A Justification of Two-Dimensional Nonlinear Viscoelastic Shells Model

Abstract

By applying formal asymptotic analysis and Laplace transformation, we obtain two-dimensional nonlinear viscoelastic shells model satisfied by the leading term of asymptotic expansion of the solution to the three-dimensional equations.

1. Introduction

In the case of pure nonlinear elasticity, Ciarlet and his collaborators have studied membrane shells, flexural shell and Koiter shell (see [1] and the references therein). The linear viscoelasticity was studied in [2–5], and Li [6–8] studied the global existence and uniqueness of weak solution, uniform rates of decay, and limit behavior of the solution to nonlinear viscoelastic Marguerre-von Kármán shallow shells. Xiao studied the time-dependent nonlinear elastic shells by the method of asymptotic analysis (see [9]).

Motivated by the above work, we deal with nonlinear viscoelastic shells and give the identification of two-dimensional variation problem satisfied by the leading term of of asymptotic expansion of the solution to the three-dimensional equations. The main contributions of this paper are the following: (a) the problem considered in this paper is nonlinear viscoelastic shells, to our knowledge this model has not been considered; (b) applying Laplace transformation, we overcome the difficulties caused by the integral term in the model; (c) the calculation and derivation are precise.

This paper is organized as follows. Section 2 begins with some preliminaries and then gives the main result. In Section 3, we give the proof of the main theorem.

2. Preliminaries and Main Results

We use the following conventions and notations throughout this work: Greek indices and exponents (except 𝜀) belong to the set {1,2}, Latin indices and exponents (except when otherwise indicated, as, e.g., when they are used to index sequences) belong to the set {1,2,3}, and the summation convention with respect to the repeated indices and exponents is systematically used. The sign ∶= indicates that the right-hand side defines the left-hand side.

Let 𝜔⊂𝐑² be a bounded connected open set with a Lipschitz boundary 𝛾, let 𝐲=(𝑦_𝛼) denote a generic point in the set 𝜔, and let 𝜕_𝛼∶=𝜕/𝜕𝑦_𝛼. Let 𝜽∶𝜔→𝐑³ be an injective mapping of 𝐶³ such that the two vectors 𝐚_𝛼(𝐲)∶=𝜕_𝛼𝜽(𝐲) are linear independent at all points 𝐲∈𝜔. They form the covariant basis of the tangent plane to the surface 𝐒=𝜽(𝜔) at the point 𝜽(𝐲); the two vectors 𝐚^𝛼(𝐲) of the same tangent plane defined by the relations 𝐚^𝛼(𝐲)⋅𝐚_𝛽(𝐲)∶=𝛿^𝛼_𝛽 constitute its contravariant basis. We also define the unit vector 𝐚₃(𝐲)=𝐚³(𝐲)=𝐚₁(𝐲)×𝐚₂(𝐲)/|𝐚₁(𝐲)×𝐚₂(𝐲)| which is normal to the 𝐒 at the point 𝜽(𝐲).

One then defines the first fundamental form, also known as metric tensor (𝑎_𝛼𝛽) or (𝑎^𝛼𝛽), the second fundamental form, also known as the curvature tensor (𝑏_𝛼𝛽) or (𝑏^𝛽_𝛼), and the Christoffel symbols Γ^𝜎_𝛼𝛽 of the surface 𝐒 by setting (whenever no confusion should arise, we henceforth drop the explicit dependence on the variable 𝐲∈𝜔) 𝑎_𝛼𝛽∶=𝐚_𝛼⋅𝐚_𝛽,𝑎^𝛼𝛽∶=𝐚^𝛼⋅𝐚^𝛽,𝑏_𝛼𝛽∶=𝐚³⋅𝜕_𝛽𝐚_𝛼,𝑏^𝛽_𝛼∶=𝑎^𝛽𝜎𝑏_𝜎𝛼,Γ^𝜎_𝛼𝛽∶=𝐚^𝜎⋅𝜕_𝛽𝐚_𝛼. (2.1) Note the symmetries 𝑎_𝛼𝛽=𝑎_𝛽𝛼,𝑏_𝛼𝛽=𝑏_𝛽𝛼, and Γ^𝜎_𝛼𝛽=Γ^𝜎_𝛽𝛼. The area element along 𝐒 is √𝑎𝑑𝑦, where 𝑎∶=det(𝑎_𝛼𝛽). All the functions defined above are at least continuous over the set 𝜔. In particular, there exists a constant 𝑎₀>0 such that 𝑎(𝐲)≥𝑎₀,forall𝐲∈𝜔.

In addition, let the covariant derivatives 𝑏^𝜎_𝛽|_𝛼 and the covariant components 𝑐_𝛼𝛽 of the third form of the surface 𝐒 be defined by 𝑏^𝜎_𝛽|_𝛼∶=𝜕_𝛼𝑏^𝜎_𝛽+Γ^𝜎_𝛼𝜏𝑏^𝜏_𝛽−Γ^𝜏_𝛼𝛽𝑏^𝜎_𝜏,𝑐_𝛼𝛽∶=𝑏^𝜎_𝛼𝑏_𝜎𝛽. (2.2)

For each 𝜀>0, we consider a shell with thickness 2𝜀 and middle surface 𝐒, whose lamé relaxation modules 𝜆(𝑡)≥0 and 𝜇(𝑡)>0(𝑡≥0) are independent of 𝜀. We define the sets Ω^𝜀∶=𝜔×(−𝜀,+𝜀),Γ^𝜀₊=𝜔×{𝜀},Γ^𝜀₋=𝜔×{−𝜀},Γ^𝜀₀=𝛾₀×[−𝜀,𝜀], (2.3) where 𝛾₀⊂𝛾 and 𝛾₀≠∅. Note that Γ^𝜀₊∪Γ^𝜀₋∪Γ^𝜀₀ constitutes a partition of the boundary of the set Ω^𝜀. Let 𝐱^𝜀=(𝑥^𝜀_𝑖) denote a generic point in the set Ω^𝜀, and let 𝜕^𝜀_𝑖∶=𝜕/𝜕𝑥^𝜀_𝑖; hence 𝑥^𝜀_𝛼=𝑦_𝛼 and 𝜕^𝜀_𝛼=𝜕_𝛼.

We then define a mapping Θ∶Ω^𝜀→𝐑³ by 𝚯𝐱^𝜀∶=𝜽(𝐲)+𝑥^𝜀₃𝐚₃(𝐲),∀𝐱^𝜀=𝐲,𝑥^𝜀₃∈Ω^𝜀, (2.4) then there exists 𝜀₀>0 such that for all 0<𝜀≤𝜀₀ the mapping Θ∶Ω^𝜀→𝐑³ is an injective mapping and the three vectors 𝐠^𝜀_𝑖(𝐱^𝜀)∶=𝜕^𝜀_𝑖Θ(𝐱^𝜀)(𝜕^𝜀_𝛼=𝜕/𝜕𝑥_𝛼,𝜕^𝜀₃=𝜕/𝜕𝑥^𝜀₃)are linear independent for each 𝐱^𝜀∈Ω^𝜀. The injectivity of the mapping Θ∶Ω^𝜀→𝐑³ ensures that the physical problem described below is meaningful.

The three vectors 𝐠^𝜀_𝑖(𝐱^𝜀) form the covariant basis at the point Θ(𝐱^𝜀), and the three vectors 𝐠^𝑖,𝜀(𝐱^𝜀) defined by 𝐠^𝑗,𝜀(𝐱^𝜀)⋅𝐠^𝜀_𝑖(𝐱^𝜀)=𝛿^𝑗_𝑖 form the contravariant. We define the metric tensor (𝑔^𝜀_𝑖𝑗) or (𝑔^{𝑖𝑗,𝜀}) and the Christoffel symbols of the manifold Θ(Ω^𝜀) by setting (we omit the explicit dependence on 𝐱^𝜀) 𝑔^𝜀_𝑖𝑗=𝐠^𝜀_𝑖⋅𝐠^𝜀_𝑗,𝑔^{𝑖𝑗,𝜀}∶=𝐠^𝑖,𝜀⋅𝐠^𝑗,𝜀,Γ^𝑘,𝜀_𝑖𝑗∶=𝐠^𝑘,𝜀⋅𝜕^𝜀_𝑖𝐠^𝜀_𝑗. (2.5) Note the symmetries 𝑔^𝜀_𝑖𝑗=𝑔^𝜀_𝑗𝑖,𝑔^{𝑖𝑗,𝜀}=𝑔^{𝑗𝑖,𝜀},Γ^𝑘,𝜀_𝑖𝑗=Γ^𝑘,𝜀_𝑗𝑖. (2.6) The volume element in the set Θ(Ω^𝜀) is 𝑔^𝜀𝑑𝑥^𝜀, where 𝑔^𝜀∶=det(𝑔^𝜀_𝑖𝑗).

For each 0<𝜀≤𝜀₀, the set Ω^𝜀∶=Θ(Ω^𝜀) is the reference configuration of a viscoelastic shell with middle surface 𝐒=𝜽(𝜔) and thickness 2𝜀. We assume that the material constituting the shell is homogeneous isotropic and Θ(Ω^𝜀) is of a nature state, so that the material is characterized by its two lamé relaxation modules 𝜆(𝑡)≥0 and 𝜇(𝑡)>0(𝑡≥0). Under the action of forces, the shell undergoes a displacement field.

Let 𝐮^𝜀(𝑡)=𝑢^𝜀_𝑖(𝑡)𝐠^𝑖,𝜀 in terms of curvilinear coordinates 𝐱^𝜀 of the reference configuration Θ(Ω^𝜀). Then, the covariant displacement field 𝐮^𝜀(𝑡)=(𝑢^𝜀_𝑖)(𝑡) satisfies the following three-dimensional equations (c.f. [1, 10]): 𝐮^𝜀(𝑡)∈𝐋^∞_𝑐−∞,𝑇;𝐖Ω^𝜀with𝐖Ω^𝜀∶=𝐯^𝜀∈𝐖^1,4Ω^𝜀,𝐯^𝜀=𝟎onΓ^𝜀₀,_Ω𝜀𝑢^𝜀_𝑖𝑡𝑡(𝑡)𝑣^𝜀_𝑖𝑔^{𝑖𝑗,𝜀}𝑔^𝜀𝑑𝑥^𝜀+_Ω𝜀𝐴^{𝑖𝑗𝑘𝑙,𝜀}(0)𝐸^𝜀_𝑘‖𝑙𝐮^𝜀(𝑡)𝐹^𝜀_𝑖‖𝑗𝐮^𝜀(𝑡),𝐯^𝜀𝑔^𝜀𝑑𝑥^𝜀+_Ω𝜀^𝑡_−∞𝐴^{′𝑖𝑗𝑘𝑙,𝜀}(𝑡−𝜏)𝐸^𝜀_𝑘‖𝑙𝐮^𝜀(𝜏)𝐹^𝜀_𝑖‖𝑗𝐮^𝜀(𝜏),𝐯^𝜀𝑔^𝜀𝑑𝜏𝑑𝑥^𝜀=_Ω𝜀𝑓^𝑖,𝜀(𝑡)𝑣^𝜀_𝑖𝑔^𝜀𝑑𝑥^𝜀+_{Γ𝜀+∪Γ𝜀−}ℎ^𝑖,𝜀(𝑡)𝑣^𝜀_𝑖𝑔^𝜀𝑑Γ^𝜀,∀𝐯^𝜀∈𝐖Ω^𝜀, (2.7) where the symbol 𝐿^∞_𝑐 denotes the subspace of 𝐿^∞>0 such that there exists a constant 𝑇 such that the functions vanish as 𝑠<−𝑇. And, 𝐴^{𝑖𝑗𝑘𝑙,𝜀}(𝑡)∶=𝜆(𝑡)𝑔^{𝑖𝑗,𝜀}𝑔^{𝑘𝑙,𝜀}+𝜇(𝑡)𝑔^{𝑖𝑘,𝜀}𝑔^{𝑗𝑙,𝜀}+𝑔^{𝑖𝑙,𝜀}𝑔^{𝑗𝑘,𝜀} (2.8) designate the contravariant components of the three-dimensional elasticity tensor, 𝐸^𝜀_𝑖‖𝑗𝑣^𝜀1∶=2𝑣^𝜀_𝑖‖𝑗+𝑣^𝜀_𝑗‖𝑖+𝑔^{𝑚𝑛,𝜀}𝑣^𝜀_𝑚‖𝑖𝑣^𝜀_𝑛‖𝑗,where𝑣^𝜀_𝑖‖𝑗∶=𝜕^𝜀_𝑗𝑣^𝜀_𝑖−Γ^𝑝,𝜀_𝑖𝑗𝑣^𝜀_𝑝, (2.9) designate the strains in the curvilinear coordinates associated with an arbitrary displacement field 𝑣^𝜀_𝑖𝑔^𝑖,𝜀 of the manifold Θ(Ω), 𝐹^𝜀_𝑖‖𝑗𝑢^𝜀(𝑡),𝑣^𝜀1∶=2𝑣^𝜀_𝑖‖𝑗+𝑣^𝜀_𝑗‖𝑖+𝑔^{𝑚𝑛,𝜀}𝑢^𝜀_𝑚‖𝑖(𝑡)𝑣^𝜀_𝑛‖𝑗+𝑢^𝜀_𝑛‖𝑗(𝑡)𝑣^𝜀_𝑚‖𝑖, (2.10) and, finally, 𝑓^𝑖,𝜀∈𝐿^∞(0,𝑇;𝐿²(Ω^𝜀)) and ℎ^𝑖,𝜀∈𝐿^∞(0,𝑇;(Γ^𝜀₊∪Γ^𝜀₋)) denote the contravariant components of the applied body and surface force densities, respectively, applied to the interior Θ(Ω^𝜀) of the shell and to its “uper” and “lower” faces Θ(Γ^𝜀₊) and Θ(Γ^𝜀₋), and designate the area element along 𝜕Ω^𝜀. We thus assume that there are no surface forces applied to the portion Θ((𝛾−𝛾₀)×[−𝜀,𝜀]) of the lateral face of the shell.

We record in passing the symmetries 𝐴^{𝑖𝑗𝑘𝑙,𝜀}=𝐴^{𝑗𝑖𝑘𝑙,𝜀}=𝐴^{𝑘𝑙𝑖𝑗,𝜀} (2.11) and the relation Γ^3,𝜀_𝛼3=Γ^𝑝,𝜀₃₃=0,𝐴^{𝛼𝛽𝜎3,𝜀}=𝐴^𝛼333,𝜀=0inΩ^𝜀. (2.12)

Our final objective consists in showing, by means of the method of formal asymptotic expansions that, if the data are of an appropriate order with respect to 𝜀 as 𝜀→0, the above three-dimensional problems are “asymptotically equivalent” to a “two-dimensional problem posed over the middle surface of the shell.” This means that the new unknown should be 𝜻^𝜀(𝑡)=(𝜁^𝜀_𝑖(𝑡)), where 𝜁^𝜀_𝑖(𝑡) are the covariant components of the displacement 𝜁^𝜀_𝑖(𝑡)𝐚_𝑖(𝐲)∶𝜔→𝐑³ of the middle surface 𝑆=𝜽(𝜔). In other words, 𝜁^𝜀_𝑖(𝑡,𝑦)𝐚_𝑖(𝐲) is the displacement of the point 𝜽(𝑦)∈𝑆.

“Asymptotic analysis” means that our objective is to study the behavior of the displacement field 𝑢^𝜀_𝑖(𝑡)𝐠^𝑖,𝜀∶Ω^𝜀→𝐑³ as 𝜀→0, an endeavour that will be a behavior as 𝜀→0 of the covariant components 𝑢^𝜀_𝑖(𝑡)∶Ω^𝜀→𝐑 of the displacement field, that is, the behavior of the unknown 𝐮^𝜀(𝑡)=(𝑢^𝜀_𝑖(𝑡))∶Ω^𝜀→𝐑³ of the three-dimensional shell problem.

Since these fields are defined on sets Ω^𝜀 that themselves vary with 𝜀, our first task naturally consists in transforming the three-dimensional problems into problems posed over a set that does not depend on 𝜀.

Furthermore, we transform problem (2.7) into an equivalent problem independent of 𝜀, posed over the domain.

Let Ω∶=𝜔×(−1,+1),Γ₀=𝛾₀×[−1,1],Γ₊∶=𝜔×{+1}, and Γ₋∶=𝜔×{−1}, and let 𝐱=(𝑥_𝑖) denote a generic point in Ω. With each point 𝐱∈Ω, we associate the point 𝐱^𝜀 through the bijection 𝜋^𝜀∶𝐱=(𝑥₁,𝑥₂,𝑥₃)∈Ω→𝐱^𝜀=(𝑥^𝜀_𝑖)=(𝑥₁,𝑥₂,𝜀𝑥₃)∈Ω^𝜀; we thus have 𝜕^𝜀_𝛼=𝜕_𝛼 and 𝜕^𝜀₃=(1/𝜀)𝜕₃. Let 𝐮^𝜀=(𝑢^𝜀_𝑖),Γ^𝜀_𝑖𝑗,𝑔^𝜀,𝐴^{𝑖𝑗𝑘𝑙,𝜀}∶Ω^𝜀→𝐑 and the vector fields 𝐯^𝜀=(𝑣^𝜀_𝑖) appearing in the three-dimensional problem (2.7) be associated with the functions Γ^𝑝_𝑖𝑗(𝜀),𝑔(𝜀),𝐴^{𝑖𝑗𝑘𝑙}(𝜀)∶Ω→𝐑 and the scaled vector fields 𝐯=(𝑣_𝑖) defined by 𝑢_𝑖(𝜀)(𝑡,𝐱)=𝑢^𝜀_𝑖𝑡,𝐱^𝜀,𝑣_𝑖(𝐱)=𝑣^𝜀_𝑖𝐱^𝜀∀𝐱^𝜀∈Ω^𝜀,Γ^𝑝_𝑖𝑗(𝜀)(𝐱)=Γ^𝑝,𝜀_𝑖𝑗𝐱^𝜀,𝑔(𝜀)(𝐱)=𝑔^𝜀𝐱^𝜀,𝐴^{𝑖𝑗𝑘𝑙}(𝜀)(𝐱)=𝐴^{𝑖𝑗𝑘𝑙,𝜀}𝐱^𝜀∀𝐱^𝜀∈Ω^𝜀. (2.13) Functions 𝑓^𝑖(𝜀)(𝑡)∶Ω→𝐑 and ℎ^𝑖(𝑡)(𝜀)∶Γ₊∪Γ₋→𝐑 are defined by setting 𝑓^𝑖(𝜀)(𝑡,𝐱)=𝑓^𝑖,𝜀𝑡,𝐱^𝜀∀𝐱^𝜀∈Ω^𝜀,ℎ^𝑖(𝜀)(𝑡,𝐱)=ℎ^𝑖,𝜀𝑡,𝐱^𝜀∀𝐱^𝜀∈Γ^𝜀₊∪Γ^𝜀₋. (2.14) Then the scaled unknown 𝐮(𝜀)(𝑡) defined above satisfies (c.f. [1]) 𝐮(𝜀)(𝑡)∈𝐋^∞_𝑐(−∞,𝑇;𝐖(Ω))with𝐖(Ω)∶=𝐯∈𝐖^1,4(Ω),𝐯=𝟎onΓ₀,_Ω𝑢_𝑖𝑡𝑡(𝜀)(𝑡)𝑣_𝑖𝑔(𝜀)𝑑𝑥+_Ω𝐴^{𝑖𝑗𝑘𝑙}(𝜀)(0)𝐸_𝑘‖𝑙(𝜀;𝐮(𝜀)(𝑡))𝐹_𝑖‖𝑗(𝜀;𝐮(𝜀)(𝑡),𝐯)𝑔(𝜀)𝑑𝑥+_Ω^𝑡_−∞𝐴^{′𝑖𝑗𝑘𝑙}(𝜀)(𝑡−𝜏)𝐸_𝑘‖𝑙(𝜀;𝐮(𝜀)(𝜏))𝐹_𝑖‖𝑗(𝜀;𝐮(𝜀)(𝜏),𝐯)𝑔(𝜀)=𝑑𝜏𝑑𝑥_Ω𝑓^𝑖(𝜀)(𝑡)𝑣_𝑖𝑔(𝜀)1𝑑𝑥+𝜀_Γ+∪Γ−ℎ^𝑖(𝜀)(𝑡)𝑣_𝑖𝑔(𝜀)𝑑Γ,∀𝐯∈𝐖(Ω), (2.15) where 𝐸_𝑖‖𝑗((𝜀𝑡1𝜀;𝐮)())∶=2𝑢_𝑖‖𝑗(𝜀)(𝑡)+𝑢_𝑗‖𝑖(𝜀)(𝑡)+𝑔^𝑚𝑛(𝜀)𝑢_𝑚‖𝑖(𝜀)(𝑡)𝑢_𝑛‖𝑗(𝜀𝑡,𝐹)()_𝑖‖𝑗((𝜀𝑡))1𝜀;𝐮)(,𝐯∶=2𝑣_𝑖‖𝑗(𝜀)+𝑣_𝑗‖𝑖(𝜀)+𝑔^𝑚𝑛(𝜀)𝑢_𝑚‖𝑖(𝜀)(𝑡)𝑣_𝑛‖𝑗(𝜀)+𝑢_𝑛‖𝑗(𝜀)(𝑡)𝑣_𝑚‖𝑖(𝜀,𝑢_𝛽‖𝛼)(𝜀)(𝑡)∶=𝜕_𝛼𝑢_𝛽(𝜀)(𝑡)−Γ^𝑝_𝛼𝛽(𝜀)𝑢_𝑝(𝜀)(𝑡),𝑣_𝛽‖𝛼(𝜀)=𝜕_𝛼𝑣_𝛽−Γ^𝑝_𝑎𝛽(𝜀)𝑣_𝑝,𝑢_3‖𝛼(𝜀)(𝑡)∶=𝜕_𝛼𝑢₃(𝜀)(𝑡)−Γ^𝜎_𝛼3(𝜀)(𝑡)𝑢_𝜎(𝜀)(𝑡),𝑣_3‖𝛼(𝜀)∶=𝜕_𝛼𝑣₃−Γ^𝜎_𝛼3(𝜀)𝑣_𝜎,𝑢_𝛼‖3(𝜀𝑡)1)(∶=𝜀𝜕₃𝑢_𝛼(𝜀)(𝑡)−Γ^𝜎_𝛼3(𝜀)𝑢_𝜎(𝜀)(𝑡),𝑣_𝛼‖3(𝜀)1∶=𝜀𝜕₃𝑣_𝛼−Γ^𝜎_𝛼3(𝜀)𝑣_𝜎,𝑢_3‖3(𝜀𝑡)1)(∶=𝜀𝜕₃𝑢₃(𝜀)(𝑡),𝑣_3‖3(𝜀)1∶=𝜀𝜕₃𝑣₃. (2.16)

The functions 𝐴^{𝑖𝑗𝑘𝑙}(𝜀) are called the contravariant components of the scaled three-dimensional elasticity tensor of the shell. The functions 𝐸_𝑖‖𝑗(𝜀;𝐮(𝜀)(𝑡)) are called the scaled strains in the curvilinear coordinates because they satisfy 𝐸_𝑖‖𝑗(𝜀;𝐮(𝜀)(𝑡))(𝑥)=𝐸^𝜀_𝑖‖𝑗𝐮^𝜀(𝑡)𝑥^𝜀∀𝑥^𝜀∈Ω^𝜀. (2.17) Note that the above definitions likewise imply that 𝐹_𝑖‖𝑗(𝜀;𝐮(𝜀)(𝑡),𝐯)(𝑥)=𝐹^𝜀_𝑖‖𝑗𝐮^𝜀(𝑡),𝐯^𝜀𝑥^𝜀,𝑢_𝑖‖𝑗(𝜀)(𝑡)(𝑥)=𝑢^𝜀_𝑖‖𝑗(𝑡)𝑥^𝜀,𝑣_𝑖‖𝑗(𝜀)(𝑥)=𝑣^𝜀_𝑖‖𝑗𝑥^𝜀∀𝑥^𝜀∈Ω^𝜀. (2.18)

For notational brevity, the point 𝐱 of some functions is suppressed where no confusion can arise.

The following two requirements constantly guide the procedures of the formal asymptotic analysis. The first requirement asserts that no restriction should be imposed on the applied forces entering the right-hand side of the equations used for determining the leading term. The second requirement asserts that, by retaining only the linear terms in any relation satisfied by terms of arbitrary order in the formal asymptotic expansion of the scaled unknown 𝐮(𝜀)(𝑡), a relation of linear theory should be recovered. For brevity, we will call it “linearization trick” (see [1]).

Theorem 2.1. Assume that the scaled unknown 𝐮(𝜀) satisfying problem (2.15) admits a formal asymptotic expansion of the form 𝐮(𝜀)(𝑡)=𝐮⁰(𝑡)+𝜀𝐮¹(𝑡)+𝜀²𝐮²(𝑡)+⋯ (2.19) with 𝐮⁰(𝑡)∈𝐿^∞(0,𝑇;𝐖(Ω)) and 𝐮¹(𝑡),𝐮²(𝑡)∈𝐿^∞(0,𝑇;𝐖^1,4(Ω)). Then in order that no restriction be put on the applied forces and that the linearization be satisfied, the components of the applied forces must be of the form 𝑓^𝑖,𝜀𝑡,𝑥^𝜀=𝑓^𝑖,0(𝑡,𝑥),𝑥^𝜀∈Ω^𝜀,ℎ^𝑖,𝜀𝑡,𝑥^𝜀=𝜀ℎ^𝑖,1(𝑡,𝑥),𝑥^𝜀∈Γ^𝜀₊∪Γ^𝜀₋, (2.20) where the functions 𝑓^𝑖,0∈𝐿^∞(0,𝑇;𝐿²(Ω)) and ℎ^𝑖,1∈Ł^∞(0,𝑇;𝐿²(Γ₊∪Γ₋)) are independent of 𝜀.
This being the case, the leading term 𝐮⁰(𝑡) is independent of the transverse variable 𝑥₃ and 𝜻⁰(𝑡)=(1/2)∫¹₋₁𝐮⁰(𝑡)𝑑𝑥₃ satisfies the following two-dimensional variation problem: 𝜻⁰(𝑡)∈𝐿^∞_𝑐((−∞,𝑇];𝐖(𝜔))with𝐖(𝜔)∶=𝜼∈𝐖^1,4(𝜔);𝜼=0𝑜𝑛𝛾₀,_𝜔𝜁⁰_𝑖𝑡𝑡(𝑡)𝜂_𝑗𝑎^𝑖𝑗√𝑎𝑑𝑦+_𝜔𝑎^{𝛼𝛽𝜎𝜏}(0)𝐸⁰_𝜎‖𝜏(𝑡)𝐹⁰_𝛼‖𝛽𝑡,𝜂√+𝑎𝑑𝑦_{𝜔𝑡−∞}𝑎^{′𝛼𝛽𝜎𝜏}(𝑡−𝜏)𝐸⁰_𝜎‖𝜏(𝜏)𝐹⁰_𝛼‖𝛽𝜏,𝜂√=𝑎𝑑𝑦𝑑𝜏_𝜔𝑝^𝑖,0(𝑡)𝜂_𝑖√𝑎𝑑𝑦,∀𝜼∈𝐖(𝜔),𝑎.𝑒.−∞<𝑡≤𝑇, (2.21) where (recall that 𝑎^𝑚𝑛=𝐚^𝑚⋅𝐚^𝑛): 𝐸⁰_𝛼‖𝛽(𝑡)1∶=2𝜁⁰_𝛼‖𝛽(𝑡)+𝜁⁰_𝛽‖𝛼(𝑡)+𝑎^𝑚𝑛𝜁⁰_𝑚‖𝛼(𝑡)𝜁⁰_𝑛‖𝛽(𝑡,𝐹_0𝛼‖𝛽)()1𝑡,𝜼∶=2𝜂_𝛼‖𝛽+𝜂_𝛽‖𝛼+𝑎^𝑚𝑛𝜁⁰_𝑚‖𝛼(𝑡)𝜂_𝑛‖𝛽+𝜁⁰_𝑛‖𝛽(𝑡)𝜂_𝑚‖𝛼,𝜂_𝛼‖𝛽=𝜕_𝛽𝜂_𝛼−Γ^𝜎_𝛼𝛽𝜂_𝜎−𝑏_𝛼𝛽𝜂₃,𝜂_3‖𝛽∶=𝜕_𝛽𝜂₃+𝑏^𝜎_𝛽𝜂_𝜎,𝑎^{𝛼𝛽𝜎𝜏}(𝑡)∶=𝑎(𝑡)𝑎^𝛼𝛽𝑎^𝜎𝜏+𝜇(𝑡)𝑎^𝛼𝜎𝑎^𝛽𝜏+𝑎^𝛼𝜏𝑎^𝛽𝜎,𝑎(𝑡)∶=𝐿⁻¹2𝜆(𝑠)𝜇(𝑠)𝜆(𝑠)(𝑠)(0)=(0)𝜇(0)+2𝜇with𝑎2𝜆𝜆(0)(0),𝑝+2𝜇_𝑖,0(𝑡)1∶=2⁺¹₋₁𝑓^𝑖,0(𝑡)𝑑𝑥₃+ℎ^𝑖,1(⋅,1)+ℎ^𝑖,1(⋅,−1), (2.22)𝜆(𝑠),𝜇(𝑠) denote Laplace transformation of 𝜆(𝑡),𝜇(𝑡), respectively, and 𝐿⁻¹ denotes the inverse Laplace transformation.

Lemma 2.2. For small 𝜀>0,it is not difficult to verify the following relations: 𝑔^𝑖𝑗(𝜀)=𝑎^𝑖𝑗+𝜀𝑥₃𝑔^𝑖𝑗,1+𝑂𝜀², (2.23) where 𝑎^𝑖𝑗∶=𝐚_𝑖⋅𝐚_𝑗,𝑔^𝛼𝛽,1∶=2𝑎^𝛼𝜎𝑏^𝛽_𝜎,𝑔^𝑖3,1Γ=0,_𝑝𝑖𝑗(𝜀)=Γ^𝑝,0_𝑖𝑗+𝜀𝑥₃Γ^𝑝,1_𝑖𝑗+𝑂𝜀², (2.24) where Γ^𝜎,0_𝛼𝛽∶=Γ^𝜎_𝛼𝛽,Γ^3,0_𝛼𝛽∶=𝑏_𝛼𝛽,Γ^𝛽,0_𝛼3∶=−𝑏^𝛽_𝛼,Γ^3,0_𝛼3=Γ^𝑝,0₃₃Γ∶=0,_{𝜎,1𝛼𝛽}∶=−𝑏^𝜎_𝛽|_𝛼,Γ^3,1_𝛼𝛽∶=−𝑏^𝜎_𝛼𝑏_𝜎𝛽,Γ^𝜎,1_𝛼3∶=−𝑏^𝜏_𝛼𝑏^𝜎_𝜏,Γ^3,1_𝛼3=Γ^𝑝,1₃₃𝐴∶=0,_{𝑖𝑗𝑘𝑙}(𝜀)(𝑡)𝑔(𝜀)=𝐴^{𝑖𝑗𝑘𝑙}(𝑡)√𝑎+𝜀𝐵^{𝑖𝑗𝑘𝑙,1}(𝑡)+𝜀²𝐵^{𝑖𝑗𝑘𝑙,2}(𝑡)+𝑜𝜀², (2.25) where 𝐴^{𝛼𝛽𝜎𝜏}(𝑡)∶=𝜆(𝑡)𝑎^𝛼𝛽𝑎^𝜎𝜏+𝜇(𝑡)𝑎^𝛼𝜎𝑎^𝛽𝜏+𝑎^𝛼𝜏𝑎^𝛽𝜎,𝐴^𝛼𝛽33(𝑡)∶=𝜆(𝑡)𝑎^𝛼𝛽,𝐴^𝛼3𝜎3(𝑡)∶=𝜇(𝑡)𝑎^𝛼𝜎,𝐴³³³³(𝑡)∶=𝜆(𝑡)+2𝜇(𝑡),𝐴^{𝛼𝛽𝜎3}(𝑡)=𝐴^𝛼333(𝑡)∶=0. (2.26)

Lemma 2.3 (see [1]). Let 𝜔 be a domain in 𝐑², and let 𝜽∈𝐶²(𝜔;𝐑³) be an injective mapping such that the two vectors 𝐚_𝛼 are linear independent at all points of 𝜔. The derivatives of the vectors of the covariant and contravariant basis are given by the formulas of Gauss 𝜕_𝛼𝐚_𝛽=Γ^𝜎_𝛼𝛽𝐚_𝜎+𝑏_𝛼𝛽𝐚₃,𝜕_𝛼𝐚^𝛽=−Γ^𝛽_𝛼𝜎𝐚^𝜎+𝑏^𝛽_𝛼𝐚³ (2.27) and Weingarten 𝜕_𝛼𝐚₃=𝜕_𝛼𝐚³=−𝑏_𝛼𝛽𝐚^𝛽=−𝑏^𝜎_𝛼𝐚_𝜎. (2.28)

Lemma 2.4. Let 𝜔∈𝐿²(Ω) be a function such that ∫_Ω𝜔𝜕₃𝜐=0 for all 𝜐∈𝐶^∞(Ω) satisfing 𝜐=0 on 𝛾×[−1,+1]. Then, 𝜔=0.

Proof. Thanks to Theorem  3.4-1 in [1].

Lemma 2.5. Assume that the scaled unknown satisfying (2.15) admits for each 0<𝜀<𝜀₀ a formal asymptotic expansion of the form 𝐮(𝜀𝑡)=1)(𝜀^𝑁𝐮^−𝑁(𝑡)+1𝜀^𝑁−1𝐮^−𝑁+1(𝑡)+⋯ (2.29) with 𝐮^−𝑁(𝑡),𝐮^−𝑁+1(𝑡)∈𝐿^∞(0,𝑇;𝐖(Ω)),𝐖(Ω)=𝐯∈𝐖^1,4(Ω),𝐯=0𝑜𝑛Γ₀,𝐮^−𝑁(𝑡)≠0, (2.30) for some integer 𝑁∈𝐙. Then, 𝑁=0.

Proof. The proof is broken into seven parts. Before beginning the proper induction in (iv), we record several useful preliminaries.
(i) Let the functions 𝐴^{𝑖𝑗𝑘𝑙}(0) be defined as in Lemma 2.2. Then, for any symmetric matrices (𝑠_𝑘𝑙) and (𝑡_𝑖𝑗), 𝐴^{𝑖𝑗𝑘𝑙}(0)𝑠_𝑘𝑙𝑡_𝑖𝑗=𝜆(0)𝑎^𝛼𝛽𝑎^𝜎𝜏+𝜇(0)𝑎^𝛼𝜎𝑎^𝛽𝜏+𝑎^𝛼𝜏𝑎^𝛽𝜎𝑠_𝜎𝜏𝑡_𝛼𝛽+4𝜇(0)𝑎^𝛼𝜎𝑠_𝛼3𝑡_𝜎3+𝜆(0)𝑎^𝛼𝛽𝑠₃₃𝑡_𝛼𝛽+𝜆(0)𝑎^𝜎𝜏𝑠_𝜎𝜏𝑡₃₃+𝜆(0)+2𝜇(0)𝑠₃₃𝑡₃₃. (2.31)
This formula, which immediately follows from the definitions, will be constantly put to use in the ensuing arguments.
(ii) Let 𝑎^𝑖𝑗∶=𝐚^𝑖⋅𝐚^𝑗. Then, for any 𝑦∈𝜔 and any matrix (𝑡_𝑖𝑗), 𝑎^𝑖𝑗𝑦𝑎^𝑚𝑛𝑦𝑡_𝑖𝑚𝑡_𝑗𝑛≥0,𝑎^𝑖𝑗𝑦𝑎^𝑚𝑛𝑦𝑡_𝑖𝑚𝑡_𝑗𝑛=0⟺𝑡_𝑖𝑗=0. (2.32) Given any 𝑦∈𝜔 and any matrix (𝑡_𝑖𝑗), let 𝐭_𝑖(𝑦)∶=𝑡_𝑖𝑚𝐚^𝑚(𝑦) and let [𝐭_𝑖(𝑦)]^𝑞 denote the 𝑞th Cartesian component of the vector 𝐭_𝑖(𝑦). We thus have 𝑎^𝑖𝑗𝑦𝑎^𝑚𝑛𝑦𝑡_𝑖𝑚𝑡_𝑗𝑛=𝑎^𝑖𝑗𝑦𝑡_𝑖𝑚𝐚^𝑚𝑦⋅𝑡_𝑗𝑛𝐚^𝑛𝑦=𝑎^𝑖𝑗𝑦𝐭_𝑖𝑦⋅𝐭_𝑗𝑦=𝑎^𝑖𝑗𝑦𝐭_𝑖𝑦^𝑝𝐭_𝑗𝑦^𝑝=𝐭_𝑖𝑦^𝑝𝐚^𝑖𝑦⋅𝐭_𝑗𝑦^𝑝𝐚^𝑗𝑦=³_𝑝=1𝐭_𝑖𝑦^𝑝𝐚^𝑖𝑦². (2.33) Hence, 𝑎^𝑖𝑗(𝑦)𝑎^𝑚𝑛(𝑦)𝑡_𝑖𝑚𝑡_𝑗𝑛≥0 and 𝑎^𝑖𝑗𝑎^𝑚𝑛𝑦𝑡_𝑖𝑚𝑡_𝑗𝑛=0⟹𝐭_𝑖𝑦^𝑝𝐚^𝑖𝑦=0,𝑝=1,2,3,⟹𝐭_𝑖𝑦=𝑡_𝑖𝑚𝐚^𝑚𝑦=0,𝑖=1,2,3,⟹𝑡_𝑖𝑚=0,𝑖,𝑚=1,2,3, (2.34) for the three vectors 𝐚^𝑖(𝑦) are linear independent.
(iii) Assume that the formal asymptotic expansion of the scaled unknown is of the form 𝐮(𝜀𝑡)=1)(𝜀^𝑁𝐮^−𝑁(𝑡)+1𝜀^𝑁−1𝐮^−𝑁+1(𝑡)+⋯forsomeinteger𝑁≥0, (2.35) with 𝐮^−𝑁∈𝐿^∞(0,𝑇;𝐖(Ω)) and 𝐮^−𝑁+1∈𝐿^∞(0,𝑇;𝐖(Ω)).
Together with the asymptotic behavior of the functions 𝑔^𝑖𝑗(𝜀) and Γ^𝑝_𝑖𝑗(𝜀) as 𝜀→0, such an expansion induces specific formal asymptotic expansions of the various functions appearing in the formulation of problem (2.15) 𝑢_𝑚‖𝛼(𝜀𝑡)=1)(𝜀^𝑁𝑢^−𝑁_𝑚‖𝛼(𝑡)+⋯,𝑢_𝑚‖3(𝜀𝑡)=1)(𝜀^𝑁+1𝑢^−𝑁−1_𝑚‖3(𝑡)+1𝜀^𝑁𝑢^−𝑁_𝑚‖3(𝑡)𝐸+⋯,_𝛼‖𝛽((𝜀𝑡=1𝜀;𝐮)())𝜀^2𝑁𝐸^−2𝑁_𝛼‖𝛽(𝑡)+⋯,𝐸_𝛼‖3((𝜀𝑡=1𝜀;𝐮)())𝜀^2𝑁+1𝐸^−2𝑁−1_𝛼‖3(𝑡)𝐸+⋯,_3‖3((𝜀𝑡=1𝜀;𝐮)())𝜀^2𝑁+2𝐸^−2𝑁−2_3‖3(𝑡)+⋯,𝐹_𝛼‖𝛽((𝜀𝑡))=1𝜀;𝐮)(,𝐯𝜀^𝑁𝐹^−𝑁_𝛼‖𝛽()𝐹𝑡,𝐯+⋯,_𝛼‖3((𝜀𝑡))=1𝜀;𝐮)(,𝐯𝜀^𝑁+1𝐹^−𝑁−1_𝛼‖3(𝑡,𝐯)+⋯,𝐹_3‖3((𝜀𝑡))=1𝜀;𝐮)(,𝐯𝜀^𝑁+2𝐹^−𝑁−2_3‖3(𝑡,𝐯)+⋯, (2.36) where, by definition, 𝑢^𝑞_𝑖‖𝑗(𝑡),𝐸^𝑞_𝑖‖𝑗(𝑡), and 𝐹^𝑞_𝑖‖𝑗(𝑡,𝐯) designate for each 𝑞∈𝐙 the coefficient of 𝜀^𝑞 in the induced expansions of 𝑢_𝑖‖𝑗(𝜀)(𝑡),𝐸_𝑖‖𝑗(𝜀;𝐮(𝜀)(𝑡)), and 𝐹_𝑖‖𝑗(𝜀;𝐮(𝜀)(𝑡),𝐯).
Note in passing that, while the functions factorizing the powers of 𝜀 are by definition independent of 𝜀, they are dependent on one or several terms 𝐮^𝑞(𝑡),𝑞≥−𝑁. In this respect, particular caution should be exercised as regards this dependence. For instance, 𝑢^−𝑁_𝑚‖𝛼(𝑡)=𝜕_𝛼𝑢_𝑚^−𝑁(𝑡)−Γ^𝑝,0_𝛼𝑚𝑢_𝑝^−𝑁(𝑡),𝑢^−𝑁_𝑚‖3(𝑡)=𝜕₃𝑢_𝑚^−𝑁+1(𝑡)−Γ^𝑝,0_𝑚3𝑢_𝑝^−𝑁(𝑡), (2.37) that is, the factor of 1/𝜀^𝑁 in 𝑢_𝑚‖𝛼(𝜀)(𝑡) depends on 𝐮^−𝑁(𝑡) but the one in 𝑢_𝑚‖3(𝜀)(𝑡) depends also on 𝐮^−𝑁+1(𝑡).
Likewise, it should be remembered that the expression of some factor may differ according to which value of 𝑁 is considered, for instance, 𝐸^−2𝑁_𝛼‖𝛽(𝑡)=12𝑎^𝑚𝑛𝑢^−𝑁_𝑚‖𝛼(𝑡)𝑢^−𝑁_𝑛‖𝛽(𝑡)1if𝑁≥1,2𝑢⁰_𝛼‖𝛽(𝑡)+𝑢⁰_𝛽‖𝛼(𝑡)+𝑎^𝑚𝑛𝑢⁰_𝑚‖𝛼(𝑡)𝑢⁰_𝑛‖𝛽(𝑡)𝐹if𝑁=0,_{−𝑁𝛼‖𝛽}()=1𝑡,𝐯2𝑎^𝑚𝑛𝑢^−𝑁_𝑚‖𝛼(𝑡)𝑣_𝑛‖𝛽+𝑢^−𝑁_𝑛‖𝛽(𝑡)𝑣_𝑚‖𝛼1if𝑁≥1,2𝑣_𝛼‖𝛽+𝑣_𝛽‖𝛼+𝑎^𝑚𝑛𝑢⁰_𝑚‖𝛼(𝑡)𝑣_𝑛‖𝛽+𝑢⁰_𝑛‖𝛽(𝑡)𝑣_𝑚‖𝛼if𝑁=0, (2.38) where 𝑣_𝑚‖𝛼∶=𝜕_𝛼𝑣_𝑚−Γ^𝑝,0_𝛼𝑚𝑣_𝑝. (2.39)
We are now in a position to start the cancellation of the factors of the successive powers of 𝜀 found in the variational equations of problem (2.15) when 𝐮(𝜀)(𝑡) is replaced by its formal expansion. In what follows, 𝐿^𝑟 designates for any integer 𝑟≥−3𝑁−4 the linear form defined by 𝐿^𝑟(𝑡,𝐯)∶=_Ω𝑓^𝑖,𝑟(𝑡)𝑣_𝑖√𝑎𝑑𝑥+_Γ+∪Γ−ℎ^𝑖,𝑟+1(𝑡)𝑣_𝑖√𝑎𝑑Γ. (2.40)
(iv) Assume that 𝑁≥0. Since the lowest power of 𝜀 in the left-hand side is 𝜀^−3𝑁−4, we are naturally led to first try 𝑓^𝑖(𝜀𝑡)=1)(𝜀^−3𝑁−4𝑓^{𝑖,−3𝑁−4}(𝑡),ℎ^𝑖(𝜀𝑡)=1)(𝜀^3𝑁+3ℎ^{𝑖,−3𝑁−3}(𝑡). (2.41)
Comparing the coefficients of 𝜀^−3𝑁−4 in (2.15) and using Lemma 2.2 and (2.36), we get the equations _Ω𝜆(0)+2𝜇(0)𝐸^−2𝑁−2_3‖3(𝑡)𝐹^−𝑁−2_3‖3(𝑡,𝐯)√𝑎𝑑𝑥+_Ω^𝑡_−∞𝜆^′(𝑡−𝜏)+2𝜇^′(𝑡−𝜏)𝐸^−2𝑁−2_3‖3(𝜏)𝐹^−𝑁−2_3‖3(𝜏,𝐯)√𝑎𝑑𝑥𝑑𝜏=𝐿^−3𝑁−4(𝑡,𝐯) (2.42) for all 𝐯∈𝐖(Ω). Since 𝐸^−2𝑁−2_3‖3(𝑡)=12𝑎^𝑚𝑛𝜕₃𝑢_𝑚^−𝑁(𝑡)𝜕₃𝑢_𝑛^−𝑁(𝑡),𝐹^−𝑁−2_3‖3(𝑡,𝐯)=𝑎^𝑚𝑛𝜕₃𝑢_𝑚^−𝑁(𝑡)𝜕₃𝑣_𝑛, (2.43) we must have 𝐿^−3𝑁−4(𝑡,𝐯)=_Ω𝑓^{𝑖,−3𝑁−4}(𝑡)𝑣_𝑖√𝑎𝑑𝑥+_Γ+∪Γ−ℎ^{𝑖,−3𝑁−3}(𝑡)𝑣_𝑖√𝑎𝑑Γ=0 (2.44) for all 𝐯∈𝐖 that are independent of 𝑥₃. Consequently, the first requirement (that there be no restriction on the applied forces) implies that we must let 𝑓^{𝑖,−3𝑁−4}(𝑡)=0,ℎ^{𝑖,−3𝑁−3}(𝑡)=0. (2.45)
By recalling (2.42)–(2.45), we have 12_Ω𝜆(0)+2𝜇(0)𝑎^𝑚𝑛𝜕₃𝑢_𝑚^−𝑁(𝑡)𝜕₃𝑢_𝑛^−𝑁(𝑡)𝑎^𝑚𝑛𝜕₃𝑢_𝑚^−𝑁(𝑡)𝜕₃𝑣_𝑛√+1𝑎𝑑𝑥2_Ω^𝑡_−∞𝜆^′(𝑡−𝜏)+2𝜇^′(𝑡−𝜏)𝑎^𝑚𝑛𝜕₃𝑢_𝑚^−𝑁(𝜏)𝜕₃𝑢_𝑛^−𝑁(𝜏)𝑎^𝑚𝑛𝜕₃𝑢_𝑚^−𝑁(𝜏)𝜕₃𝑣_𝑛√𝑎𝑑𝑥𝑑𝜏=0, (2.46) that is, 𝑑𝑑𝑡^𝑡_−∞Ω𝜆(𝑡−𝜏)+2𝜇(𝑡−𝜏)𝑎^𝑚𝑛𝜕₃𝑢_𝑚^−𝑁(𝜏)𝜕₃𝑢_𝑛^−𝑁(𝜏)𝑎^𝑚𝑛𝜕₃𝑢_𝑚^−𝑁(𝜏)𝜕₃𝑣_𝑛𝑑𝑥𝑑𝜏=0. (2.47) Therefore, ^𝑡_−∞Ω𝜆(𝑡−𝜏)+2𝜇(𝑡−𝜏)𝑎^𝑚𝑛𝜕₃𝑢_𝑚^−𝑁(𝜏)𝜕₃𝑢_𝑛^−𝑁(𝜏)𝑎^𝑚𝑛𝜕₃𝑢_𝑚^−𝑁(𝜏)𝜕₃𝑣_𝑛𝑑𝑥𝑑𝜏=const, (2.48) which implies ^𝑡_−∞Ω𝜆(𝑡−𝜏)+2𝜇(𝑡−𝜏)𝑎^𝑚𝑛𝜕₃𝑢_𝑚^−𝑁(𝜏)𝜕₃𝑢_𝑛^−𝑁(𝜏)𝑎^𝑚𝑛𝜕₃𝑢_𝑚^−𝑁(𝜏)𝜕₃𝑣_𝑛𝑑𝑥𝑑𝜏=0. (2.49)
Letting 𝐯=𝐮^−𝑁(𝜏) in (2.49) shows that ^𝑡_−∞Ω𝜆(𝑡−𝜏)+2𝜇(𝑡−𝜏)𝑎^𝑚𝑛𝜕₃𝑢_𝑚^−𝑁(𝜏)𝜕₃𝑢_𝑛^−𝑁(𝜏)²𝑑𝑥𝑑𝜏=0. (2.50)
Since the symmetric (𝑎^𝑖𝑗) is positive definite, we conclude that 𝜕₃𝐮^−𝑁(𝑡)=𝜕₃𝑢_𝑚^−𝑁(𝑡)=0inΩ, (2.51) that is, 𝐮^−𝑁(𝑡) is independent of 𝑥₃. Inserting (2.51) into (2.43) yields 𝐸^−2𝑁−2_3‖3(𝑡)=0,𝐹^−𝑁−2_3‖3(𝑡,𝐯)=0∀𝐯∈𝐖(Ω). (2.52) A usual, any function defined on Ω that is independent of 𝑥₃ is identified with a function defined on 𝜔, and (2.36) and (2.52) imply 𝐮^−𝑁(𝑡)∈𝐿^∞(0,𝑇;𝐖(𝜔)),𝐖(𝜔)∶=𝜼∈𝐖^1,4(𝜔);𝜼=0on𝛾₀𝐸_{−2𝑁−2𝑖‖𝑗},(𝑡)=0inΩ,𝐹^−𝑁−2_𝑖‖𝑗(𝑡,𝐯)=0∀𝐯∈𝐖(Ω) (2.53)(2.54)
Noting (2.36) and (2.51), we also have 𝐸^−2𝑁−1_𝑖‖𝑗(𝑡)=0inΩ. (2.55) Since 𝐸^−2𝑁−1_𝛼‖𝛽(𝑡)=0 (the leading term in the formal expansion of 𝐸_𝛼‖𝛽(𝜀;𝐮(𝜀)(𝑡)) is order of −2𝑁) and 𝐸^−2𝑁−1_𝑖‖3(𝑡)=0 (since 𝜕₃𝐮^−𝑁(𝑡)=0, each factor of 1/𝜀^2𝑁+1 in the expansion of 𝐸_𝛼‖3(𝜀;𝐮(𝜀)(𝑡)) vanishes because it contains some derivative 𝜕₃𝑢_𝑚^−𝑁 and the leading term in the expansion of 𝐸_3‖3(𝜀;𝐮(𝜀)(𝑡)) is of order strictly higher than (−2𝑁−1)), our next try is thus 𝑓^𝑖(𝜀𝑡)=1)(𝜀^3𝑁+3𝑓^{𝑖,−3𝑁−3}(𝑡),ℎ^𝑖(𝜀𝑡)=1)(𝜀^3𝑁+2ℎ^{𝑖,−3𝑁−2}(𝑡). (2.56) Comparing the coefficient of 𝜀^−3𝑁−3 in (2.15) then yields equations (the functions 𝐴^{𝑖𝑗𝑘𝑙}(0) are defined in Lemma 2.2) _Ω𝐴^{𝑖𝑗𝑘𝑙}(0)𝐸^−2𝑁−1_𝑘‖𝑙(𝑡)𝐹^−𝑁−2_𝑖‖𝑗(𝑡,𝐯)√𝑎𝑑𝑥+_Ω^𝑡_−∞𝐴^{𝑖𝑗𝑘𝑙}(0)𝐸^−2𝑁−1_𝑘‖𝑙(𝜏)𝐹^−𝑁−2_𝑖‖𝑗(𝜏,𝐯)√𝑎𝑑𝑥𝑑𝜏=𝐿^−3𝑁−3(𝑡,𝐯) (2.57) for all 𝐯∈𝐖(Ω). But since (2.55), we must let 𝑓^{𝑖,−3𝑁−3}(𝑡)=0 and ℎ^{𝑖,−3𝑁−2}(𝑡)=0 (first requirement) and accordingly try 𝑓^𝑖(𝜀𝑡)=1)(𝜀^3𝑁+2𝑓^{𝑖,−3𝑁−2}(𝑡),ℎ^𝑖(𝜀𝑡)=1)(𝜀^3𝑁+2ℎ^{𝑖,−3𝑁−1}(𝑡). (2.58) In which case the cancellation of the coefficient of 𝜀^−3𝑁−2 in (2.15) yields the equations _Ω𝐴^{𝑖𝑗𝑘𝑙}(0)𝐸^−2𝑁_𝑘‖𝑙(𝑡)𝐹^−𝑁−2_𝑖‖𝑗(𝑡,𝐯)√𝑎𝑑𝑥+_Ω^𝑡_−∞𝐴^{′𝑖𝑗𝑘𝑙}(𝑡−𝜏)𝐸^−2𝑁_𝑘‖𝑙(𝜏)𝐹^−𝑁−2_𝑖‖𝑗(𝜏,𝐯)√𝑎𝑑𝑥𝑑𝜏=𝐿^−3𝑁−2(𝑡,𝐯) (2.59) for all 𝐯∈𝐖(Ω). But since (2.54), we must let 𝑓^{𝑖,−3𝑁−2}(𝑡)=0 and ℎ^{𝑖,−3𝑁−1}(𝑡)=0 (first requirement).
(v) Assume that 𝑁≥1. Our next try being thus 𝑓^𝑖(𝜀𝑡)=1)(𝜀^3𝑁+1𝑓^{𝑖,−3𝑁−1},ℎ^𝑖(𝜀)=1𝜀^3𝑁+2ℎ^{𝑖,−3𝑁}, (2.60) the cancellation of the coefficient of 𝜀^−3𝑁−1 in the variational equations of problem (2.15) then yields the equations _Ω𝐴^{𝑖𝑗𝑘𝑙}(0)𝐸^−2𝑁_𝑘‖𝑙(𝑡)𝐹^−𝑁−1_𝑖‖𝑗(𝑡,𝐯)√𝑎𝑑𝑥+_Ω^𝑡_−∞𝐴^{′𝑖𝑗𝑘𝑙}(𝑡−𝜏)𝐸^−2𝑁_𝑘‖𝑙(𝜏)𝐹^−𝑁−1_𝑖‖𝑗(𝜏,𝐯)√𝑎𝑑𝑥𝑑𝜏=𝐿^−3𝑁−1(𝑡,𝐯) (2.61) for all 𝐯∈𝐖(Ω), where 𝐸^−2𝑁_𝛼‖𝛽(𝑡)=12𝑎^𝑚𝑛𝑢^−𝑁_𝑚‖𝛼(𝑡)𝑢^−𝑁_𝑛‖𝛽(𝑡),𝐹^−𝑁−1_𝛼‖𝛽()𝐸𝑡,𝐯=0,_{−2𝑁𝛼‖3}(𝑡)=12𝑎^𝑚𝑛𝑢^−𝑁_𝑚‖𝛼(𝑡)𝑢^−𝑁_𝑛‖3(𝑡),𝐹^−𝑁−1_𝛼‖3()=1𝑡,𝐯2𝑎^𝑚𝑛𝑢^−𝑁_𝑚‖𝛼(𝑡)𝜕₃𝑣_𝑛,𝐸^−2𝑁_3‖3(𝑡)=12𝑎^𝑚𝑛𝑢^−𝑁_𝑚‖3(𝑡)𝑢^−𝑁_𝑛‖3(𝑡),𝐹^−𝑁−1_3‖3(𝑡,𝐯)=𝑎^𝑚𝑛𝑢^−𝑁_𝑚‖3(𝑡)𝜕₃𝑣_𝑛, (2.62) the functions 𝑢^−𝑁_𝑚‖𝑖(𝑡) being those defined in (2.36).
Letting 𝐯∈𝐖(Ω) be independent of 𝑥₃ then shows that we must let 𝑓^{𝑖,−3𝑁−1}(𝑡)=0 and ℎ^{𝑖,−3𝑁}=0; hence, _Ω𝐴^{𝑖𝑗𝑘𝑙}(0)𝐸^−2𝑁_𝑘‖𝑙(𝑡)𝐹^−𝑁−1_𝑖‖𝑗(𝑡,𝐯)√𝑎𝑑𝑥+_Ω^𝑡_−∞𝐴^{′𝑖𝑗𝑘𝑙}(𝑡−𝜏)𝐸^−2𝑁_𝑘‖𝑙(𝜏)𝐹^−𝑁−1_𝑖‖𝑗(𝜏,𝐯)√𝑎𝑑𝑥𝑑𝜏=0 (2.63) for all 𝐯∈𝐖(Ω). Let the field 𝐰^𝑁=(𝑤^𝑁_𝑚) be defined for all (𝑦,𝑥₃)∈Ω by 𝑤^𝑁_𝑚(𝑡)∶=𝑢_𝑚^−𝑁+1(𝑡)−1+𝑥₃Γ^𝑝,0_𝑚3𝑢_𝑝^−𝑁(𝑡). (2.64) Then, 𝐰^𝑁(𝑡)∈𝐿^∞(0,𝑇;𝐖(Ω)) because both 𝐮^−𝑁(𝑡) and 𝐮^−𝑁+1(𝑡) are assumed to be in the space 𝐿^∞(0,𝑇;𝐖(Ω)).
Furthermore, 𝜕₃𝑤^𝑁_𝑚(𝑡)=𝑢^−𝑁_𝑚‖3(𝑡), so that 𝐹^−𝑁−1_𝛼‖3𝑡,𝐰^𝑁(𝑡)=𝐸^−2𝑁_𝛼‖3(𝑡),𝐹^−𝑁−1_3‖3𝑡,𝐰^𝑁(𝑡)=2𝐸^−2𝑁_3‖3(𝑡). (2.65)
Using Lemma 2.2 and (2.65), we get 𝐴^{𝑖𝑗𝑘𝑙}(0)𝐸^−2𝑁_𝑘‖𝑙(𝑡)𝐹^−𝑁−1_𝑖‖𝑗𝑡,𝐰^𝑁=2𝜆(0)𝑎^𝑚𝑛𝐸^−2𝑁_𝑚‖𝑛(𝑡)𝐸^−2𝑁_3‖3(𝑡)+4𝜇(0)𝑎^𝑚𝑛𝐸^−2𝑁_𝑚‖3(𝑡)𝐸^−2𝑁_𝑛‖3(𝑡). (2.66)
Since 𝑎^𝑚𝑛𝐸^−2𝑁_𝑚‖𝑛(𝑡)=12𝑎^𝑖𝑗𝑎^𝑚𝑛𝑢^−𝑁_𝑖‖𝑚(𝑡)𝑢^−𝑁_𝑗‖𝑛(𝑡)≥0inΩ, (2.67) (by (ii)) 𝐸^−2𝑁_3‖3(𝑡)=12𝑎^𝑚𝑛𝑢^−𝑁_𝑚‖3(𝑡)𝑢^−𝑁_𝑛‖3(𝑡)≥0,𝑎^𝑚𝑛𝐸^−2𝑁_𝑚‖3(𝑡)𝐸^−2𝑁_𝑛‖3(𝑡)≥0inΩ (2.68) (the matrix (𝑎^𝑚𝑛) is positive definite), in a similar way as in (iv), we can obtain from (2.63) that ^𝑡_−∞Ω𝐴^{𝑖𝑗𝑘𝑙}(𝑡−𝜏)𝐸^−2𝑁_𝑘‖𝑙(𝜏)𝐹^−𝑁−1_𝑖‖𝑗(𝜏,𝐯)𝑑𝑥𝑑𝜏=0 (2.69) for all 𝐯∈𝐖(Ω).
Letting 𝐯=𝐰^𝑁(𝜏) in (2.69) and noting (2.66)–(2.68), we conclude that 𝑎^𝑚𝑛𝐸^−2𝑁_𝑚‖3(𝑡)𝐸^−2𝑁_𝑛‖3(𝑡)=0inΩ; (2.70) hence (the matrix (𝑎^𝑚𝑛) is positive definite) 𝐸^−2𝑁_𝑚‖3(𝑡)=0inΩ. (2.71)
In particular then, 𝐸^−2𝑁_3‖3(𝑡)=(1/2)𝑎^𝑚𝑛𝑢^−𝑁_𝑚‖3(𝑡)𝑢^−𝑁_𝑛‖3(𝑡)=0 (by (2.62)) and thus (the matrix (𝑎^𝑚𝑛) is positive definite) 𝑢^−𝑁_𝑚‖3(𝑡)=0. (2.72)
(vi) Assume that 𝑁≥2 (the case 𝑁=1 is considered separately, c.f. (viii)). Our next try being thus 𝑓^𝑖(𝜀𝑡)=1)(𝜀^3𝑁𝑓^{𝑖,−3𝑁}(𝑡),ℎ^𝑖(𝜀𝑡)=1)(𝜀^3𝑁−1ℎ^{𝑖,−3𝑁+1}(𝑡), (2.73) the cancellation of the coefficient of 𝜀^−3𝑁 in the variational equations of problem (2.15) then yields the equations (note that two terms are needed here from the expansions of the functions 𝐴^{𝑖𝑗𝑘𝑙}(𝜀)𝑔(𝜀), c.f. Lemma 2.2)) _Ω𝐴^{𝑖𝑗𝑘𝑙}(0)𝐸^−2𝑁_𝑘‖𝑙(𝑡)𝐹^−𝑁_𝑖‖𝑗(𝑡,𝐯)+𝐸^−2𝑁+1_𝑘‖𝑙(𝑡)𝐹^−𝑁−1_𝑖‖𝑗(𝑡,𝐯)√𝑎𝑑𝑥+_Ω𝐵^{𝑖𝑗𝑘𝑙,1}𝐸^−2𝑁_𝑘‖𝑙(𝑡)𝐹^−𝑁−1_𝑖‖𝑗()+𝑡,𝐯𝑑𝑥_Ω𝑡−∞𝐴^{′𝑖𝑗𝑘𝑙}(𝑡−𝜏)𝐸^−2𝑁_𝑘‖𝑙(𝜏)𝐹^−𝑁_𝑖‖𝑗(𝜏,𝐯)+𝐸^−2𝑁+1_𝑘‖𝑙(𝜏)𝐹^−𝑁−1_𝑖‖𝑗(𝜏,𝐯)√+𝑎𝑑𝑥𝑑𝜏_Ω𝐵^{′𝑖𝑗𝑘𝑙,1}(𝑡−𝜏)𝐸^−2𝑁_𝑘‖𝑙(𝜏)𝐹^−𝑁−1_𝑖‖𝑗(𝜏,𝐯)𝑑𝑥𝑑𝜏=𝐿^−3𝑁(𝑡,𝐯) (2.74) for all 𝐯∈𝐖(Ω), where (by (2.62) and (2.72)) 𝐸^−2𝑁_𝛼‖𝛽(𝑡)=12𝑎^𝑚𝑛𝑢^−𝑁_𝑚‖𝛼(𝑡)𝑢^−𝑁_𝑚‖𝛽(𝑡),𝐸^−2𝑁_𝑖‖3(𝑡)𝐹=0,_{−𝑁−1𝛼‖𝛽}(𝑡,𝐯)=0,𝐹^−𝑁−1_𝛼‖3()=1𝑡,𝐯2𝑎^𝑚𝑛𝑢^−𝑁_𝑚‖𝛼(𝑡)𝜕₃𝑣_𝑛,𝐹^−𝑁−1_3‖3()𝐹𝑡,𝐯=0,_{−𝑁𝛼‖𝛽}()=1𝑡,𝐯2𝑎^𝑚𝑛𝑢^−𝑁_𝑚‖𝛼(𝑡)𝑣_𝑛‖𝛽+𝑢^−𝑁_𝑛‖𝛽(𝑡)𝑣_𝑚‖𝛼,𝐹^−𝑁_3‖3=𝑎^𝑚𝑛𝑢^−𝑁+1_𝑚‖3(𝑡)𝜕₃𝑣_𝑛, (2.75) the last expression of 𝐹^−𝑁_3‖3(𝑡,𝐯) being valid only if 𝑁≥2 (the expressions of 𝐹^−𝑁_𝛼‖3(𝑡,𝐯) are not needed since 𝐴^{𝛼3𝜎𝜏}(0)=0 by Lemma 2.2 and 𝐸^−2𝑁_𝛼‖3(𝑡)=0 by (2.71)).
Noting that 𝐹^−𝑁−1_𝛼||3(𝐯)=𝐹^−𝑁_3‖3=0 if 𝜕₃𝐯=0, we thus conclude that the variational equations (2.74) reduce to _Ω𝐴^{𝛼𝛽𝜎𝜏}(0)𝐸^−2𝑁_𝜎‖𝜏(𝑡)𝐹^−𝑁_𝛼‖𝛽(𝑡,𝐯)√𝑎𝑑𝑥+_Ω^𝑡_−∞𝐴^{′𝛼𝛽𝜎𝜏}(𝑡−𝜏)𝐸^−2𝑁_𝜎‖𝜏(𝜏)𝐹^−𝑁_𝛼‖𝛽(𝜏,𝐯)√𝑎𝑑𝑥𝑑𝜏=𝐿^−3𝑁(𝑡,𝐯) (2.76) for all 𝐯∈𝐖(Ω) that are independent of 𝑥₃.
Since each term in the sum 𝐴^{𝛼𝛽𝜎𝜏}(0)𝐸^−2𝑁_𝜎‖𝜏(𝑡)𝐹^−𝑁_𝛼‖𝛽(𝑡,𝐯) is cubic with respect to the functions 𝑢^−𝑁_𝑚‖𝛼(𝑡), the linearization trick (second requirement) implies that 𝐿^−3𝑁(𝑡,𝐯)=0 for all 𝐯∈𝐖(Ω) that are independent of 𝑥₃. Hence, we must let 𝑓^{𝑖,−3𝑁}(𝑡)=0 and ℎ^{𝑖,−3𝑁+1}(𝑡)=0. Hence _Ω𝐴^{𝛼𝛽𝜎𝜏}(0)𝐸^−2𝑁_𝜎‖𝜏(𝑡)𝐹^−𝑁_𝛼‖𝛽(𝑡,𝐯)√𝑎𝑑𝑥+_Ω^𝑡_−∞𝐴^{′𝛼𝛽𝜎𝜏}(𝑡−𝜏)𝐸^−2𝑁_𝜎‖𝜏(𝜏)𝐹^−𝑁_𝛼‖𝛽(𝜏,𝐯)√𝑎𝑑𝑥𝑑𝜏=0. (2.77)
In a similar way as in (iv), we can obtain from (2.77) that ^𝑡_−∞Ω𝐴^{𝛼𝛽𝜎𝜏}(𝑡−𝜏)𝐸^−2𝑁_𝜎‖𝜏(𝜏)𝐹^−𝑁_𝛼‖𝛽(𝜏,𝐯)𝑑𝑥𝑑𝜏=0.