On Initial-Boundary Value Problems for Hyperbolic Equations in Domains with Conical Points

Abstract

We consider an initial-boundary value problem for general higher-order hyperbolic equation in an infinite cylinder with the base containing conical points on the boundary. We establish several results on the unique solvability, the regularity, and the asymptotic behaviour of the solution near the conical points.

1. Introduction

A large number of investigations have been devoted to boundary value problems in nonsmooth domains with conical points. Up to now, elliptic boundary value problems in domains with point singularities have been thoroughly investigated (see, e.g., [1–3]).

We are concerned with hyperbolic equations in domains with conical points. This topic has been investigated in many works with different approaches. For example, [4, 5] the Cauchy-Dirichlet and Cauchy-Neumann problems for second-order hyperbolic systems with the coefficients independent of the time variable were treated in which the asymptotics of the solutions were established with explicit formulas for the coefficients. In [6–9], initial-boundary value problems for general higher-order hyperbolic equations and systems with the coefficients depending on both spatial and time variable in a domain containing conical points were studied in which the unique solvability, the regularity, and the asymptotic behaviour of the solutions near the conical points were obtained. In the present paper, these results are extended to initial-boundary value problems for general higher-order hyperbolic equations with more general boundary conditions in infinite cylinders with the bases containing conical points. Such boundary conditions have been considered for elliptic equations in [10, 11] and for parabolic equations in [12, 13].

Our paper is organized as follows. Section 2 is devoted to some notations and the formulation of the problem. The main results will be stated in Section 3. The proofs of the main theorems will be given in Sections 4 and 5.

2. Notations and the Formulation of the Problem

Let 𝐺 be a bounded domain in ℝ^𝑛(𝑛⩾2) with the boundary 𝜕𝐺. We suppose that Γ=𝜕𝐺⧵{0} is a smooth manifold and 𝐺 in a neighborhood 𝑈 of the origin 0 coincides with the cone 𝐾={𝑥∶𝑥/|𝑥|∈Ω}, where Ω is a smooth domain on the unit sphere 𝑆^𝑛−1 in ℝ^𝑛. For each 𝑇, 0<𝑇⩽∞, denote 𝑄_𝑇=𝐺×(0,𝑇),  𝐺_𝑇=𝐺×{𝑇}. Specifically, we set 𝑄=𝑄_∞=𝐺×ℝ^∗₊, 𝑆=Γ×ℝ₊, where ℝ^∗₊=(0,∞), ℝ₊=[0,∞). For each multiindex 𝛼=(𝛼₁,…,𝛼_𝑛)∈ℕ^𝑛, set |𝛼|=𝛼₁+⋯+𝛼_𝑛 and 𝜕^𝛼_𝑥=𝜕^𝛼_𝑥=𝜕^𝛼1_𝑥1⋯𝜕^𝛼𝑛_𝑥𝑛. For a nonnegative integer 𝑘 we write 𝑢_𝑡𝑘 instead of 𝜕^𝑘_𝑡𝑢.

We introduce the following differential operator: 𝐿𝑢=𝐿(𝑥,𝑡,𝜕)𝑢=_{|𝛼|,|𝛽|⩽𝑚}(−1)^|𝛼|𝜕^𝛼𝑎_𝛼𝛽𝜕^𝛽𝑢, (1) where 𝑎_𝛼𝛽=𝑎_𝛼𝛽(𝑥,𝑡) are bounded complex-valued functions defined in 𝑄. We assume that 𝑎_𝛼𝛽=(−1)^{|𝛼|+|𝛽|}𝑎_𝛽𝛼 for all |𝛼|,|𝛽|⩽𝑚. This means the differential operator 𝐿 is formally self-adjoint. We assume further that there exists a positive constant 𝜇₀ such that _{|𝛼|=|𝛽|=𝑚}𝑎_𝛼𝛽(𝑥,𝑡)𝜂_𝛽𝜂_𝛼≥𝜇_0|𝛼|=𝑚𝜂_𝛼², (2) for all 𝜂_𝛼∈ℂ, |𝛼|=𝑚 and all (𝑥,𝑡)∈𝑄.

We introduce also a system of boundary operators: 𝐵_𝑗=𝐵_𝑗𝑥,𝑡,𝜕_𝑥=_{|𝛼|⩽𝜇𝑗}𝑏_𝑗,𝛼(𝑥,𝑡)𝜕^𝛼_𝑥,𝑗=1,…,𝑚, (3) on 𝑆 with smooth coefficients in 𝑄. Suppose that ord𝐵_𝑗=𝜇_𝑗⩽𝑚−1for𝑗=1,…,𝜒,𝑚⩽ord𝐵_𝑗=𝜇_𝑗⩽2𝑚−1for𝑗=𝜒+1,…,𝑚, (4) and the coefficients of 𝐵_𝑗 are independent of 𝑡 if ord𝐵_𝑗<𝑚, where ord𝐵_𝑗 stands for the order of the differential operator 𝐵_𝑗. Suppose that {𝐵_𝑗(𝑥,𝑡,𝜕_𝑥)}^𝑚_𝑗=1 is a normal system on 𝑆 for all 𝑡∈ℝ₊; that is, the two following conditions are satisfied:(i)𝜇_𝑗≠𝜇_𝑘 for 𝑗≠𝑘,(ii)𝐵^∘_𝑗(𝑥,𝑡,𝜈(𝑥))≠0 for all (𝑥,𝑡)∈𝑆, 𝑗=1,…,𝑚. Here 𝜈(𝑥) is the unit outer normal to 𝑆 at point 𝑥, and 𝐵^∘_𝑗(𝑥,𝑡,𝜕_𝑥) is the principal part of 𝐵_𝑗(𝑥,𝑡,𝜕_𝑥), 𝐵^∘_𝑗=𝐵^∘_𝑗𝑥,𝑡,𝜕_𝑥=_{|𝛼|=𝜇𝑗}𝑏_𝑗,𝛼(𝑥,𝑡)𝜕^𝛼_𝑥,𝑗=1,…,𝑚. (5) Furthermore, we assume that 𝐵^∘_𝑗(0,𝑡,𝜈(𝑥))≠0 for all 𝑥∈𝑆 sufficiently close to the origin 0.

To be able to reduce the problem considered to variational form we assume that there are boundary operators Φ_𝑗 on 𝑆, 𝑗=1,…,𝑚, such that 𝐵(𝑡,𝑢,𝑣)=_𝐺𝐿𝑢𝑣𝑑𝑥+^𝜒_𝑗=1𝑆Φ_𝑗𝑢𝐵_𝑗+𝑣𝑑𝑠_{𝑚𝑗=𝜒+1𝑆}𝐵_𝑗𝑢Φ_𝑗𝑣𝑑𝑠, (6) for all 𝑢,𝑣∈𝐶^∞₀(𝐺⧵{0}) and a.e. 𝑡∈ℝ₊. Here 𝐵(𝑡,𝑢,𝑣)=^𝑚_{|𝛼|,𝛽=0𝐺}𝑎_𝛼𝛽(⋅,𝑡)𝜕^𝛽_𝑥𝑢𝜕^𝛼_𝑥𝑣𝑑𝑥,𝑡∈ℝ₊, (7) is the bilinear form associated with the operator 𝐿(𝑥,𝑡,𝜕_𝑥). Of course this is an essential restriction on the structure of the boundary operators in (3). However, if the system of boundary operators in (3) is a Dirichlet system (then all ord of 𝐵_𝑗 are less than 𝑚) or a generalized Neumann system (then 𝑚⩽ord𝐵_𝑗⩽2𝑚−1 for all 𝑗=1,…,𝑚), the equality (6) holds for a suitable system Φ_𝑗,𝑗=1,…,𝑚 (see [10, Section I.7.]).

In this paper, we consider the following problem: 𝑢_𝑡𝑡𝐵+𝐿𝑢=𝑓inQ,_𝑗𝑢=0on𝑆,𝑗=1,…,𝑚,𝑢∣_𝑡=0=𝑢_𝑡∣_𝑡=0=0in𝐺. (8)(9)

Before giving the definition of generalized solutions to this problem, let us introduce some needed functional spaces.

Let us denote by 𝐻^𝑙(𝐺)(𝑙∈ℕ) the usual Sobolev space of all functions 𝑢 defined in 𝐺 with the norm ‖𝑢‖_𝐻^𝑙_(𝐺)=_{𝐺|𝛼|⩽𝑙}𝐷^𝛼𝑢²𝑑𝑥^1/2<∞. (10) If 𝑙⩾1, by 𝐻^𝑙−(1/2)(Γ) we denote the space of traces of functions in 𝐻^𝑙(𝐺) on Γ with the norm ‖𝑢‖_𝐻^𝑙−(1/2)_(Γ)=inf‖𝑣‖_𝐻^𝑙_(𝐺)∶𝑣∈𝐻^𝑙(𝐺),𝑣∣_Γ=𝑢. (11) We set 𝐻^𝑚_𝐵(𝐺)=𝑢∈𝐻^𝑚(𝐺)∶𝐵_𝑗𝑢=0on𝑆for𝑗=1,…,𝜒, (12) with the same norm as in 𝐻^𝑚(𝐺). By 𝑉^𝑙_2,𝛼(𝐺)(𝛼∈ℝ) we denote the closure of 𝐶^∞₀(𝐺⧵{0}) with respect to the norm ‖𝑢‖_𝑉^𝑙_2,𝛼(𝐺)=_{|𝑝|⩽𝑙𝐺}𝑟^{2(𝛼+|𝑝|−𝑙)}𝐷^𝑝𝑢²𝑑𝑥^1/2, (13) where 𝑟=|𝑥|=(∑^𝑛_𝑘=1𝑥²_𝑘)^1/2. By 𝐻^𝑙_𝛼(𝐺) we denote the weighted Sobolev space of functions 𝑢 defined in 𝐺 with the norm ‖𝑢‖_𝐻^𝑙_𝛼(𝐺)=_{|𝑝|⩽𝑙𝐺}𝑟^2𝛼𝐷^𝑝𝑢²𝑑𝑥^1/2<+∞. (14) If 𝑙⩾1, then 𝑉_𝛼^𝑙−1/2(Γ),𝐻_𝛼^𝑙−1/2(Γ) denote the spaces consisting of traces of functions from respective spaces 𝑉^𝑙_2,𝛼(𝐺),𝐻^𝑙_𝛼(𝐺) on the boundary Γ with the respective norms ‖𝑢‖_𝑉𝛼^𝑙−1/2_(Γ)=inf‖𝑣‖_𝑉^𝑙_2,𝛼(𝐺)∶𝑣∈𝑉^𝑙_2,𝛼(𝐺),𝑣∣_Γ,‖𝑢‖=𝑢_{𝐻𝛼𝑙−1/2(Γ)}=inf‖𝑣‖_𝐻^𝑙_𝛼(𝐺)∶𝑣∈𝐻^𝑙_𝛼(𝐺),𝑣∣_Γ=𝑢. (15)

Let 𝑋,𝑌 be Banach spaces, 0<𝑇⩽∞. We denote by 𝐿₂((0,𝑇),𝑋) the space of all functions 𝑢∶(0,𝑇)→𝑋 such that ‖𝑢‖_{𝐿2((0,𝑇),𝑋)}=^𝑇₀‖𝑢(𝑡)‖²_𝑋𝑑𝑡^1/2<∞, (16) and by 𝐻¹((0,𝑇),𝑋,𝑌) the space of all functions 𝑢∈𝐿₂((0,𝑇),𝑋) such that 𝑢_𝑡∈𝐿₂((0,𝑇),𝑌). The norm in 𝐻¹((0,𝑇),𝑋,𝑌) is defined by ‖𝑢‖_𝐻¹_{((0,𝑇),𝑋,𝑌)}=‖𝑢‖²_{𝐿2))((0,𝑇,𝑋}+𝑢_𝑡²_{𝐿2))((0,𝑇,𝑌}^1/2. (17) For some 𝛾∈ℝ, we denote by 𝐿₂(ℝ^∗₊,𝑋,𝛾) the space of all functions 𝑢∶ℝ^∗₊→𝑋 such that ‖𝑢‖_𝐿2(ℝ^∗_+,𝑋,𝛾)=^∞₀‖𝑢(𝑡)‖²_𝑋𝑒^−2𝛾𝑡𝑑𝑡^1/2<∞, (18) and by 𝐻¹(ℝ^∗₊,𝑋,𝑌,𝛾) the space of all functions 𝑢∈𝐿₂(ℝ^∗₊,𝑋,𝛾) such that 𝑢_𝑡∈𝐿₂(ℝ^∗₊,𝑌,𝛾) with the norm ‖𝑢‖_𝐻¹_(ℝ^∗_{+,𝑋,𝑌,𝛾)}=‖𝑢‖²_𝐿2ℝ^∗_+,𝑋,𝛾+𝑢_𝑡²_{𝐿2ℝ∗+,𝑌,𝛾}^1/2. (19) For shortness, we set 𝐻_𝐵^𝑚,1𝑄_𝑇=𝐻¹(0,𝑇),𝐻^𝑚_𝐵(𝐺),𝐿₂(𝐺,𝐿₂)𝑄,𝛾=𝐿₂ℝ^∗₊,𝐿₂(𝐺),𝐻,𝛾_𝑙,0𝑄,𝛾=𝐿₂ℝ^∗₊,𝐻^𝑙(𝐺),𝐻,𝛾_𝑙,1𝑄,𝛾=𝐻¹ℝ^∗₊,𝐻^𝑙(𝐺),𝐿₂(𝐺),𝑉,𝛾_𝑙,02,𝛽𝑄,𝛾=𝐿₂ℝ^∗₊,𝑉^𝑙_2,𝛽(𝐺),𝐻,𝛾_𝛽𝑙,0𝑄,𝛾=𝐿₂ℝ^∗₊,𝐻^𝑙_𝛽(𝐺),𝐻,𝛾_𝛽𝑙,1𝑄,𝛾=𝐻¹ℝ^∗₊,𝐻^𝑙_𝛽(𝐺),𝐿₂(𝐺),𝛾. (20)

Now the definition of generalized solutions of the problem (8)-(9) is given as follows.

Definition 1. Let 𝑓 be a given function defined on 𝑄 which belongs to 𝐿₂(𝑄_𝑇) for each 𝑇>0. A function 𝑢∈𝐻_𝐵^𝑚,1(𝑄,𝛾) for some 𝛾∈ℝ is called a generalized solution of the problem (8)-(9) if and only if 𝑢∣_𝑡=0=0, and for each 𝑇>0 the equality −^𝑇_0𝐺𝑢_𝑡𝜂_𝑡𝑑𝑡+^𝑚_{|𝛼|,𝛽=0}^𝑇_0𝐺𝑎_𝛼𝛽(𝑥,𝑡)𝜕^𝛽_𝑥𝑢𝜕^𝛼_𝑥=𝜂𝑑𝑥𝑑𝑡_𝑇0𝐺𝑓𝜂𝑑𝑡, (21) holds for all 𝜂∈𝐻_𝐵^𝑚,1(𝑄_𝑇) satisfying 𝜂(⋅,𝑇)=0.

3. Statements of the Main Results

The unique solvability of the problem is given by the following theorem.

Theorem 2. There exists a positive real number 𝛾₀ such that for each 𝛾>𝛾₀, if 𝑓∈𝐿₂(𝑄,𝜎) for some real number 𝜎, the problem (8)-(9) has a unique generalized solution 𝑢 in the space 𝐻_𝐵^𝑚,1(𝑄,𝛾+𝜎) and ‖𝑢‖²_{𝐻𝐵𝑚,1𝑄,𝛾+𝜎}≤𝐶𝑓²_{𝐿2()𝑄,𝜎}, (22) where 𝐶 is a constant independent of 𝑢 and 𝑓.

The following theorem states on the regularity of the generalized solution in weighted Sobolev spaces.

Theorem 3. Let ℎ be a positive integer and 𝛾₀ be the number as in Theorem 2. Suppose that the function 𝑓 satisfies the following conditions for some real number 𝜎:(i)𝑓_𝑡𝑘∈𝐿₂(𝑄,𝑘𝛾₀+𝜎),𝑘⩽ℎ, (ii)𝑓_𝑡𝑘(𝑥,0)=0,0≤𝑘≤ℎ−1. Let 𝑢∈𝐻_𝐵^𝑚,1(𝑄,𝛾+𝜎) for some 𝛾>𝛾₀ be the generalized solution of the problem (8)-(9). Then 𝑢_𝑡𝑘∈𝐻_𝑚^2𝑚,1(𝑄,(𝑘+2)𝛾+𝜎) for 𝑘=0,1,…,ℎ−1 and ^ℎ−1_𝑘=0𝑢_𝑡𝑘²_{𝐻𝑚2𝑚,1()𝑄,𝑘+2𝛾+𝜎}⩽𝐶^ℎ_𝑘=0𝑓_𝑡𝑘²_{𝐿2𝑄,𝑘𝛾0+𝜎}, (23) where 𝐶 is a constant independent of 𝑢 and 𝑓.

The proofs of these theorems will be given in Section 4. The number 𝛾₀ will be defined by formula (66). It is natural that this number should be chosen as small as possible.

The remainder of this section is devoted to construct the theorem on the asymptotic behaviour of the solution near the conical point.

Let 𝑟=|𝑥| and 𝜔 be an arbitrary local coordinate system on 𝑆^𝑛−1. Let 𝛿 be a positive real constant. A differential operator 𝑃𝑥,𝑡,𝜕_𝑥=_{|𝛼|⩽𝑙}𝑝_𝛼(𝑥,𝑡)𝜕^𝛼_𝑥, (24) is called 𝛿-admissible operator of order 𝑙 near the conical point 0 if the coefficients 𝑝_𝛼(𝑥,𝑡) are infinitely differentiable in 𝑄 and there is representation in a neighborhood 𝑈 of the conical point 0: 𝑝_𝛼(𝑥,𝑡)=𝑟^{|𝛼|−𝑙}𝑝_𝛼⁽⁰⁾(𝜔,𝑡)+𝑟^{|𝛼|−𝑙+𝛿}𝑝_𝛼⁽¹⁾(𝜔,𝑟,𝑡), (25) where 𝑝_𝛼⁽⁰⁾∈𝐶^∞(Ω×ℝ₊) and the functions 𝑝_𝛼⁽¹⁾ are infinitely differentiable functions in Ω×ℝ₊×ℝ₊ such that 𝑟𝜕_𝑟^𝑗𝜕^𝛽_𝜔𝜕^𝑘_𝑡𝑝_𝛼⁽¹⁾(𝜔,𝑟,𝑡)<𝑐_𝑗𝑘𝛽, (26) for every multiindex 𝛽 and every pair of nonnegative integers 𝑗,𝑘. Here the constants 𝑐_𝑗𝑘𝛽 do not depend on 𝜔,𝑟 and 𝑡. The leading part 𝒫 of the operator 𝑃 at the point 0 is defined by 𝒫𝑥,𝑡,𝜕_𝑥=_{|𝛼|⩽𝑙}𝑟^{|𝛼|−𝑙}𝑝_𝛼⁽⁰⁾(𝜔,𝑡)𝜕^𝛼_𝑥. (27) It can be directly verified that the derivative 𝜕^𝛼_𝑥 has the form 𝜕^𝛼_𝑥=𝑟^{−|𝛼||𝛼|}_𝑝=0𝑝_𝛼𝑗𝜔,𝜕_𝜔(𝑟𝜕𝑟)^𝑗, (28) where 𝑝_𝛼𝑗(𝜔,𝜕_𝜔) are differential operators of order ⩽|𝛼|−𝑗 with smooth coefficients on Ω. Thus the operator 𝒫(𝑥,𝑡,𝜕_𝑥) can be represented as 𝒫𝑥,𝑡,𝜕_𝑥=𝑟^𝑙−𝑙_𝑗=0𝑝_𝑗𝜔,𝑡,𝜕_𝜔𝑟𝜕_𝑟^𝑗. (29)

For convenience we rewrite the operator 𝐿(𝑥,𝑡,𝜕_𝑥) in the form 𝐿=𝐿𝑥,𝑡,𝜕_𝑥=_{|𝛼|⩽2𝑚}𝑎_𝛼(𝑥,𝑡)𝜕^𝛼_𝑥. (30) Let ℒ=ℒ(𝑥,𝑡,𝜕_𝑥),ℬ_𝑗=ℬ_𝑗(𝑥,𝑡,𝜕_𝑥)(𝑗=1,…,𝑚) be the leading parts of 𝐿(𝑥,𝑡,𝜕_𝑥), 𝐵_𝑗(𝑥,𝑡,𝜕_𝑥) at the point 𝑥=0. Since the coefficients of the operators 𝐿(𝑥,𝑡,𝜕_𝑥) and 𝐵_𝑗(𝑥,𝑡,𝜕_𝑥) are smooth, it is verified easily that ℒ𝑥,𝑡,𝜕_𝑥=_|𝛼|=2𝑚𝑎_𝛼(0,𝑡)𝜕^𝛼_𝑥,ℬ_𝑗𝑥,𝑡,𝜕_𝑥=_{|𝛼|=𝜇𝑗}𝑏_𝑗,𝛼(0,𝑡)𝜕^𝛼_𝑥, (31) and the operators 𝐿(𝑥,𝑡,𝜕_𝑥) and 𝐵_𝑗(𝑥,𝑡,𝜕_𝑥) are 1-admissible. Rewrite ℒ(𝑥,𝑡,𝜕_𝑥),ℬ_𝑗(𝑥,𝑡,𝜕_𝑥) in the form ℒ𝑥,𝑡,𝜕_𝑥=𝑟^−2𝑚𝓛𝜔,𝑡,𝜕_𝜔,𝑟𝜕_𝑟,ℬ_𝑗𝑥,𝑡,𝜕_𝑥=𝑟^−𝜇𝑗𝓑_𝑗𝜔,𝑡,𝜕_𝜔,𝑟𝜕_𝑟. (32) We introduce the operator 𝓤()=𝓛𝜆,𝑡𝜔,𝑡,𝜕_𝜔,𝜆,𝓑₁𝜔,𝑡,𝜕_𝜔,𝜆,…,𝓑_𝑚𝜔,𝑡,𝜕_𝜔,𝜆, (33) (𝜆∈ℂ,𝑡∈ℝ₊) of the parameter-depending elliptic boundary value problem 𝓛𝜔,𝑡,𝜕_𝜔𝓑,𝜆𝑢=𝑓inΩ,_𝑗𝜔,𝑡,𝜕_𝜔,𝜆𝑢=𝑔_𝑗on𝜕Ω,𝑗=1,…,𝑚. (34) This is a pencil of continuous operators from 𝒳≡𝑊^𝑙₂(Ω)into𝓎≡𝑊₂^𝑙−2𝑚(Ω)×^𝑚_𝑗=1𝑊^{𝑙−𝜇𝑗}₂^−(1/2)(𝜕Ω)(𝑙⩾2𝑚), (35) depending polynomially on 𝜆∈ℂ.

We mention now some well-known definitions [3]. Let 𝑡₀∈ℝ₊ fixed. If 𝜆₀∈ℂ, 𝜑₀∈𝒳 such that 𝜑₀≠0,𝓤(𝜆₀,𝑡₀)𝜑₀=0, then 𝜆₀ is called an eigenvalue of 𝓤(𝜆,𝑡₀) and 𝜑₀∈𝒳 is called an eigenvector corresponding to 𝜆₀. Λ=dimker𝓤(𝜆₀,𝑡₀) is called the geometric multiplicity of the eigenvalue 𝜆₀.

If the elements 𝜑₁,…,𝜑_𝑠 of 𝒳 satisfy the equations ^𝜎_𝑞=01𝜕𝑞!𝑑𝜆^𝛽𝑞𝓤𝜆,𝑡₀∣_𝜆=𝜆0𝜑_𝜎−𝑞=0for𝜎=1,…,𝑠, (36) then the ordered collection 𝜑₀,𝜑₁,…,𝜑_𝑠 is said to be a Jordan chain corresponding to the eigenvalue 𝜆₀ of the length 𝑠+1. The rank of the eigenvector 𝜑₀ (rank𝜑₀) is the maximal length of the Jordan chains corresponding to the eigenvector 𝜑₀.

A canonical system of eigenvectors of 𝓤(𝜆₀,𝑡₀) corresponding to the eigenvalue 𝜆₀ is a system of eigenvectors 𝜑_1,0,…,𝜑_Λ,0 such that rank𝜑_1,0 is maximal among the rank of all eigenvectors corresponding to 𝜆₀ and rank𝜑_𝑗,0 is maximal among the rank of all eigenvectors in any direct complement in ker𝓤(𝜆₀,𝑡₀) to the linear span of the vectors 𝜑_1,0,…,𝜑_𝑗−1,0(𝑗=2,…,Λ). The numbers 𝜅_𝑗=rank𝜑_𝑗,0(𝑗=1,…,Λ) are called the partial multiplicities and the sum 𝜅=𝜅₁+⋯+𝜅_Λ is called the algebraic multiplicity of the eigenvalue 𝜆₀.

The eigenvalue of 𝜆₀ is called simple if its algebraic multiplicity is equal to one.

For each fixed 𝑡∈ℝ₊ the set of all complex number 𝜆 such that 𝓤(𝜆,𝑡) is not invertible is called the spectrum of 𝓤(𝜆,𝑡). It is known that the spectrum of 𝓤(𝜆,𝑡) is an enumerable set of its eigenvalues (see [3, Theorem 5.2.1]). Moreover, there are constants 𝛿,𝑅 such that 𝓤(𝜆,𝑡) is invertible for all 𝑡∈ℝ₊ and all 𝜆 in the set 𝒟∶={𝜆∈ℂ∶|Re𝜆|⩽𝛿|Im𝜆|,|𝜆|⩾𝑅}, (37) (see [3, Theorem 3.6.1]).

To receive asymptotic formulas of the solutions with the coefficients regular with respect to the variable 𝑡 we require later that eigenvalues and eigenvectors of the pencil 𝓤(𝜆,𝑡) satisfy the following assumption.

Let 𝑙₁,𝑙₂ be nonnegative integers, and let 𝛽₁,𝛽₂ be real numbers such that 𝑙₁−𝛽₁<𝑙₂−𝛽₂. We say that the assumption (𝐇) for numbers 𝑙₁,𝑙₂,𝛽₁,𝛽₂ is fulfilled if the following conditions are satisfied.(i) The lines Re𝜆=−𝛽_𝑖+𝑙_𝑖−𝑛/2(𝑖=1,2) do not contain eigenvalues of the pencil 𝓤(𝜆,𝑡), and the strip −𝛽₁+𝑙₁−𝑛/2<Re𝜆<−𝛽₂+𝑙₂−𝑛/2 contains the eigenvalues 𝜆_𝜇(𝑡),𝜇=1,…,𝑁, with the geometric multiplicities Λ_𝜇 and the partial multiplicities 𝜅_𝜇,𝑘, 𝜇=1,…,𝑁,𝑘=1,…,Λ_𝜇, not depending on 𝑡∈ℝ₊. These eigenvalues are smooth functions on ℝ₊.(ii) A canonical system 𝜑^(𝜇)_𝑘,𝑠(𝜔,𝑡),𝑘=1,…,Λ_𝜇,𝑠=1,…,𝜅_𝜇,𝑘, (38) of Jordan chains of 𝓤(𝜆,𝑡) corresponding to the eigenvalue 𝜆_𝜇(𝑡) (𝜇∈{1,…,𝑁}) can be chosen, which consists of functions that are smooth for 𝑡∈ℝ₊ for all 𝜔∈Ω.

Theorem 4. Let ℎ be a positive integer. Let 𝜖,𝛽 be real numbers such that 0⩽𝜖<1/2, 0<𝛽⩽𝑚 and 𝜖+𝑛/2, 𝛽+𝑛/2 are not integers in the case 𝑚⩾𝑛/2. Suppose that the assumptions of Theorem 3 and the assumption (𝐇) for 𝑙₁=𝑙₂=2𝑚, 𝛽₁=𝑚+𝜖, 𝛽₂=𝛽 are fulfilled. Let 𝑢∈𝐻^𝑚,1(𝑄,𝛾)(𝛾>𝛾₀) be the generalized solution of the problem (8)-(9). Suppose further that the operators 𝐿(𝑥,𝑡,𝜕_𝑥) and 𝐵_𝑗(𝑥,𝑡,𝜕_𝑥) are (𝑚+𝜖−𝛽)- admissible near the origin 0. Then,(i) for the case 𝑚<𝑛/2, the solution 𝑢 admits the decomposition 𝑢(𝑥,𝑡)=^𝑁Λ_𝜇=1^𝜇𝜅_𝑘=1^{𝜇,𝑘−1}_𝑠=0𝑐_{𝜇,𝑘,𝑠}(𝑡)𝑟^{𝜆𝜇(𝑡)}×^𝑠_𝜎=01()𝜎!ln𝑟^𝜎𝜑^(𝜇)_𝑘,𝑠(𝜔,𝑡)+𝑤(𝑥,𝑡), (39) where 𝑤, 𝑐_{𝜇,𝑘,𝑠} are functions satisfying 𝑤_𝑡𝑝∈𝑉^2𝑚,0_2,𝛽(𝑄,(𝑝+2)𝛾+𝜎), (𝑐_{𝜇,𝑘,𝑠})_𝑡𝑝∈𝐿₂(ℝ^∗₊,(𝑝+2)𝛾+𝜎) for 𝑝=0,…,ℎ,(ii) for the case 𝑚⩾𝑛/2, we assume further that if 𝜆_𝑘(𝑡)=𝑞 is integer for some 𝑡∈ℝ₊ and some 𝑘,1⩽𝑘⩽𝑁, then 𝜆_𝑘(𝑡)=𝑞 for all 𝑡∈ℝ₊; then the solution 𝑢 admits the decomposition 𝑢(𝑥,𝑡)=_{|𝛼|⩽2𝑚−ℓ−1}𝑐_𝛼(𝑡)𝑥^𝛼+^{2𝑚−ℓ1−1}_{𝑘=2𝑚−ℓ}𝑟^𝑘𝜅𝑘_𝑠=01()𝑠!ln𝑟^𝑠𝜑_𝑘,𝑠()+𝜔,𝑡_{𝑁Λ𝜇=1𝜇𝜅𝑘=1𝜇,𝑘−1𝑠=0}𝑐_{𝜇,𝑘,𝑠}(𝑡)𝑟^{𝜆𝜇(𝑡)}×^𝑠_𝜎=01()𝜎!ln𝑟^𝜎𝜑^(𝜇)_𝑘,𝑠(𝜔,𝑡)+𝑤(𝑥,𝑡), (40) where 𝑤,𝑐_𝛼,𝑐_{𝜇,𝑘,𝑠} are functions satisfying 𝑤_𝑡𝑝∈𝑉^2𝑚,0_2,𝛽(𝑄,(𝑝+2)𝛾+𝜎), (𝑐_𝛼)_𝑡𝑝,(𝑐_{𝜇,𝑘,𝑠})_𝑡𝑝∈𝐿₂(ℝ^∗₊,(𝑝+2)𝛾+𝜎), for 𝑝=0,…,ℎ, and ℓ₁ is the integer with ℓ₁−𝑛/2<𝛽<ℓ₁+1−𝑛/2, 𝜅_𝑘=0 if 𝑘 is not an eigenvalue of 𝓤(𝜆,𝑡); otherwise 𝜅_𝑘 is the maximal partial multiplicity of the eigenvalue 𝑘.

4. Proofs of Theorems 2 and 3

First, let us introduce some more notations. For functions 𝑢,𝑣 defined in 𝐺 and 𝑘∈ℕ, we set |𝑢|_𝑘,𝐺=_{𝐺|𝛼|=𝑘}𝜕^𝛼𝑢²𝑑𝑥^1/2,(𝑢,𝑣)_𝐺=_𝐺𝑢𝑣𝑑𝑥. (41) For functions 𝑢 and 𝑣 defined in 𝑄, we set |𝑢|_{𝑘,𝑄𝜏}=^𝜏₀|𝑢(𝑡)|²_𝑘,𝐺𝑑𝑡^1/2,𝐵_𝑡(𝑡,𝑢,𝑣)=_{|𝛼|,|𝛽|⩽𝑚𝐺}𝜕𝑎_𝛼𝛽(𝑡)𝜕𝜕𝑡^𝛽𝑢(𝑡)𝜕^𝛼𝑣(𝑡)𝐵𝑑𝑥,_𝜏𝑡(𝑢,𝑣)=^𝜏₀𝐵_𝑡(𝑡,𝑢,𝑣)𝑑𝑡. (42) Here and hereafter, we use 𝑢(𝑡) instead of 𝑢(⋅,𝑡) for the shortness.

To prove Theorem 2, it is needed to introduce the Gronwall-Bellman and interpolation inequalities stated in the following lemmas.

Lemma 5 (see [14, Lemma 3.1]). Assume 𝑢,𝛼,𝛽 are real-valued continuous on an interval [𝑎,𝑏], 𝛽 is nonnegative and integrable on [𝑎,𝑏] and 𝛼 is nondecreasing satisfying 𝑢(𝜏)⩽𝛼(𝜏)+^𝜏_𝑎𝛽(𝑡)𝑢(𝑡)𝑑𝑡∀𝑎⩽𝜏⩽𝑏. (43) Then 𝑢(𝜏)⩽𝛼(𝜏)exp^𝜏_𝑎𝛽(𝑡)𝑑𝑡∀𝑎⩽𝜏⩽𝑏. (44)

Lemma 6 (see [15] Theorem4.14). For each positive real number 𝜖 and each integer 𝑗, 0<𝑗<𝑚, there exists a positive real number 𝐶=𝐶(𝐺,𝑚,𝜖) which is dependent only on 𝐺, 𝑚, and 𝜖 such that the inequality |𝑢|²_𝑗,𝐺⩽𝜖|𝑢|²_𝑚,𝐺+𝐶|𝑢|²_0,𝐺, (45) holds for all 𝑢∈𝐻^𝑚(𝐺).

Proof of Theorem 2. The theorem is proved by repeating almost word for word the proof of Theorem 3.3 of [7]. Here we present the proof of the existence to show that the restriction of negativeness of 𝜎 in that theorem can be omitted.
Let {𝜑_𝑘}^∞_𝑘=1 be a basis of 𝐻^𝑚_𝐵(𝐺) which is orthonormal in 𝐿₂(𝐺). Put 𝑢^𝑁(𝑥,𝑡)=^𝑁_𝑘=1𝑐^𝑁_𝑘(𝑡)𝜑_𝑘(𝑥), (46) where (𝑐^𝑁_𝑘(𝑡))^𝑁_𝑘=1 are the solution of the system of the following ordinary differential equations of second order: 𝑢^𝑁_𝑡𝑡(𝑡),𝜑_𝑙𝐺+𝐵𝑡,𝑢^𝑁,𝜑_𝑙=𝑓(𝑡),𝜑_𝑙𝐺,𝑙=1,…,𝑁, (47) with the initial conditions 𝑐^𝑁_𝑘(0)=𝑑𝑐^𝑁_𝑘(0)𝑑𝑡=0,𝑘=1,…,𝑁. (48) Let us multiply (47) by (𝑑/𝑑𝑡)𝑐^𝑁_𝑘(𝑡). Take the sum with respect to 𝑙 from 1 to 𝑁 to receive 𝑢^𝑁_𝑡𝑡(𝑡),𝑢^𝑁_𝑡(𝑡)_𝐺+𝐵𝜏,𝑢^𝑁,𝑢^𝑁_𝑡=𝑓(𝑡),𝑢^𝑁_𝑡(𝑡)_𝐺. (49) Now adding this equality to its complex conjugate with noting that 𝐵(𝜏,𝑢,𝑣)=𝐵(𝜏,𝑣,𝑢), (50) by the formally self-adjointness of the operator 𝐿, then integrating the obtained equality with respect to 𝑡 from 0 to 𝜏 with using the integration by parts and (48), we arrive at 𝑢^𝑁_𝑡(𝜏)²_0,𝐺+𝐵𝜏,𝑢^𝑁,𝑢^𝑁=𝐵^𝜏_𝑡𝑢^𝑁,𝑢^𝑁+2Re𝑓,𝑢^𝑁_𝑡𝑄𝜏. (51) Noting that 𝑢^𝑁(𝜏)²_0,𝐺=2Re𝑢^𝑁,𝑢^𝑁_𝑡𝑄𝜏, (52) we have from (51) that 𝑢^𝑁_𝑡(𝜏)²_0,𝐺+𝐵₀𝜏,𝑢^𝑁,𝑢^𝑁+𝜌|𝑢(𝜏)|²_0,𝐺=𝐵^𝜏_𝑡𝑢^𝑁,𝑢^𝑁+2𝜌Re𝑢^𝑁,𝑢^𝑁_𝑡𝑄𝜏−_{|𝛼|,|𝛽|⩽𝑚|𝛼|+|𝛽|<2𝑚−1𝐺}𝑎_𝛼𝛽(𝜏)𝜕^𝛽𝑢^𝑁(𝜏)𝜕^𝛼𝑢^𝑁(𝜏)𝑑𝑥+2Re𝑓,𝑢^𝑁_𝑡𝑄𝜏. (53)
Now we give estimations for the terms of (53). Firstly, by (2), we see that the left-hand side of (53) is greater than 𝑢^𝑁_𝑡(𝜏)²_0,𝐺+𝜇₀𝑢^𝑁(𝜏)²_𝑚,𝐺+𝜌|𝑢(𝜏)|²_0,𝐺. (54) We write 𝐵^𝜏_𝑡 as the sum of the two following terms: 𝐼₁=_{|𝛼|=|𝛽|=𝑚𝑄𝜏}𝜕𝑎_𝛼𝛽𝜕𝜕𝑡^𝛽𝑢^𝑁𝜕^𝛼𝑢^𝑁𝐼𝑑𝑥𝑑𝑡,₂=_{|𝛼|,|𝛽|⩽𝑚|𝛼|+|𝛽|⩽2𝑚−1𝑄𝜏}𝜕𝑎_𝛼𝛽𝜕𝜕𝑡^𝛽𝑢^𝑁𝜕^𝛼𝑢^𝑁𝑑𝑥𝑑𝑡. (55) Put 𝜇₁=sup𝜕𝑎_𝛼𝛽(∶|𝛼|=𝛽()∈𝜕𝑡𝑥,𝑡)=𝑚,𝑥,𝑡𝑄,𝑚^′=_|𝛼|=𝑚1. (56) Then, by the Cauchy inequality, we have 𝐼₁⩽𝜇_{1|𝛼|=|𝛽|=𝑚}12𝜕^𝛽𝑢^𝑁2_0,𝑄𝜏+𝜕^𝛼𝑢^𝑁2_0,𝑄𝜏⩽𝑚^′𝜇₁𝑢^𝑁2_{𝑚,𝑄𝜏}. (57) By the Cauchy inequality and the interpolation inequality (45), for an arbitrary positive number 𝜖₁, we have 𝐼₂⩽𝜖₁𝑢^𝑁2_{𝑚,𝑄𝜏}+𝐶₁𝑢^𝑁2_0,𝑄𝜏, (58) where 𝐶₁=𝐶₁(𝜖₁) is a nonnegative constant independent of 𝑢^𝑁, 𝑓, and 𝜏. Now using again the Cauchy and interpolation inequalities, for an arbitrary positive number 𝜖₂ with 𝜖₂<𝜇₀, it holds that _{|𝛼|,|𝛽|⩽𝑚|𝛼|+|𝛽|<2𝑚−1𝐺}𝑎_𝛼𝛽𝜕^𝛽𝑢^𝑁𝜕^𝛼𝑢^𝑁𝑑𝑥⩽𝜖₂𝑢^𝑁(𝜏)²_𝑚,𝐺+𝐶₂𝑢^𝑁(𝜏)²_0,𝐺, (59) where 𝐶₂=𝐶₂(𝜖₂) is a nonnegative constant independent of 𝑢^𝑁, 𝑓, and 𝜏. Also by Cauchy inequality we have 2𝜌Re𝑢^𝑁,𝑢^𝑁_𝑡𝑄𝜏⩽𝜇₀−𝜖₂𝜌²𝑚^′𝜇₁+𝜖₁𝑢^𝑁2_0,𝑄𝜏+𝑚^′𝜇₁+𝜖₁𝜇₀−𝜖₂𝑢^𝑁_𝑡²_0,𝑄𝜏. (60)
Now, to deal with the last term of (53), let us consider first the case 𝜎⩾0. In this case, we use the following inequality: 2Re𝑓,𝑢^𝑁_𝑡𝑄𝜏⩽𝜖₃𝑢^𝑁_𝑡²_0,𝑄𝜏+1𝜖₃𝑓²_0,𝑄𝜏, (61) where 𝜖₃>0 arbitrary. Combining the above estimations we get from (53) that 𝑢^𝑁_𝑡(𝜏)²_0,𝐺+𝜇₀−𝜖₂𝑢^𝑁(𝜏)²_𝑚,𝐺+𝜌−𝐶₂𝑢^𝑁(𝜏)²_0,𝐺⩽𝑚^′𝜇₁+𝜖₁𝑢^𝑁2_{𝑚,𝑄𝜏}+𝐶₁+𝜇₀−𝜖₂𝜌²𝑚^′𝜇₁+𝜖₁𝑢^𝑁2_0,𝑄𝜏+𝑚^′𝜇₁+𝜖₁𝜇₀−𝜖₂+𝜖₃𝑢^𝑁_𝑡²_0,𝑄𝜏+1𝜖₃𝑓²_0,𝑄𝜏. (62) Now fix 𝜖₁, 𝜖₂ and consider the function 𝑔𝜌=𝐶₁+𝜇₀−𝜖₂𝜌²/𝑚^′𝜇₁+𝜖₁𝜌−𝐶₂for𝜌>𝐶₂. (63) We see that 𝑑𝑔=𝑑𝜌𝐴𝜌²−2𝐴𝐶₂𝜌−𝐶₁𝜌−𝐶₂²𝜇with𝐴=⁰−𝜖₂𝑚^′𝜇₁+𝜖₁. (64) We see that the function 𝑔 has a unique minimum at 𝜌₀=𝜌₀𝜖₁,𝜖₂=𝐶₂+𝐶²₂+𝐶₁𝐴>𝐶₂. (65) Let us denote 𝛾₀=12inf_{𝜖1>00<𝜖2<𝜇0}𝑚max^′𝜇₁+𝜖₁𝜇₀−𝜖₂,𝑔𝜌₀. (66) Now we take real numbers 𝛾, 𝛾₁ arbitrarily satisfying 𝛾₀<𝛾₁<𝛾. Then there are positive real numbers 𝜖₁, 𝜖₂ (𝜖₂<𝜇₀), 𝜌 (𝜌>𝐶₂(𝜖₁,𝜖₂)), and 𝜖₃ such that 𝑚^′𝜇₁+𝜖₁𝜇₀−𝜖₂+𝜖₃<2𝛾₁,𝐶₁𝜖₁,𝜖₂+𝜇₀−𝜖₂𝜌²/𝑚^′𝜇₁+𝜖₁𝜌−𝐶₂𝜖₁,𝜖₂<2𝛾₁. (67) From now to the end of the present proof, we fix such constants 𝜖₁, 𝜖₂, 𝜖₃, and 𝜌 and denote by |‖𝑢^𝑁(𝜏)‖|²_𝐺 the left-hand side of (62). It follows from (62) and (67) that 𝑢^𝑁(𝜏)²_𝐺⩽2𝛾₁^𝜏₀|‖𝑢(𝑡)‖|²_𝐺+1𝑑𝑡𝜖₃^𝜏₀𝑓(𝑡)²_0,𝐺𝑑𝑡∀𝜏⩾0. (68) By the Gronwall-Bellman inequality (44), we deduce from (68) that 𝑢^𝑁(𝜏)²_𝐺⩽1𝜖₃𝑒^{2𝛾1𝜏𝜏}₀𝑓(𝑡)²_0,𝐺⩽1𝑑𝑡𝜖₃𝑒^{2(𝛾1+𝜎)𝜏}𝑓²_{𝐿2()𝑄,𝜎}, (69) for all 𝜏⩾0. Here we used the fact that, for 𝜎⩾0, ^𝜏₀𝑓(⋅,𝑡)²_0,𝐺𝑑𝑡=𝑒^{2𝜎𝜏𝜏}₀𝑒^−𝜎𝜏𝑓(⋅,𝑡)²_0,𝐺𝑑𝑡⩽𝑒^2𝜎𝜏𝑓²_{𝐿2()𝑄,𝜎}. (70) In the case of 𝜎<0, instead of (61), we give the following inequality 2Re𝑓,𝑢^𝑁_𝑡𝑄𝜏=2Re𝑒^−𝜎𝑡𝑓,𝑒^𝜎𝑡𝑢^𝑁_𝑡𝑄𝜏⩽𝜖₃𝑒^𝜎𝑡𝑢^𝑁_𝑡²_0,𝑄𝜏+1𝜖₃𝑒^−𝜎𝑡𝑓²_0,𝑄𝜏⩽𝜖₃𝑢^𝑁_𝑡²_0,𝑄𝜏+1𝜖₃𝑒^−𝜎𝑡𝑓²_0,𝑄𝜏∀𝜏⩾0. (71) Thus, by repeating the above arguments we receive (68) with the last term replaced by the last term of (71), and, therefore, we also get (69) for every real number 𝜎.
Now multiplying both sides of this inequality by 𝑒^{−2(𝛾+𝜎)𝜏}, then integrating them with respect to 𝜏 from 0 to ∞, we arrive at 𝑢^𝑁2_{𝑄,𝛾+𝜎}⩽1𝜖₃𝑓²_{𝐿2()𝑄,𝜎}, (72) where we used the notation 𝑢^𝑁2_{𝑄,𝛾+𝜎}=^∞₀𝑒^{−2(𝛾+𝜎)𝜏}𝑢^𝑁(𝜏)²_𝐺𝑑𝜏. (73) It is clear that |‖⋅‖|_{𝑄,𝛾+𝜎} is a norm in 𝐻_𝐵^𝑚,1(𝑄,𝛾+𝜎) which is equivalent to the norm ‖⋅‖_𝐻𝐵^𝑚,1_{(𝑄,𝛾+𝜎)}. Thus, it follows from (72) that 𝑢^𝑁2_{𝐻𝐵𝑚,1(𝑄,𝛾+𝜎)}⩽𝐶𝑓²_{𝐿2()𝑄,𝜎}, (74) where 𝐶 is a constant independent of 𝑓 and 𝑁.
From the inequality (74), by standard weak convergence arguments (see, e.g., [16, Ch. 7]), we can conclude that the sequence {𝑢^𝑁}^∞_𝑁=1 possesses a subsequence convergent to a function 𝑢∈𝐻_𝐵^𝑚,1(𝑄,𝛾+𝜎) which is a generalized solution of problem (8)-(9). Moreover, it follows from (74) that the inequality (22) holds.

Now we are going to prove Theorem 3. First, we give some needed auxiliary lemmas. The first lemma deals with the regularity of the solution with respect to time variable. It is proved by repeating almost word for word the proof of Theorem 3.4 of [7] with noting that, as in Theorem 2, the assumption 𝜎⩾0 in [7] can be removed.

Lemma 7. Suppose that all the assumptions of Theorem 3 are fulfilled. Then 𝑢_𝑡𝑘∈𝐻_𝐵^𝑚,1(𝑄,(𝑘+1)𝛾+𝜎) for 𝑘=0,1,…,ℎ and ^ℎ_𝑘=0𝑢_𝑡𝑘²_𝐻𝐵^𝑚,1_{()𝑄,𝑘+1𝛾+𝜎}≤𝐶^ℎ_𝑘=0𝑓_𝑡𝑘²_{𝐿2𝑄,𝑘𝛾0+𝜎}, (75) where 𝐶 is a constant independent of 𝑢 and 𝑓.

From the proof of [12, Lemma 5.3] we have the following lemma.

Lemma 8. Let 𝑓∈𝐻⁰_𝑚(𝐺), 𝑔_𝑗∈𝐻^{2𝑚−𝜇𝑗}_𝑚^−1/2(Γ), 𝑗=1,…,𝑚, and 𝑡₀∈ℝ^∗₊. Suppose that 𝑢∈𝐻^𝑚_𝐵(𝐺)∩𝐻^2𝑚_loc(𝐺⧵{0}) is a solution of the following problem: 𝐿𝑥,𝑡₀𝐵,𝜕𝑢=𝑓𝑖𝑛𝐺,_𝑗𝑥,𝑡₀,𝜕𝑢=𝑔_𝑗𝑜𝑛Γ,𝑗=1,…,𝑚. (76) Then 𝑢∈𝐻_𝑚^2𝑚(𝐺) and ‖𝑢‖²_{𝐻𝑚2𝑚(𝐺)}⩽𝐶𝑓²_{𝐻0𝑚(𝐺)}+^𝑚_𝑗=1𝑔_𝑗²_𝐻𝑗𝑚^{2𝑚−𝜇−1/2}_(Γ)+‖𝑢‖²_𝐻^𝑚_𝐵(𝐺), (77) where the constant 𝐶 is independent of 𝑢,𝑓, and 𝑡₀.

Proof of Theorem 3. According to Lemma 7 we have 𝑢_𝑡𝑘∈𝐻_𝐵^𝑚,1𝑄,(𝑘+1)𝛾+𝜎,𝑘⩽ℎ. (78) Moreover, as in proof of Theorem 4.1 of [7], we have 𝑢_𝑡𝑘(⋅,𝑡)∈𝐻^2𝑚_loc𝐺⧵{0}fora.e.𝑡∈ℝ^∗₊andall𝑘⩽ℎ−1. (79)
Now we prove the theorem by induction on ℎ. By (78), 𝑢_𝑡𝑡∈𝐿₂(𝑄,2𝛾+𝜎). Thus, from (21) we have 𝐵𝑡,𝑢,𝜂=𝑓(𝑡)−𝑢_𝑡𝑡(𝑡),𝜂, (80) for all 𝜂∈𝐻^𝑚_𝐵(𝐺) and a.e. 𝑡∈ℝ^∗₊. Since 𝑓₁(𝑡)∶=𝑓(𝑡)−𝑢_𝑡𝑡(𝑡)∈𝐿₂(𝐺)⊂𝐻⁰_𝑚(𝐺) for a.e. 𝑡∈ℝ^∗₊, by Lemma 8, we get from (80) that 𝑢(𝑡)∈𝐻_𝑚^2𝑚(𝐺) for a.e. 𝑡∈ℝ^∗₊ and ‖𝑢(𝑡)‖²_{𝐻𝑚2𝑚(𝐺)}⩽𝐶𝑓₁(𝑡)²_𝐿2(𝐺)+‖𝑢(𝑡)‖²_𝐻^𝑚_(𝐺), (81) where 𝐶 is a constant independent of 𝑢,𝑓₁, and 𝑡. Now multiplying both sides of (81) with 𝑒^{−2(2𝛾+𝜎)𝑡}, then integrating with respect to 𝑡 from 0 to ∞ and using estimates from Lemma 7, we obtain ‖𝑢‖²_{𝐻𝑚2𝑚,1𝑄,2𝛾+𝜎}⩽𝐶𝑓²_{𝐿2()𝑄,𝜎}, (82) where 𝐶 is a constant independent of 𝑢 and 𝑓. Hence, the theorem is valid for ℎ=1.
Assume that the theorem is true for some nonnegative ℎ−2. We will prove it for ℎ−1. Differentiating ℎ−1 times both sides of (80) with respect to 𝑡 we have 𝐵𝑡,𝑢_𝑡ℎ−1,𝜂=𝑓_𝑡ℎ−1(𝑡),𝜂−𝑢_𝑡ℎ+1(𝑡)−,𝜂_{ℎ−2𝑘=0}ℎ𝑘𝐵_{𝑡ℎ−𝑘−1}𝑡,𝑢_𝑡𝑘,𝜂, (83) for all 𝜂∈𝐻^𝑚_𝐵(𝐺), a.e. 𝑡∈ℝ^∗₊. From (6) it follows that 𝐵_𝑡𝑘𝑡,𝑣,𝜂=𝐿_𝑡𝑘(𝑥,𝑡,𝜕)𝑣,𝜂, (84) for all 𝑣∈𝐻^𝑚_𝐵(𝐺)∩𝐻_𝑚^2𝑚(𝐺) and 𝜂∈𝐻^𝑚_𝐵(𝐺). Thus, from (83) we deduce 𝐵𝑡,𝑢_𝑡ℎ,𝜂=𝐹_ℎ(⋅,𝑡),𝜂, (85) for all 𝜂∈𝐻^𝑚_𝐵(𝐺) and a.e. 𝑡∈ℝ₊, where 𝐹_ℎ=𝑓_𝑡ℎ(⋅,𝑡)−𝑢_𝑡ℎ+2()−⋅,𝑡_{ℎ−1𝑘=0}ℎ𝑘𝐿_{𝑡ℎ−𝑘}(⋅,𝑡,𝐷)𝑢_𝑡𝑘∈𝐻⁰_𝑚(𝐺). (86) By the induction assumption, it holds that 𝑢_𝑡𝑘∈𝐻_𝑚^2𝑚,1𝑄,(𝑘+2)𝛾+𝜎⊂𝐻_𝑚^2𝑚,1𝑄,(ℎ+1)𝛾+𝜎,𝑘=0,…,ℎ−2. (87) Moreover, 𝑓_𝑡ℎ−1∈𝐿₂𝑄,(ℎ−1)𝛾+𝜎⊂𝐿₂𝑄,(ℎ+1)𝛾+𝜎 (88) by the assumption of the theorem, and 𝑢_𝑡ℎ+1∈𝐿₂𝑄,(ℎ+1)𝛾+𝜎 (89) by Lemma 7. Thus, for a.e. 𝑡∈ℝ^∗₊, we have 𝑓(𝑡)∈𝐻⁰_𝑚(𝐺), 𝑔(𝑡)∈𝐻_𝛼^𝑗−(1/2)(Γ), and 𝑓(𝑡)²_{𝐻0𝑚(𝐺)}⩽𝐶𝑓_𝑡ℎ−1(𝑡)²+𝑢_𝑡ℎ+2(𝑡)²_𝐿2(𝐺)+^ℎ−2_𝑘=0𝑢_𝑡𝑘(𝑡)²_{𝐻𝑚2𝑚(𝐺)}, (90) where 𝐶 is the constant independent of 𝑢, 𝑓 and 𝑡. Now we can repeat the arguments above to conclude that 𝑢_𝑡ℎ−1∈𝐻_𝑚^2𝑚,1(𝑄,(ℎ+1)𝛾+𝜎) with the estimate (23) for 𝑘=ℎ−1. The proof is completed.

5. The Proof of Theorem 4

Let us first give some auxiliary lemmas.

Lemma 9. Let 𝑙 be a nonnegative integer, 𝑙⩾2𝑚, and let 𝛽₁, 𝛽₂ be real numbers, 𝛽₁>𝛽₂. Suppose that the assumption (𝐇) is fulfilled for the numbers 𝑙₁=𝑙₂=𝑙,𝛽₁,𝛽₂. Let ℎ∈ℕ, let 𝛾₀⩽𝛾₁⩽⋯⩽𝛾_ℎ be real numbers, and let 𝑓 and 𝑔_𝑗 be functions satisfying 𝑓_𝑡𝑝∈𝑉^{𝑙−2𝑚,0}_2,𝛽2(𝑄,𝛾_𝑝), (𝑔_𝑗)_𝑡𝑝∈𝑉^{𝑙−𝜇𝑗−(1/2),0}_2,𝛽2(𝑆,𝛾_𝑝) for 𝑝=0,…,ℎ. Suppose that 𝑢∈𝑉^𝑙,0_2,𝛽1(𝑄,𝛾₀) is a solution of the problem ℒ𝑡,𝜕_𝑥ℬ𝑢=𝑓𝑖𝑛𝑄,_𝑗𝑡,𝜕_𝑥𝑢=𝑔_𝑗𝑜𝑛𝑆,𝑗=1,…,𝑚. (91) Then 𝑢 admits the following representation: 𝑢(𝑥,𝑡)=^𝑁Λ_𝜇=1^𝜇𝜅_𝑘=1^{𝜇,𝑘−1}_𝑠=0𝑐_{𝜇,𝑘,𝑠}(𝑡)𝑟^{𝜆𝜇(𝑡)}×^𝑠_𝜎=01()𝜎!ln𝑟^𝜎𝜑^(𝜇)_𝑘,𝑠(𝜔,𝑡)+𝑤(𝑥,𝑡), (92) where 𝑤 and 𝑐_{𝜇,𝑘,𝑠} are functions satisfying 𝑤_𝑡𝑝∈𝑉^𝑙,0_2,𝛽2(𝑄,𝛾_𝑝), (𝑐_{𝜇,𝑘,𝑠})_𝑡𝑝∈𝐿₂(ℝ^∗₊,𝛾_𝑝) for 𝑝=0,…,ℎ.

Proof. From the proof of Lemma 4.5 of [13] (see also [17, Lemma 4.1]), it is known that, for each fixed 𝑡∈ℝ^∗₊, the solution 𝑢(𝑥,𝑡) admits the representation (92) and the following inequality max𝑤_𝑡𝑝(⋅,𝑡)²_{𝑉𝑙22,𝛽(𝐺)},𝑐_{𝜇,𝑘,𝑠𝑡𝑞}(𝑡)⩽𝐶^𝑝_𝑞=0𝑓_𝑡𝑞(⋅,𝑡)²_{𝑉2𝑙−2𝑚2,𝛽(𝐺)}+^𝑚𝑝_𝑗=1𝑞=0𝑔_𝑗𝑡𝑞(⋅,𝑡)²_{𝑉𝑗2𝑙−𝜇−1/22,𝛽(Γ)}, (93) holds for all 𝑡∈ℝ^∗₊, 𝑝=0,1,…,ℎ, where 𝐶 is a constant independent of 𝑢, 𝑓, 𝑔_𝑗, and 𝑡. Now multiplying both sides of (93) by 𝑒^{−2𝛾𝑝𝑡}, then integrating with respect to 𝑡 from 0 to ∞, we see that 𝑤_𝑡𝑝∈𝑉^𝑙2,0_2,𝛽2(𝑄,𝛾_𝑝), (𝑐_{𝜇,𝑘,𝑠})_𝑡𝑝∈𝐿₂(ℝ^∗₊,𝛾_𝑝). The lemma is proved.

By applying Lemma 9 and repeating the arguments in the proof of [13, Lemma4.6], we get the following lemma.

Lemma 10. Suppose that all assumptions of Lemma 9 are fulfilled. Suppose further that the operators 𝐿(𝑥,𝑡,𝜕_𝑥) and 𝐵_𝑗(𝑥,𝑡,𝜕_𝑥) are 𝛿-admissible near the origin 0, where 𝛿=𝛽₁−𝛽₂. Let 𝑢∈𝑉^𝑙1,0_2,𝛽1(𝑄,𝛾₀) be a solution of the problem 𝐿𝑥,𝑡,𝜕_𝑥𝐵𝑢=𝑓𝑖𝑛𝑄,_𝑗𝑥,𝑡,𝜕_𝑥𝑢=𝑔_𝑗𝑜𝑛𝑆,𝑗=1,…,𝑚. (94) Then 𝑢 admits a representation of the form (92).

The following lemma follows directly from the proof of [13, Lemma 4.7].

Lemma 11. Let ℎ∈ℕ, and let 𝛾₀⩽𝛾₁⩽⋯⩽𝛾_ℎ be real numbers. Let 𝑓=𝑟^{𝜆0(𝑡)−2𝑚}𝑔𝑓,_𝑗=𝑟^{𝜆0(𝑡)−𝜇𝑗}𝑔_𝑗,𝑗=1,…,𝑚, (95) where 𝑓 and 𝑔_𝑗 are given functions defined on Ω×ℝ^∗₊, 𝜕Ω×ℝ₊, respectively, satisfying (𝑓)_𝑡𝑝∈𝐿₂(Ω×ℝ^∗₊,𝛾_𝑝),(𝑔_𝑗)_𝑡𝑝∈𝐿₂(𝜕Ω×ℝ₊,𝛾_𝑝), 𝑗=1,…,𝑚, 𝑝=0,…,ℎ. Suppose that if 𝜆₀(𝑡) is an eigenvalue of 𝓤(𝜆,𝑡) for some 𝑡, then it is an eigenvalue of 𝓤(𝜆,𝑡) for all 𝑡∈ℝ₊ with the geometric multiplicity and the partial multiplicities not depending on 𝑡∈ℝ₊. Then there exists a solution 𝑢 of the problem (91) which has the form 𝑢=𝑟^{𝜆0𝜅(𝑡)}_𝜎=01()𝜎!ln𝑟^𝜎𝑢_𝜅−𝜎, (96) where 𝑢_𝜎 are functions defined on Ω×ℝ^∗₊ satisfying (𝑢_𝜎)_𝑡𝑝∈𝐿₂(ℝ^∗₊,𝐻^2𝑚(Ω),𝛾_𝑝),𝜎=0,…,𝜅. Here 𝜅=0 if 𝜆₀(𝑡) is not an eigenvalue of 𝓤(𝜆,t); otherwise 𝜅 is the maximal partial multiplicity of 𝜆₀(𝑡).

Proof of Theorem 4. According to Theorem 3, we have 𝑢_𝑡𝑝∈𝐻_𝑚^2𝑚,1𝑄,(𝑘+2)𝛾+𝜎⊂𝐻^2𝑚,1_𝑚+𝜖𝑄,(𝑘+2)𝛾+𝜎, (97) for 𝑝=0,…,ℎ−1.
Rewrite (8) in the form 𝐿𝑢=𝑓₁∶=𝑓−𝑢_𝑡𝑡𝐵in𝑄,_𝑗𝑢=0on𝑆,𝑗=1,…,𝑚. (98)(99) Since 𝑢_𝑡𝑝+2∈𝐿₂(𝑄,(𝑝+2)𝛾+𝜎) by (75) and 𝑓_𝑡𝑝∈𝐿₂(𝑄,𝑝𝛾+𝜎) by the assumption, we have 𝑓_1𝑡𝑝∈𝐿₂𝑄,𝑝+2𝛾+𝜎≡𝑉^0,0_2,0𝑄,𝑝+2𝛾+𝜎⊂𝑉^0,0_2,𝛽𝑄,𝑝+2𝛾+𝜎, (100) for 𝑝=0,…,ℎ−1.
Now the assertions of the theorem are obtained by applying Lemma 10 and repeating almost word for word the proof of [13, Theorem 4.8].

Acknowledgment

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no. 101.01-2011.30.