑇‖‖𝑣^𝑚−1‖‖^𝑊2𝑁_1(𝑇)+^𝑡0𝐸_𝑚(𝑠)𝑑𝑠.(3.106)
Combining (3.101), (3.103)–(3.106), we then obtain𝐸_𝑚(𝑡)≤≡𝑇𝑇𝜉𝜉₄₄‖‖𝑣_𝑚−1‖‖^𝑊2𝑁_1(𝑇)+𝜉₁+𝜉₂+𝜉₃+𝜉₄^𝑡0𝐸_𝑚(𝑠)𝑑𝑠‖‖𝑣_𝑚−1‖‖^𝑊2𝑁_1(𝑇)+2𝜁^𝑀(1)^𝑡0𝐸_𝑚(𝑠)𝑑𝑠.(3.107)
By using Gronwall's lemma, we obtain from (3.107), that‖‖𝑣_𝑚‖‖_𝑊1(𝑇)≤𝜇_𝑇‖‖𝑣_𝑚−1‖‖^𝑁𝑊_1(𝑇),(3.108) where 𝜇_𝑇 is the constant given by 𝜇_𝑇=1+1√𝑏_∗𝐶₀√𝑇𝜉₄exp𝑇𝜁^𝑀(1).(3.109)
Hence, we obtain from (3.108) that‖‖𝑢_𝑚−𝑢_𝑚+𝑝‖‖_𝑊1(𝑇)≤1−𝑘_𝑇⁻¹𝜇_𝑇^{−1/(𝑁−1)}𝑘_𝑇^𝑁𝑚,(3.110) for all 𝑚 and 𝑝. We take 𝑇>0 small enough, such that 𝑘_𝑇=(𝜇_𝑇)^1/(𝑁−1)𝑀<1. It follows that {𝑢_𝑚} is a Cauchy sequence in 𝑊₁(𝑇). Then there exists 𝑢∈𝑊₁(𝑇) such that 𝑢_𝑚→𝑢 strongly in 𝑊₁(𝑇). By the same argument used in the proof of Theorem 3.1, 𝑢∈𝑊₁(𝑀,𝑇) is a unique weak solution of the problem (1.1). Passing to the limit as 𝑝→+∞ for 𝑚 fixed, we obtain the estimate (3.98) from (3.110) and Theorem 3.6 follows.

Acknowledgments

The authors wish to express their sincere thanks to the referees for the suggestions and valuable comments. The authors are also extremely grateful for the support given by Vietnam's National Foundation for Science and Technology Development (NAFOSTED) under Project 101.01-2010.15.