Exact Asymptotic Expansion of Singular Solutions for the (𝟐+1)-D Protter Problem

Abstract

We study three-dimensional boundary value problems for the nonhomogeneous wave equation, which are analogues of the Darboux problems in ℝ². In contrast to the planar Darboux problem the three-dimensional version is not well posed, since its homogeneous adjoint problem has an infinite number of classical solutions. On the other hand, it is known that for smooth right-hand side functions there is a uniquely determined generalized solution that may have a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic light cone and does not propagate along the cone. The present paper describes asymptotic expansion of the generalized solutions in negative powers of the distance to this singular point. We derive necessary and sufficient conditions for existence of solutions with a fixed order of singularity and give a priori estimates for the singular solutions.

1. Introduction

In the present paper some boundary value problems (BVPs) formulated by M. H. Protter for the wave equation with two space and one time variables are studied as a multidimensional analogue of the classical Darboux problem in the plane. While the Darboux BVP in ℝ² is well posed the Protter problem is not and its cokernel is infinite dimensional. Therefore the problem is not Fredholm and the orthogonality of the right-hand side function 𝑓 to the cokernel is one necessary condition for existence of classical solution. Alternatively, to avoid infinite number of conditions the notion of generalized solution is introduced that allows the solution to have singularity on a characteristic part of the boundary. It is known that for smooth right-hand side functions there is unique generalized solution and it may have a strong power-type singularity that is isolated at one boundary point. In the present paper we prove asymptotic expansion formula for the generalized solutions in negative powers of the distance to the singular point in the case when 𝑓 is trigonometric polynomial. We leave for the next section the precise formulation of the paper’s main results and the comparisons with recent publications concerning Protter problems, including a semi-Fredholm solvability result in the general case of smooth 𝑓 but for somewhat easier (3+1)-D wave equation problem. First we give here a short historical survey.

Protter arrived at the multidimensional problems for hyperbolic equations while examining BVPs for mixed type equations, starting with planar problems with strong connection to transonic flow phenomena. In the plane, the problems of Tricomi, Frankl, and Guderley-Morawetz are the classical boundary-value problems that appear in hodograph plane for 2D transonic potential flows (see, e.g., the survey of Morawetz [1]). The first two of these problems are relevant to flows in nozzles and jets, and the third problem occurs as an approximation to a respective “exact” boundary-value problem in the study of flows around airfoils. For the Gellerstedt equation of mixed type, Protter [2] proposes a 3D analogue to the two-dimensional Guderley-Morawetz problem. At the same time, he formulates boundary value problems in the hyperbolic part of the domain, which is bounded by two characteristics and one noncharacteristic surfaces of the equation. The planar Guderley-Morawetz mixed-type problem is well studied. Existence of weak solutions and uniqueness of strong solutions in weighted Sobolev spaces were first established by Morawetz by reducing the problem to a first order system which then gives rise to solutions to the scalar equation in the presence of sufficient regularity. The availability of such sufficient regularity follows from the work of Lax and Phillips [3] who also established that the weak solutions of Morawetz are strong. On the other hand, for the 3D Protter mixed-type problems a general understanding of the situation is not at hand—even the question of well posedness is surprisingly subtle and not completely resolved. One has uniqueness results for quasiregular solutions, a class of solutions introduced by Protter, but there are real obstructions to existence in this class. To investigate the situation, we study a simpler problem—the Protter problems in the hyperbolic part Ω of the domain for the mixed-type problem. For the wave equation □𝑢≡𝑢_𝑥1𝑥1+𝑢_𝑥2𝑥2−𝑢_𝑡𝑡=𝑓(𝑥,𝑡), (1.1) this is the set Ω∶=𝑥₁,𝑥₂1,𝑡∶0<𝑡<2,𝑡<𝑥²₁+𝑥²₂<1−𝑡. (1.2) It is bounded, see Figure 1, by two characteristic cones of (1.1) 𝑆₁=𝑥₁,𝑥₂1,𝑡∶0<𝑡<2,𝑥²₁+𝑥²₂,𝑆=1−𝑡₂=𝑥₁,𝑥₂1,𝑡∶0<𝑡<2,𝑥²₁+𝑥²₂=𝑡, (1.3) and the disk 𝑆₀={(𝑥₁,𝑥₂,𝑡)∶𝑡=0,𝑥²₁+𝑥²₂<1}, centered at the origin 𝑂(0,0,0).

Figure 1 

The domain Ω.

One could think of the Protter problems in Ω as three-dimensional variant of the planar Darboux problem. The classic Darboux problem involves a hyperbolic equation in a characteristic triangle bounded by two characteristic and one noncharacteristic segments. The data are prescribed on the noncharacteristic part of the boundary and one of the characteristics. Actually, the set Ω could be produced via rotation around the 𝑡-axis in ℝ³ of the flat triangle Ω₂∶={(𝑥₁,𝑡)∶0<𝑡<1/2;𝑡<𝑥₁<1−𝑡}⊂ℝ²—a characteristic triangle for the corresponding string equation 𝑢_𝑥1𝑥1−𝑢_𝑡𝑡=𝑔𝑥₁,𝑡. (1.4) As mentioned before, the classical Darboux problem for (1.4) is to find solution in Ω₂ with data prescribed on {𝑡=0} and {𝑡=1−𝑥₁}, for example. In conformity with this planar BVP, Protter [2, 4] formulated and studied the following problems.

Problems (P1) and (P2)
Find a solution of the wave equation (1.1) in which Ω satisfies one of the following boundary conditions: 𝑢|_𝑆0=0,𝑢|_𝑆1=0, (P1) or 𝑢_𝑡|_𝑆0=0,𝑢|_𝑆1=0. (P2)

Nowadays, it is known that the Protter Problems (P1) and (P2) are not well posed, in contrast to the planar Darboux problem. In fact, in 1957 Tong [5] proved the existence of infinite number nontrivial classical solutions to the corresponding homogeneous adjoint problem P1^∗. The adjoint BVPs to Problems (P1) and (P2) were also introduced by Protter.

Problems P1^∗ and P2^∗
Find a solution of the wave equation (1.1) in Ω which satisfies the boundary conditions: 𝑢|_𝑆0=0,𝑢|_𝑆2=0(adjointtoProblem(P1)), P1^∗ or 𝑢_𝑡|_𝑆0=0,𝑢|_𝑆2=0(adjointtoProblem(P2)). P2^∗

Since [5], for each of the homogeneous Problems P1^∗ and P2^∗ (i.e., 𝑓≡0 in (1.1)), an infinite number of classical solutions has been found (see Popivanov, Schneider [6], Khe [7]). According to this fact, a necessary condition for classical solvability of Problem (P1) or (P2) is the orthogonality in 𝐿₂(Ω) of the right-hand side function 𝑓(𝑥,𝑡) to all the solutions of the corresponding homogenous adjoint problem P1^∗ or P2^∗. Although Garabedian proved [8] the uniqueness of a classical solution of Problem (P1) (for its analogue in ℝ⁴), generally, Problems (P1) and (P2) are not classically solvable. Instead, Popivanov and Schneider [6] introduced the notion of generalized solution. It allows the solution to have singularity on the inner cone 𝑆₂ and by this the authors avoid the infinite number of necessary conditions in the frame of the classical solvability. In [6] some existence and uniqueness results for the generalized solutions are proved and some singular solutions of Protter Problems (P1) and (P2) are constructed.

In the present paper we study the properties of the generalized solution for Protter Problem (P2) in ℝ³. From the results in [6] it follows that for 𝑛∈ ℕ there exists a smooth right-hand side function 𝑓∈ 𝐶^𝑛(Ω), such that the corresponding unique generalized solution of Problem (P2) has a strong power-type singularity at the origin 𝑂 and behaves like 𝑟^−𝑛(𝑃,𝑂) there. This feature deviates from the conventional belief that such BVPs are classically solvable for very smooth right-hand side functions 𝑓. Another interesting aspect is that the singularity is isolated only at a single point the vertex 𝑂 of the characteristic light cone, and does not propagate along the bicharacteristics which makes this case different from the traditional case of propagation of singularity (see, e.g., Hörmander [9], Chapter 24.5).

The Protter problems have been studied by different authors using various types of techniques like Wiener-Hopf method, special Legendre functions, a priori estimates, nonlocal regularization, and others. For recent known results concerning Protter’s problems see the paper [6] and references therein. For further publications in this area see [7, 10–16]. On the other hand, Bazarbekov gives in Ω another analogue of the classical Darboux problem (see [17]) and analogously in ℝ⁴ (see [18]) in the corresponding four-dimensional domain Ω. Some different statements of Darboux type problems in ℝ³ or connected with them Protter problems for mixed type equations (also studied in [2]) can be found in [19–25]. Some results concerning the nonexistence principle for nontrivial solution of semilinear mixed type equations in multidimensional case, can be found in [26]. For recent existence results concerning closed boundary-value problems for mixed type equations see for example [27], and also [28] that studies an elliptic-hyperbolic equation which arises in models of electromagnetic wave propagation through zero-temperature plasma. The existence of bounded or unbounded solutions for the wave equation in ℝ³ and ℝ⁴, as well as for the Euler-Poisson-Darboux equation has been studied in [7, 13–16, 29].

Further, we aim to find some exact a priori estimates for the singular solutions of Problem (P2) and to outline the exact structure and order of singularity. For some other Protter problems necessary and sufficient conditions for existence of solutions with fixed order of singularity were found (see [15] in ℝ³ and [16] in ℝ⁴) and an asymptotic formula for the solution of Problem (P1) in ℝ⁴ was obtained in [30].

Considering Protter Problems, Popivanov and Schneider [6] proved the existence of singular solutions for both wave and degenerate hyperbolic equation. First a priori estimates for singular solutions of Protter Problems, involving the wave equation in ℝ³, were obtained in [6]. In [10] Aldashev mentioned the results of [6] and, for the case of the wave equation in ℝ^𝑚+1, he notes the existence of solutions in the domain Ω_𝜀 (Ω_𝜀→Ω and 𝑆_2,𝜀 approximates 𝑆₂ if 𝜀→0), which blows up on the cone 𝑆_2,𝜀 like 𝜀^{−(𝑛+𝑚−2)}, when 𝜀→0. It is obvious that for 𝑚=2 this results can be compared to the estimates in Corollary 2.4 here. Finally, we point out that in the case of an equation, which involves the wave operator and nonzero lower terms, Karatoprakliev [24] obtained a priori estimates, but only for the sufficiently smooth solutions of Protter Problem.

Regarding the ill-posedness of the Protter Problems, there have appeared some possible regularization methods in the case of the wave equation, involving either lower order terms ([11, 31]), or some other type perturbations, like integrodifferential term, or nonlocal one ([12]).

In Section 2 the result of the existence of infinite number of classical solutions to the homogeneous Problem P2^∗ (Lemma 2.1) and the definition of generalized solution of Problem (P2) are given. The main results of the paper, concerning the asymptotic expansion of the unique generalized solution 𝑢(𝑥,𝑡) of Problem (P2) (Theorem 2.3) are formulated and discussed. The expansion of 𝑢(𝑃) is given in negative powers of the distance 𝑟(𝑃,𝑂) to the point 𝑂 of singularity. An estimate for the remainder term and the exact behavior of the singularity under the orthogonality conditions imposed on the right-hand side function of the wave equation is found. Necessary and sufficient conditions for the existence of only bounded solutions are given in Corollary 2.4. In Section 3, the auxiliary 2D boundary value Problems (P2.1) and (P2.2), which correspond to the (2+1)-D Problem (P2), are considered. Actually, these 2D problems are transferred to an integral Volterra equation, which is invertible. Using the special Legendre functions 𝑃_𝜈, some exact formulas for the solution of the Problem (P2.2) are derived in Lemma 3.4. Some figures showing the effects appearing near the singularity point are also presented. Section 4 contains the most technical part of the paper. In this section the results concerning the asymptotic expansions of the generalized solution of the 2D Problem (P2.1) are proved and the proof of the main Theorem 2.3 is given.

2. Main Results on (2+1)-D Protter’s Problem 6

Define the functions 𝐸^𝑛_𝑘(𝑥,𝑡)=^𝑘_𝑖=0𝐵^𝑘_𝑖𝑥²₁+𝑥²₂−𝑡^{2𝑛−1/2−𝑘−𝑖}𝑥²₁+𝑥²₂^𝑛−𝑖,𝑛,𝑘∈ℕ∪{0}, (2.1) where the coefficients are 𝐵^𝑘_𝑖∶=(−1)^𝑖(𝑘−𝑖+1)_𝑖(𝑛+1/2−𝑘−𝑖)_𝑖𝑖!(𝑛−𝑖)_𝑖,𝐵^𝑘₀=1, (2.2) with (𝑎)_𝑖∶=𝑎(𝑎+1)⋯(𝑎+𝑖−1),  (𝑎)₀∶=1. Then for the functions 𝑊^𝑛_𝑘,1(𝑥,𝑡)∶=𝐸^𝑛_𝑘(𝑥,𝑡)Re𝑥₁+𝑖𝑥₂^𝑛,𝑊^𝑛_𝑘,2(𝑥,𝑡)∶=𝐸^𝑛_𝑘(𝑥,𝑡)Im𝑥₁+𝑖𝑥₂^𝑛, (2.3) we have the following lemma.

Lemma 2.1 (see [29]). Let 𝑛∈ℕ, 𝑛≥4. For 𝑘=0,…,[(𝑛−3)/2] and 𝑖=1,2 the functions 𝑊^𝑛_𝑘,𝑖(𝑥,𝑡) are classical 𝐶²(Ω)∩𝐶^∞(Ω) solutions to the homogeneous Problem P2^∗.

A necessary condition for the existence of classical solution for Problem (P2) is the orthogonality of the right-hand side function 𝑓 to all functions 𝑊^𝑛_𝑘,𝑖(𝑥,𝑡), which are solutions of the homogeneous adjoint Problem P2^∗. To avoid these infinite number necessary conditions in the framework of classical solvability, one needs to introduce some generalized solutions of Problems (P2) with possible singularities on the characteristic cone 𝑆₂, or only at its vertex 𝑂. Popivanov and Schneider in [6] give the following definition.

Definition 2.2. A function 𝑢=𝑢(𝑥₁,𝑥₂,𝑡) is called a generalized solution of the Problem (P2) in Ω if:(1)𝑢∈𝐶¹(Ω⧵𝑂), 𝑢_𝑡|_𝑆0⧵𝑂=0, 𝑢|_𝑆1=0,(2)the identity _Ω𝑢_𝑡𝑤_𝑡−𝑢_𝑥1𝑤_𝑥1−𝑢_𝑥2𝑤_𝑥2−𝑓𝑤𝑑𝑥₁𝑑𝑥₂𝑑𝑡=0 (2.4)holds for all 𝑤∈𝐶¹(Ω),𝑤_𝑡=0 on 𝑆₀, and 𝑤=0 in a neighborhood of 𝑆₂.

The uniqueness of the generalized solution of Problem (P2) and existence results for 𝑓∈𝐶¹(Ω) can be found in [6].

Further, we fix the right-hand side function 𝑓 as a trigonometric polynomial of order 𝑙 with respect to the polar angle: 𝑓𝑥₁,𝑥₂,𝑡=Re^𝑙_𝑛=2𝑓_𝑛(|𝑥|,𝑡)𝑥₁+𝑖𝑥₂^𝑛, (2.5) with some complex-valued function-coefficients 𝑓_𝑛(|𝑥|,𝑡). For 𝑛=0,…,𝑙;𝑘=0,…,[𝑛/2] and 𝑖=1,2, denote by 𝛽^𝑛_𝑘,𝑖 the constants 𝛽^𝑛_𝑘,𝑖∶=_Ω𝑊^𝑛_𝑘,𝑖(𝑥,𝑡)𝑓(𝑥,𝑡)𝑑𝑥𝑑𝑡. (2.6)

Note that actually 𝛽⁰_𝑘,𝑖=0 and 𝛽¹_𝑘,𝑖=0 in cases of 𝑛=0 and 𝑛=1, due to the special form of the functions 𝑊^𝑛_𝑘,𝑖 and the fact that in the representation (2.5) of the function 𝑓 the sum starts from 𝑛=2.

The main result is as follows.

Theorem 2.3. Suppose that the function 𝑓(𝑥,𝑡)∈𝐶¹(Ω) is a trigonometric polynomial (2.5). Then there exist functions 𝐹^𝑛(𝑥,𝑡), 𝐹^𝑛_𝑘,𝑖(𝑥,𝑡), 𝐹(𝑥,𝑡)∈𝐶²(Ω⧵𝑂) with the following properties:(i)the unique generalized solution 𝑢(𝑥,𝑡) of Problem (P2) exists, belongs to 𝐶²(Ω⧵𝑂) and has the asymptotic expansion at the origin 𝑂: 𝑢(𝑥,𝑡)=^𝑙_𝑚=0|𝑥|²+𝑡^2−𝑚/2𝐹^𝑚(𝑥,𝑡)+|𝑥|²+𝑡^21/4𝐹(𝑥,𝑡)ln|𝑥|²+𝑡², (2.7)(ii)for the coefficient functions 𝐹^𝑚(𝑥,𝑡) the representation 𝐹^𝑚(𝑥,𝑡)=^{[(𝑙−𝑚)/2]2}_𝑘=0𝑖=1𝛽^𝑚+2𝑘_𝑘,𝑖𝐹^𝑚+2𝑘_𝑘,𝑖(𝑥,𝑡),𝑚=0,…,𝑙, (2.8) holds, where the functions 𝐹^𝑛_𝑘,𝑖(𝑥,𝑡) are bounded and independent of 𝑓,(iii)if in the expression (2.8) for 𝐹^𝑚(𝑥,𝑡) at least one of the constants 𝛽^𝑚+2𝑘_𝑘,𝑖 is different from zero (i.e., the corresponding orthogonality condition is not fulfilled), then there exists a direction (𝛼₁,𝛼₂,1) with (𝛼₁,𝛼₂,1)𝑡∈ 𝑆₂ for 0<𝑡<1/2, such that lim_𝑡→+0𝐹^𝑚(𝛼₁𝑡,𝛼₂𝑡,𝑡)=𝑐_𝑚=𝑐𝑜𝑛𝑠𝑡≠0,(iv)if in the expression (2.8) for 𝐹⁰(𝑥,𝑡) at least one of the constants 𝛽^2𝑘_𝑘,𝑖 is different from zero (i.e., the corresponding orthogonality condition is not fulfilled), then the generalized solution is not continuous at 𝑂,(v)for the function 𝐹(𝑥,𝑡) the estimate |𝐹(𝑥,𝑡)|≤𝐶max_Ω𝑓(𝑥,𝑡)+max_Ω𝑓_𝑡(𝑥,𝑡),(𝑥,𝑡)∈Ω, (2.9) holds with a constant 𝐶 independent of 𝑓.

As a consequence of Theorem 2.3 one gets the following results that highlight the two extreme cases of the assertion. The first part gives rough estimate of the expansion (2.7) and describes the “worst” possible singularity. The second part shows that one could control the solution by making some of the defined by (2.6) constants 𝛽^𝑛_𝑘,𝑖 in (2.8) to be zero, that is, by taking 𝑓 to be orthogonal in 𝐿₂(Ω) to the corresponding functions 𝑊^𝑛_𝑘,𝑖 defined in (2.3).

Corollary 2.4. Suppose that 𝑓∈𝐶¹(Ω) has the form (2.5).(i)Without any orthogonality conditions imposed, the unique generalized solution 𝑢 of Problem (P2) satisfies the a priori estimate |𝑢(𝑥,𝑡)|≤𝐶|𝑥|²+𝑡^2−𝑙/2𝑓_𝐶1(Ω),(𝑥,𝑡)∈Ω. (2.10)(ii)Let the orthogonality conditions, 𝛽^𝑛_𝑘,𝑖≡_Ω𝑊^𝑛_𝑘,𝑖(𝑥,𝑡)𝑓(𝑥,𝑡)𝑑𝑥𝑑𝑡=0, (2.11) be fulfilled for all 𝑛=2,…,𝑙;𝑘=0,…,[(𝑛−1)/2] and 𝑖=1,2. Then the generalized solution 𝑢(𝑥,𝑡) belongs to 𝐶²(Ω⧵𝑂), is bounded and the a priori estimate sup_Ω|𝑢|≤𝐶𝑓_𝐶(Ω)+𝑓_𝑡𝐶(Ω) (2.12) holds.(iii)In addition to (ii), if the conditions (2.11) are fulfilled for 𝑘=[𝑛/2] also, then 𝑢∈𝐶(Ω) is a classical solution and 𝑢(𝑂)=0.

Let us point out that in the case (ii), the generalized solution 𝑢 is bounded if and only if the conditions (2.11) are fulfilled for 𝑘≤[(𝑛−1)/2] due to Theorem 2.3(iii). In addition, if all the conditions (2.11) are fulfilled for 𝑘≤[(𝑛−1)/2], but for some 𝑘=[𝑛/2] the corresponding orthogonality condition is not satisfied, then 𝑢 is not continuous at 𝑂, according to Theorem 2.3 (iv). Such a solution is illustrated in Figure 4.

Notice that some of the functions 𝑊^𝑛_𝑘,𝑖(𝑥,𝑡) involved in the orthogonality conditions in Corollary 2.4(ii) and (iii) are not classical solutions of the homogenous adjoint Problem P2^∗ in view of Lemma 2.1, although they satisfy the homogenous wave equation in Ω. In fact, for some 𝑘, 𝑊^𝑛_𝑘,𝑖 or their derivatives may be discontinuous at 𝑆₂. For example when 𝑛 is an odd number and 𝑘=(𝑛−1)/2, the functions 𝑊^𝑛_𝑘,𝑖 are not continuous at the origin 𝑂. On the other hand, when 𝑛 is even and 𝑘=𝑛/2, 𝑊^𝑛_𝑛/2,𝑖 are singular on the cone 𝑆₂ and do not satisfy the homogeneous adjoint boundary condition there. However, this singularity is integrable in the domain Ω.

To explain the results in Theorem 2.3 and Corollary 2.4 we construct Table 1. It illustrates the connection between the singularity of the generalized solution and the functions 𝑊^𝑛_𝑘,𝑖.

Table 1 

The order of singularity of the solution and the functions 𝑊^𝑛_𝑘,𝑖.

Both functions 𝑊^𝑛_𝑘,𝑖, 𝑖=1,2 are located in column number 𝑛 and row number (𝑛−2𝑘) in Table 1. Thus, 𝑊^𝑛_0,𝑖 form the rightmost diagonal, the next one is empty—we put in these cells “diamonds” ⋄, 𝑊^𝑛_1,𝑖 constitute the third one, and so on. The row number designates the order of singularity of the generalized solution.

Corollary 2.4 shows that the generalized solution 𝑢(𝑥,𝑡) is bounded, when the right-hand side function 𝑓 is orthogonal to the functions in Table 1, except the ones in row number 0. If 𝑓 is orthogonal to all the functions in the Table 1 (including the row 0), then 𝑢 is continuous in Ω. When the right-hand side 𝑓 satisfies orthogonal conditions (2.11) for all the functions from the rows in Table 1 with row-number larger then 𝑚, 0<𝑚<𝑙, but there is a function 𝑊^𝑝_𝑞,𝑖 with 𝑝−2𝑞=𝑚 from 𝑚 th row which is not orthogonal to 𝑓 (i.e., 𝛽^𝑝_𝑞,𝑖≠0), then the solution behaves like 𝑟^−𝑚 at the origin, according to the expansion (2.7). If there are no orthogonality conditions, then the worst case with singularity 𝑟^−𝑙 appears.

Figures 2–5 are created using MATLAB and represent some numerical computations for singular solutions of Problem (P2) (actually the behaviour in (𝑟,𝑡)-domain 𝐷₁, not including the terms sin𝑛𝜑 and cos𝑛𝜑). They illustrate different cases according to the main results for the existence of a singularity at the origin 𝑂 depending on orthogonality conditions. Figure 2 is related to Corollary 2.4(i)—it gives the graph of the solution for the worst case without any orthogonality conditions fulfilled and the solution is going to −∞ at the singular point 𝑂. In Figure 3, only one of orthogonality conditions (2.11) for 𝑘≤[(𝑛−1)/2] is not fulfilled and the solution tends to ±∞. Figures 4 and 5 are connected to Corollary 2.4(ii) and (iii): Figure 4 presents the case when all the orthogonality conditions (2.11) for 𝑘≤[(𝑛−1)/2] are satisfied and the solution is bounded but not continuous at (0,0), while Figure 5 concerns the last part (iii) from Corollary 2.4, when conditions (2.11) are additionally fulfilled for 𝑘=[𝑛/2] and the solution is continuous.

Figure 2 

No orthogonality conditions.

Figure 3 

One orthogonality condition is not fulfilled.

Figure 4 

Orthogonality conditions fulfilled for 𝑘≤[(𝑛−1)/2].

Figure 5 

All orthogonality conditions fulfilled for 𝑘≤[𝑛/2].

Remark 2.5. We mention some differences between the results given here for the Problem (P2) and some other results in ℝ³, but for the Problem (P1), like that from [15]. (i)In [15], assuming the right-hand side function 𝑓 is smooth enough (i.e., 𝑓∈𝐶^𝑙) only the behavior of the singularities was studied using some weighted norms (analogous to the weighted Sobolev norms in corner domains). In the present paper we need only 𝑓∈𝐶¹ and find in addition the explicit asymptotic expansion of the generalized solution. The bounded but not continuous at the origin solutions are also studied here. (ii)Comparing the power of singularity of the generalized solution for Problem (P2) here and for Problem (P1) in [15] for the worst case without any orthogonality conditions one can see that the power in the estimate (2.10) from Corollary 2.4(i) is (|𝑥|²+𝑡²)^−𝑙/2, while in the analogous estimate in Conclusion 1 [15] it is (|𝑥|²+𝑡²)^{−(𝑙−1)/2}.(iii)It is interesting to compare the results [14, 15], published in 2002. Going in a different way in both cases the authors asked for singular solutions of Problem (P1) in ℝ³. However, in [14] there are absent any analogues to the orthogonality conditions presented in [15], and in contrast to [15] in [14] the dependence of the exact order of singularity on the data is not clarified.

Remark 2.6. Let us also compare the present expansion and the results in [30], where an asymptotic expansion of Problem (P1) is found for somewhat easier four-dimensional case.(i)Both for Problem (P2) in ℝ³ here and Problem (P1) in ℝ⁴ as in [30], the study is based on the properties of the special Legendre functions. Instead of Legendre functions 𝑃_𝜈 with non-integer indices 𝜈=𝑛−1/2 here, in the four-dimensional case one has to deal with integer indices 𝜈=𝑛, that is, simply with the Legendre polynomials 𝑃_𝑛. One can easily modify both these techniques to obtain similar results for the (𝑚+1)-dimensional problems: for even 𝑚 (analogous to the present case ℝ³) or for odd 𝑚 (similarly to ℝ⁴ case). Some different kind of results for the Protter problems in ℝ^𝑚+1 are presented in [10, 11]. (ii)For the four-dimensional Problem (P1) in [30], the Corollary 3.3 gives only that the solution is bounded, it could be discontinuous at the origin. On the other hand, here Theorem 2.3 gives us also the control over the bounded but not continuous parts of the generalized solution (through the coefficient 𝐹⁰(𝑥,𝑡) for 𝑚=0 in the expansion formula (2.7)). As a sequence, Corollary 2.4(iii) guarantees that the solution is continuous.(iii)Based on the formulae and the computations from [30], the general case in ℝ⁴ is also treated, when the right-hand side 𝑓 is smooth enough, but not a finite harmonic polynomial analogous to (2.5). The results are announced and published in [32, 33]. For right-hand side functions 𝑓∈𝐶¹⁰(Ω) in [33] the necessary and sufficient conditions for the existence of bounded solution are found. They involve infinite number of orthogonality conditions for 𝑓 that comes from the fact that this is not a Fredholm problem. On the other hand, the results from [33] show that the linear operator mapping the generalized solution 𝑢 into 𝑓 is a semi-Fredholm operator in 𝐶¹⁰(Ω). Let us recall that a semi-Fredholm operator is a bounded operator that has a finite dimensional kernel or cokernel and closed range. Additionally, in [32] a right-hand side function is constructed such that the unique generalized solution of Protter Problem (P1) in ℝ⁴ has exponential type singularity. One expects that similar results could also be obtained for the Problem (P2) in ℝ³ studied here. These questions correspond to the Open Problem (1) below.

Remark 2.7. Let us mention one obvious consequence of Theorem 2.3 and all the arguments above, concerning construction of functions orthogonal to the solutions 𝑊^𝑛_𝑘,𝑖 of the homogeneous adjoint Problem P2^∗. Take an arbitrary 𝐶²(Ω) function 𝑈(𝑥,𝑡) satisfying the boundary conditions (P2). Then the function 𝐹∶=□𝑈 with the wave operator □, is orthogonal to all the functions 𝑊^𝑛_𝑘,𝑖, 𝑛=1,2,…..

Finally, we formulate some still open questions, that naturally arise from the previous works on the Protter problem and the discussions above.

Open Problems. (1) To study the more general case when the right-hand side function 𝑓∈𝐶^𝑘(Ω), for an appropriate 𝑘. The smooth function 𝑓 could be represented as a Fourier series rather than, the finite trigonometric polynomial (2.5) in the discussions here.(i)Find some appropriate conditions for the function 𝑓 under which there exists a generalized solution of the Protter problem (P2). (ii)What kind of singularity can the generalized solution have? The a priori estimates, obtained in [6, 31], which indicate that the generalized solutions of Problem (P2) (including the singular ones), can have at most an exponential growth as 𝜌→0. The natural question is as follows: is there a singular solution of these problems with exponential growth as 𝜌→0 or do all such solutions have only polynomial growth?(iii)Is it possible to prove some a priori estimates for generalized solutions of the Problem (P2) with smooth function 𝑓 which is not a harmonic polynomial? (iv)Find some appropriate conditions for the function 𝑓 under which the Problem (P2) has only regular, bounded solutions, or even classical solutions.
(2) To study the Protter problems for degenerate hyperbolic equations. Up to now it is only known that some singular solutions exist.(i)We do not know what is the exact behavior of the singular solution even when the right-hand side function 𝑓 is a finite sum like (2.5). Can we prove some a priori estimates for generalized solutions?(ii)Is it possible to find some orthogonality conditions for the function 𝑓, as here, under which only bounded solutions exist?
(3) Why does there appear a singularity for such smooth right-hand side even for the wave equation? Can we numerically model this phenomenon?
(4) What happens with the ill-posedness of the Protter problems in a more general domain (as in [2, 4]) when the maximal symmetry is lost if the cone 𝑆₂ is replaced by another light characteristic one with the vertex away from the origin.

3. Preliminaries

We have a relation between the functions 𝑊^𝑛_𝑘,𝑖 and the Legendre functions 𝑃_𝜈. For 𝜈>−1/2, the functions 𝑃_𝜈 could be defined by the equality (Section 3.7, formula (6), from Erdélyi et al. [34]), 𝑃_𝜈(𝑧)=1𝜋^𝜋₀𝑧+√𝑧²−1cos𝑡^𝜈𝑑𝑡,𝑧≥0, (3.1) where for 𝑧<1 in this formula √𝑧²−1∶=𝑖√1−𝑧².

Let (𝜌, 𝜑,𝑡) be the cylindrical coordinates in ℝ³, that is, 𝑥₁=𝜌cos𝜑, 𝑥₂=𝜌sin𝜑. For simplicity, define the function 𝐸^𝑛_𝑘(𝜌,𝑡)∶=𝐸^𝑛_𝑘(𝑥,𝑡)|𝑥|^𝑛. The following result is in connection with Lemma 5.1 from [15]. Actually, to prove this result one could formally follows the arguments of Lemma 2.3 from [16], where the four-dimensional analogue of Problem (P2) is treated.

Lemma 3.1. For 𝑛∈ℕ and 𝜈=𝑛−1/2 define the functions ℎ^𝜈_𝑘𝜉,𝜂∶=^𝜉_𝜂𝑠^𝑘𝑃_𝜈𝜉𝜂+𝑠²𝑠𝜉+𝜂𝑑𝑠, (3.2) for 0<𝜂<𝜉. Then in {𝜌>𝑡} the equality 𝜌^−1/2𝜕ℎ𝜕𝑡^{𝜈𝜈−2𝑘}𝜌+𝑡2,𝜌−𝑡2=𝑎^𝑛_𝑘𝐸^𝑛_𝑘𝜌,𝑡 (3.3) holds for 𝑘=0,1,…,[𝑛/2] with some constants 𝑎^𝑛_𝑘≠0.

Proof. Lemma 5.1 from [15] for 𝑘≥0 gives 𝜌^−1/2ℎ^𝜈_{𝜈−2𝑘−2}𝜌+𝑡2,𝜌−𝑡2=𝐶^𝑛_𝑘𝐻^𝑛_𝑘𝜌,𝑡, (3.4) where 𝐶^𝑛_𝑘=const≠0 and according to Lemma 2.2 from [29] 𝜕𝐻𝜕𝑡^𝑛𝑘𝜌,𝑡=2(𝑛−𝑘−1)𝐸^𝑛_𝑘+1𝜌,𝑡. (3.5) Therefore the equality (3.3) holds for 𝑘≥1. We have to prove it for 𝑘=0. In the proof of Lemma 5.1 from [15] the integrals ℎ^𝜈_𝑘 were calculated using the Mellin transform. In order to compute ℎ^𝜈_𝑘(𝜉,𝜂) in the same way let us first introduce the variables 𝑥 and 𝑧: 𝜉=𝜌+𝑡2;𝜂=𝜌−𝑡2𝜌;𝑥=²𝜌²−𝑡²𝜌;𝑧=²−𝑡^21/22. (3.6) As a consequence after some calculations, formulas (2.2.(4)), (1.10), and (1.4) from Samko et al. [35], show that 𝑧^−𝜈−1ℎ^𝜈_𝜈𝜉,𝜂=𝐶_𝜈𝑥^(𝜈+1)/2𝐼¹₀₊𝑥^{−𝜈−3/2}(𝑥−1)₊^1/2(𝑥), (3.7) where 𝐼^𝛼₀₊(𝑢)(𝑠) is the Riemann-Liouville fractional integral (for its properties see e.g., [34, 35]); in our case we have 𝐼¹₀(𝑢)(𝑠)=∫^𝑠₀𝑢(𝜏)𝑑𝜏. As usual, we denote also 𝜆₊(𝑠)∶=𝜆(𝑠) for 𝑠>0,𝜆₊(𝑠)∶=0 for 𝑠≤0. The substitution of (3.6) in (3.7) shows that 𝜕𝑧𝜕𝑡^𝜈+1𝑥^(𝜈+1)/2𝐼¹₀₊𝑥^{−𝜈−3/2}(𝑥−1)₊^−1/2=𝜕𝑧𝜕𝑡^𝜈+1𝑥^{(𝜈+1)/2𝑥}₀𝜏^{−𝜈−3/2}(𝜏−1)₊^−1/2=𝜕𝑑𝜏𝜌𝜕𝑡^𝜈+12^{−𝜈−1𝑥}₁𝜏^{−𝜈−3/2}(𝜏−1)^−1/2𝑑𝜏=2^−𝜈−1𝜌^𝜈+1𝑥^{−𝜈−3/2}(𝑥−1)^−1/2𝜕𝑥𝜕𝑡=2^−𝜈−1𝜌^𝜈+1𝜌²−𝑡^2𝜈+3/2𝜌^2𝜈+3𝜌²−𝑡^21/2𝑡2𝑡𝜌²𝜌²−𝑡²²=2^−𝜈𝜌²−𝑡^2𝜈𝜌^𝜈 (3.8) and thus 𝜌^−1/2𝜕ℎ𝜕𝑡^𝜈𝜈𝜌+𝑡2,𝜌−𝑡2=𝑎^𝑛₀𝐸^𝑛₀𝜌,𝑡. (3.9)

The next result is crucial for construction of solutions of Problem (P2) in the discussions later.

Lemma 3.2. Let 𝜈∈𝑅, 𝜈>1/2 and the functions 𝐹(𝜉)∈𝐶¹(0,1/2] satisfy 𝐹(1/2)=0. Then all solutions 𝜆∈𝐶¹(0,1/2] of the Volterra integral equation of first kind ^𝜉_1/2𝜆^′𝜉₁𝑃_𝜈𝜉𝜉₁𝑑𝜉₁=𝐹(𝜉) (3.10) are 𝜆(𝜉)1=𝜆2+𝐹(𝜉)+_𝜉^1/2𝑃^′_𝜈𝜉₁𝜉𝐹𝜉₁𝜉₁𝑑𝜉₁. (3.11)

Proof. Formulas (35.17) and (35.28) from Samko et al. [35] state that the solution of the integral equation (3.10) is given by 𝜆^′(𝜉)𝑑=−𝜉²𝑑𝜉²𝜉_𝜉^1/2𝑃_𝜈𝜉₁𝜉𝐹𝜉₁𝜉²₁𝑑𝜉₁𝑑=−𝜉𝑑𝜉²𝑑𝑑𝜉_𝜉^1/2𝑃_𝜈𝜉₁𝜉𝐹𝜉₁𝜉²₁𝑑𝜉₁. (3.12) Then, using that 𝐹(1/2)=0, an integration gives (3.11).

One could use the Mellin transform to calculate the following integral.

Lemma 3.3 (see [16]). Let 𝜈∈𝑅, 𝜈>−1/2, then _𝜉^1/2𝑃_𝜈𝜉₁𝜉𝑃_𝜈2𝜉₁𝜉²₁𝑑𝜉₁=1−2𝜉𝜉. (3.13)

According to the existence and uniqueness results in [6], it is sufficient to study Problem (P2) when the right-hand side 𝑓 of the wave equation is simply 𝑓𝜌,𝑡,𝜑=𝑓¹_𝑛𝜌,𝑡cos𝑛𝜑+𝑓²_𝑛𝜌,𝑡sin𝑛𝜑,𝑛∈ℕ∪{0}. (3.14) Then we seek solutions for the wave equation of the same form: 𝑢𝜌,𝑡,𝜑=𝑢¹_𝑛𝜌,𝑡cos𝑛𝜑+𝑢²_𝑛𝜌,𝑡sin𝑛𝜑. (3.15) Thus Problem (P2) reduces to the following one.

Problem (P2.1)
Solve the equation 𝑢_𝑛𝜌𝜌+1𝜌𝑢_𝑛𝜌−𝑢_𝑛𝑡𝑡−𝑛²𝜌²𝑢_𝑛=𝑓_𝑛𝜌,𝑡 (3.16) in 𝐷₁={0<𝑡<1/2;𝑡<𝜌<1−𝑡}⊂ℝ² with the boundary conditions 𝑢_𝑛𝑡𝜌,0=0for0<𝜌≤1,𝑢_𝑛𝜌,1−𝜌=0for12≤𝜌≤1. (P2.1)
Let us now introduce new coordinates 𝜉=𝜌+𝑡2;𝜂=𝜌−𝑡2, (3.17) and set 𝑣𝜉,𝜂=𝜌^1/2𝑢_𝑛𝜌,𝑡;𝑔𝜉,𝜂=𝜌^1/2𝑓_𝑛𝜌,𝑡. (3.18) Denoting 𝜈=𝑛−1/2, one transforms Problem (P2.1) into the following.

Problem (P2.2)
Find a solution 𝑣(𝜉,𝜂) of the equation 𝑣_𝜉𝜂−𝜈(𝜈+1)𝜉+𝜂²𝑣=𝑔𝜉,𝜂 (3.19) in the domain 𝐷={0<𝜉<1/2;0<𝜂<𝜉} with the following boundary conditions: 𝑣_𝜉−𝑣_𝜂1𝜂,𝜂=0,𝑣2,𝜂=0for1𝜂∈0,2. (P2.2)
Problems (P2.1) and (P2.2) were introduced in [6], although the change of coordinates 𝜉=1−𝜌−𝑡 and 𝜂=1−𝜌+𝑡 was used there instead of (3.17). Of course, because the solution of Problem (P2) may be singular, the same is true for the solutions of (P2.1) and (P2.2). For that reason, Popivanov and Schneider [6] defined and proved the existence and uniqueness of generalized solutions of Problems (P2.1) and (P2.2), which correspond to the generalized solution of Problem (P2). Further, by “solution” of Problem (P2.1) or (P2.2) we mean exactly this unique generalized solution.

Lemma 3.4. Let 𝜈∈𝑅,𝜈>1/2 and 𝑔∈𝐶¹(𝐷). Then the solution 𝑣(𝜉,𝜂) of Problem (P2.2) is given by the following formula: 𝑣𝜉,𝜂=𝜏(𝜉)+_𝜉^1/2𝜏𝜉₁𝜕𝜕𝜉₁𝑃_𝜈𝜉−𝜂𝜉₁+2𝜉𝜂𝜉₁𝜉+𝜂𝑑𝜉₁−_𝜉^1/2𝜂₀𝑃_𝜈𝜉−𝜂𝜉₁−𝜂₁+2𝜉₁𝜂₁+2𝜉𝜂𝜉₁+𝜂₁𝑔𝜉𝜉+𝜂¹,𝜂₁𝑑𝜂₁𝑑𝜉₁, (3.20) where 𝜏(𝜉)=^𝜉_1/2𝑃_𝜈𝜉₁𝜉𝐺𝜉₁𝑑𝜉₁,𝐺(𝜉)=_𝜉^1/2𝜉₀𝑃_𝜈𝜉₁𝜂₁+𝜉²𝜉𝜉₁+𝜂₁𝜕𝜕𝜉₁−𝜕𝜕𝜂₁𝑔𝜉₁,𝜂₁𝑑𝜂₁𝑑𝜉₁−^𝜉₀𝑃_𝜈𝜂₁+2𝜉²𝜉2𝜂₁𝑔1+12,𝜂₁𝑑𝜂₁−_𝜉^1/2𝑃_𝜈𝜉𝜉₁𝑔𝜉₁,0𝑑𝜉₁. (3.21)(3.22)

Proof. Notice that the function 𝑅𝜉₁,𝜂₁;𝜉,𝜂=𝑃_𝜈𝜉−𝜂𝜉₁−𝜂₁+2𝜉₁𝜂₁+2𝜉𝜂𝜉₁+𝜂₁𝜉+𝜂 (3.23) is a Riemann function for (3.19) (Copson [36]). Therefore, following Aldashev [10], we can construct the function 𝑣(𝜉,𝜂) as a solution of a Goursat problem in 𝐷 with boundary conditions 𝑣(1/2,𝜂)=0 and 𝑣(𝜉,0)=𝜏(𝜉) with some unknown function 𝜏(𝜉)∈𝐶²(0,1/2], which will be determined later: 𝑣𝜉,𝜂=𝜏(𝜉)+_𝜉^1/2𝜏𝜉₁𝜕𝜕𝜉₁𝑅𝜉₁,0;𝜉,𝜂𝑑𝜉₁−_𝜉^1/2𝜂₀𝑅𝜉₁,𝜂₁;𝜉,𝜂𝑔𝜉₁,𝜂₁𝑑𝜂₁𝑑𝜉₁. (3.24) Now, the boundary condition 𝜕−𝜕𝜕𝜉𝑣𝜕𝜂_𝜂=𝜉=0. (3.25) gives an integral equation for 𝜏(𝜉). For that reason, let us define the function 𝐺(𝜉): 𝐺(𝜉)𝜕∶=−𝜕𝜕𝜉𝜕𝜂_𝜉^1/2𝜂₀𝑅𝜉₁,𝜂₁;𝜉,𝜂𝑔𝜉₁,𝜂₁𝑑𝜂₁𝑑𝜉_1𝜂=𝜉=_𝜉^1/2𝜉₀𝑃^′_𝜈𝜉₁𝜂₁+𝜉²𝜉𝜉₁+𝜂₁𝜉₁−𝜂₁𝜉𝜉₁+𝜂₁𝑔𝜉₁,𝜂₁𝑑𝜂₁𝑑𝜉₁−^𝜉₀𝑔𝜉,𝜂₁𝑑𝜂₁−_𝜉^1/2𝑔𝜉₁,𝜉𝑑𝜉₁=−^𝜉₀𝑔𝜉,𝜂₁𝑑𝜂₁−_𝜉^1/2𝑔𝜉₁,𝜉𝑑𝜉₁−_𝜉^1/2𝜉₀𝑔𝜉₁,𝜂₁𝜕𝜕𝜉₁−𝜕𝜕𝜂₁𝑃_𝜈𝜉₁𝜂₁+𝜉²𝜉𝜉₁+𝜂₁𝑑𝜂₁𝑑𝜉₁=_𝜉^1/2𝜉₀𝑃_𝜈𝜉₁𝜂₁+𝜉²𝜉𝜉₁+𝜂₁𝜕𝜕𝜉₁−𝜕𝜕𝜂₁𝑔𝜉₁,𝜂₁𝑑𝜂₁𝑑𝜉₁−^𝜉₀𝑃_𝜈𝜂₁+2𝜉²𝜉2𝜂₁𝑔1+12,𝜂₁𝑑𝜂₁−_𝜉^1/2𝑃_𝜈𝜉𝜉₁𝑔𝜉₁,0𝑑𝜉₁. (3.26) Obviously, 𝐺∈𝐶²[0,1/2]. The condition (3.25) leads us to the following equation: 𝜏^′(𝜉)−1𝜉𝜏(𝜉)𝑃^′_𝜈(1)−_𝜉^1/2𝜏𝜉₁𝜉²₁𝑃_𝜈^′′𝜉𝜉₁𝑑𝜉₁=𝐺(𝜉). (3.27) Then, using 𝜏(1/2)=𝑣(1/2,0)=0, we have ^𝜉_1/2𝑑𝑑𝜉₁𝜉²₁𝜏^′𝜉₁𝑃_𝜈𝜉𝜉₁𝑑𝜉₁=𝜉²𝐺(𝜉)−𝜏^′(1/2)4𝑃_𝜈(2𝜉). (3.28) A necessary solvability condition for the unknown function 𝜏∈𝐶²(0,1/2] is: 𝜏^′(1/2)=𝐺(1/2). One could solve this Volterra integral equation of the first kind, using Lemma 3.2. The result is 𝜉²𝜏^′(𝜉)−14𝜏^′12=𝜉²𝐺(𝜉)−14𝜏^′12𝑃_𝜈(2𝜉)+_𝜉^1/2𝑃^′_𝜈𝜉₁𝜉4𝜉²₁𝐺𝜉₁−𝜏^′(1/2)𝑃_𝜈2𝜉₁4𝜉₁𝑑𝜉₁. (3.29) Integrate, we find 𝜏(𝜉)=^𝜉_1/2𝐺(𝑧)+1𝑧²_𝑧^1/2𝑃^′_𝜈𝜉₁𝑧𝜉₁𝐺𝜉₁𝑑𝜉₁+𝜏𝑑𝑧_′(1/2)4^𝜉_1/21𝑧²−𝑃_𝜈(2𝑧)𝑧²−1𝑧^2𝑧_1/2𝑃^′_𝜈𝜉₁𝑧𝑃_𝜈2𝜉₁𝜉₁⁻¹𝑑𝜉₁𝑑𝑧. (3.30) Now, using Lemma 3.3 and the equality ^𝜉_1/21𝑧²_𝑧^1/2𝑃^′_𝜈𝜉₁𝑧𝐹𝜉₁𝑑𝜉₁𝑑𝑧=^𝜉_1/2𝑃_𝜈𝜉₁𝜉𝐹𝜉−1¹𝜉₁𝑑𝜉₁, (3.31) for 𝐹(𝜉)= 𝑃_𝜈(2𝜉)𝜉⁻¹ one finds that the coefficient of 𝜏^′(1/2) in (3.30) is zero. Using again (3.31) for 𝐹(𝜉)= 𝜉𝐺(𝜉), 𝜏 is simply 𝜏(𝜉)=^𝜉_1/2𝑃_𝜈𝜉₁𝜉𝐺𝜉₁𝑑𝜉₁. (3.32) Obviously, 𝜏∈𝐶²(0,1/2] and 𝜏(1/2)=0,𝜏^′(1/2)=𝐺(1/2). Finally, the solution of Problem (P2.2) is given by the formulae (3.20), (3.21), and (3.22).

4. Proofs of Main Results

In order to study the behavior of the generalized solution of Problem (P2), in view of relations (3.18) and Lemma 3.4, we will examine the function 𝑣(𝜉,𝜂) defined by the formulae (3.20), (3.21), and (3.22). It is not hard to see that the part “responsible” for the singularity is the integral in (3.21) for the function 𝜏(𝜉). In fact, 𝜏(𝜉) blows up at 𝜉=0, since the argument 𝜉₁/𝜉 and thus the values of the Legendre function 𝑃_𝜈 in (3.21) go to infinity when 𝜉→0. Actually, 𝑃_𝜈(𝑧) grows like |𝑧|^𝜈 at infinity. In the next lemma we find the dependance of the exact order of singularity of 𝜏(𝜉) on the function 𝐺(𝜉). It is governed by the constants 𝛾_𝑘∶=₀^1/2𝜉^𝜈−2𝑘𝐺(𝜉)𝑑𝜉for𝑘=0,…,𝜈+12. (4.1) Actually, these numbers are closely connected to the constants 𝛽^𝑛_𝑘,𝑖 from Theorem 2.3. We will clarify this relation later in Lemma 4.1 and the proof of Theorem 2.3.

Lemma 4.1. Let 𝜈=𝑛−1/2, where 𝑛∈ℕ, 𝑛≥2, and let the function 𝐺(𝜉)∈𝐶¹[0,1/2]. Then the function 𝜏(𝜉)=^𝜉_1/2𝑃_𝜈𝜉₁𝜉𝐺𝜉₁𝑑𝜉₁ (4.2) belongs to 𝐶²(0,1/2] and satisfies the representation 𝜏(𝜉)=^[(𝜈+1)/2]_𝑘=0𝐶^𝜈_𝑘𝛾_𝑘𝜉^{−(𝜈−2𝑘)}(𝜉)1+𝜓,𝜉∈0,2, (4.3) where the function 𝜓(𝜉)∈𝐶²(0,1/2], |𝜓(𝜉)|≤𝐶𝜉max{|𝐺(𝜉)|∶𝜉∈0,1/2]} and the nonzero constants 𝐶^𝜈_𝑘 and 𝐶 are independent of 𝐺(𝜉).

Proof. The argument of the Legendre function 𝑃_𝜈 in (4.2) satisfies the inequality 𝜉₁/𝜉≥1, which allows us to apply the representation (3.1): 𝜏(𝜉)=1𝜋1𝜉^𝜈𝜉_1/2^𝜋₀𝜉₁+𝜉²₁−𝜉²cos𝑡^𝜈𝐺𝜉₁𝑑𝑡𝑑𝜉₁. (4.4) We will study the expansion at 𝜉=0 of the function 𝐹(𝜉)∶=_𝜉^1/2𝜋₀𝜉₁+𝜉²₁−𝜉²cos𝑡^𝜈𝐺𝜉₁𝑑𝑡𝑑𝜉₁. (4.5) Let us define the functions 𝑀^𝜈_𝑘𝜉₁,𝜉∶=(−1)^𝑘(𝜈−2𝑘+1)_2𝑘2^𝑘(1/2)_𝑘^𝜋0𝜉₁+𝜉²₁−𝜉²cos𝑡^𝜈−2𝑘sin^2𝑘𝑡𝑑𝑡, (4.6) for 𝜉≤𝜉₁≤1/2. Then, obviously 𝐹(𝜉)=_𝜉^1/2𝑀^𝜈₀𝜉₁,𝜉𝐺𝜉₁𝑑𝜉₁. (4.7) First, we will examine the properties of the functions 𝑀^𝜈_𝑘(𝜉₁,𝜉) and their derivatives with respect to 𝜉. We start with the equality 𝑀^𝜈_𝑘(𝜉,𝜉)=𝑎^𝜈_𝑘𝜉^𝜈−2𝑘,𝑎^𝜈_𝑘≠0. (4.8) Further, the index 𝑘 will be less than 𝜈. Notice that for 𝑘<𝜈+1/2 the integrals ∫^𝜋₀(1±cos𝑡)^𝜈−2𝑘sin^2𝑘𝑡𝑑𝑡 are convergent. Then, for 𝜉≤𝜉₁ we have the equality 𝑀^𝜈_𝑘𝜉₁,0=𝑏^𝜈_𝑘𝜉₁^𝜈−2𝑘,𝑏^𝜈_𝑘≠0, (4.9) and the inequality 𝑀^𝜈_𝑘𝜉₁,𝜉≤𝑐^𝜈_𝑘𝜉₁^𝜈−2𝑘. (4.10)
Differentiating with respect to 𝜉 one finds (−1)^𝑘+1𝜕𝑀𝜕𝜉^𝜈𝑘𝜉₁=(),𝜉𝜈−2𝑘+1_2𝑘(𝜈−2𝑘)2^𝑘(1/2)_𝑘^𝜋0𝜉₁+𝜉²₁−𝜉²cos𝑡^{𝜈−2𝑘−1}𝜉𝜉²₁−𝜉²sin^2𝑘=()𝑡cos𝑡𝑑𝑡𝜈−2𝑘_2𝑘+1(𝜈−2𝑘−1)2^𝑘(1/2)_𝑘(2𝑘+1)^𝜋0𝜉𝜉₁+𝜉²₁−𝜉²cos𝑡^{𝜈−2𝑘−2}sin^2𝑘+2=()𝑡𝑑𝑡−1^𝑘+1𝜉𝑀^𝜈_𝑘+1𝜉₁,𝜉. (4.11)
Therefore, for the derivatives of 𝑀^𝜈₀ we find by induction 𝜕^𝑘𝑀^𝜈₀𝜕𝜉^𝑘𝜉₁,𝜉=^[𝑘/2]_𝑖=0𝐶^𝑘_𝑖𝜉^𝑘−2𝑖𝑀^𝜈_𝑘−𝑖𝜉₁,𝜉, (4.12) where the coefficients 𝐶^𝑘_𝑖 are positive constants. We want to evaluate these derivatives of 𝑀^𝜈₀ at 𝜉=0. Let us estimate the terms in the last sum for 𝑘<𝜈:(i)when 𝑖 is such that 𝜈−2(𝑘−𝑖)<0 the inequality (4.10) gives the estimate 𝜉^𝑘−2𝑖𝑀^𝜈_𝑘−𝑖𝜉₁,𝜉≤𝜉^𝑘−2𝑖𝑐^𝜈_𝑘𝜉₁^{𝜈−2(𝑘−𝑖)}≤𝑐^𝜈_𝑘𝜉^𝜈−𝑘; (4.13)(ii)when 𝜈−2(𝑘−𝑖)≥0 and 𝑘/2>𝑖, we have 𝜉^𝑘−2𝑖𝑀^𝜈_𝑘−𝑖𝜉₁,𝜉≤𝑐^𝜈_𝑘𝜉^𝑘−2𝑖. (4.14)Hence 𝜉^𝑘−2𝑖𝑀^𝜈_𝑘−𝑖(𝜉₁,𝜉)|_𝜉=0=0 for 2𝑖<𝑘. Therefore, at the point 𝜉=0 the only one nonzero term in the sum (4.12) is for 2𝑖=𝑘, that is, 𝜕^𝑘𝑀^𝜈₀𝜕𝜉^𝑘𝜉₁,0=0,if𝑘isodd𝐶^𝑘_𝑘/2𝑏^𝜈_𝑘/2𝜉₁^𝜈−𝑘,if𝑘iseven. (4.15) The last observation is that (4.8) and (4.12) imply 𝜕^𝑘𝑀^𝜈₀𝜕𝜉^𝑘_𝜉1=𝜉=𝑑^𝜈_𝑘𝜉^𝜈−𝑘, (4.16) where 𝑑^𝜈_𝑘=∑^[𝑘/2]_𝑖=0𝐶^𝑘_𝑖𝑎^𝜈_𝑘−𝑖 are constants.
Now, we go back to the function 𝐹(𝜉). We want to differentiate [𝜈] times and evaluate at 𝜉=0. Differentiating (4.7) we find the following:𝐹^′(𝜉)=−𝑎^𝜈₀𝐺(𝜉)𝜉^𝜈+_𝜉^1/2𝜕𝑀𝜕𝜉^𝜈0𝜉₁,𝜉𝐺𝜉₁𝑑𝜉₁. (4.17) Next, since the assertion for 𝐺(𝜉) is only 𝐺(𝜉)∈𝐶¹[0,1/2], instead of 𝐹^′(𝜉) we will differentiate the function 𝐹₁(𝜉)∶=𝐹^′(𝜉)+𝑎^𝜈₀𝐺(𝜉)𝜉^𝜈=_𝜉^1/2𝜕𝑀𝜕𝜉^𝜈0𝜉₁,𝜉𝐺𝜉₁𝑑𝜉₁. (4.18) Notice that it belongs to 𝐶[0,1/2]∩𝐶¹(0,1/2] and the derivative is 𝐹^′₁(𝜉)=−𝑑^𝜈₁𝐺(𝜉)𝜉^𝜈−1+_𝜉^1/2𝜕²𝜕𝜉²𝑀^𝜈₀𝜉₁,𝜉𝐺𝜉₁𝑑𝜉₁. (4.19) In the same way, after denoting 𝐹₀(𝜉)≡𝐹(𝜉), define for 𝑗=1,…,[𝜈] the functions 𝐹_𝑗(𝜉)∶=𝐹^′_𝑗−1(𝜉)+𝑑^𝜈_𝑗−1𝐺(𝜉)𝜉^{𝜈−𝑗+1} (4.20) with the constants 𝑑^𝜈_𝑗 from (4.16). Then, using (4.16), it follows by induction that 𝐹_𝑗 is continuous in [0,1/2] and 𝐹_𝑗(𝜉)=_𝜉^1/2𝜕^𝑗𝜕𝜉^𝑗𝑀^𝜈₀𝜉₁,𝜉𝐺𝜉₁𝑑𝜉₁,𝑗=0,…,[𝜈]. (4.21) On the other hand, 𝐹_𝑗(0)=₀^1/2𝜕^𝑗𝜕𝜉^𝑗𝑀^𝜈₀𝜉₁,0𝐺𝜉₁𝑑𝜉₁,𝑗=0,…,[𝜈]−1. (4.22) Hence, according to (4.15), for 𝑗≤[𝜈]−1, 𝐹_𝑗(0)=0,if𝑗isodd𝛾^′_𝑖,if𝑗iseven. (4.23)
The next step is to evaluate the integral 𝐹_[𝜈]. Using (4.12), one could rewrite it in the form 𝐹_[𝜈](𝜉)=_𝜉^1/2𝜕^[𝜈]𝜕𝜉^[𝜈]𝑀^𝜈₀𝜉₁,𝜉𝐺𝜉₁𝑑𝜉₁=^[[𝜈]/2]_𝑖=0𝐶_𝑖^[𝜈]_𝜉^1/2𝜉^{[𝜈]−2𝑖}𝑀^𝜈_{[𝜈]−𝑖}𝜉₁,𝜉𝐺𝜉₁𝑑𝜉₁. (4.24) For all the terms in the last sum, except one, the estimate is straightforward.
(1) When 𝑖 is such that [𝜈]−2𝑖≥2, for the corresponding terms we have 𝜉^{[𝜈]−2𝑖}𝑀^𝜈_{[𝜈]−𝑖}𝜉₁,𝜉≤𝜉²𝜉₁^{[𝜈]−2𝑖−2}𝑐^𝜈_{[𝜈]−𝑖}𝜉₁^{𝜈−2([𝜈]−𝑖)}≤𝑐^𝜈_{[𝜈]−𝑖}𝜉²𝜉₁^−3/2, (4.25) and, therefore, _𝜉^1/2𝜉^{[𝜈]−2𝑖}𝑀^𝜈_{[𝜈]−𝑖}𝜉₁,𝜉𝐺𝜉₁𝑑𝜉₁≤𝑐^𝜈_{[𝜈]−𝑖}𝐴𝜉²_𝜉^1/2𝜉₁^−3/2𝑑𝜉₁≤𝐶𝐴𝜉^3/2, (4.26) where 𝐴∶=max{|𝐺(𝜉)|∶𝜉∈0,1/2]}.
(2) For the last term in (4.24) with 𝑖=[[𝜈]/2] there are two cases:(2a) when [𝜈]=2𝑚 is an even number this is the integral _𝜉^1/2𝑀^𝜈_𝑚𝜉₁,𝜉𝐺𝜉₁𝑑𝜉₁; (4.27)(2b) when [𝜈]=2𝑚−1 is an odd number, the integral is _𝜉^1/2𝜉𝑀^𝜈_𝑚𝜉₁,𝜉𝐺𝜉₁𝑑𝜉₁. (4.28) For simplicity let us define some constants 𝛾^′_𝑖𝛾^′_𝑖∶=₀^1/2𝑀^𝜈_𝑖𝜉₁,0𝐺𝜉₁𝑑𝜉₁, (4.29) related to the constants 𝛾_𝑖 given by (4.1). Indeed, due to (4.9) the equality 𝛾^′_𝑖=𝑏^𝜈_𝑖𝛾_𝑖 holds. Let us begin with the following case.(2a):[𝜈]=2𝑚,thatis,𝜈=2𝑚+1/2. We will evaluate the difference: _𝜉^1/2𝑀^𝜈_𝑚𝜉₁,𝜉𝐺𝜉₁𝑑𝜉₁−𝛾^′_𝑚=_𝜉^1/2𝑀^𝜈_𝑚𝜉₁,𝜉𝐺𝜉₁𝑑𝜉₁−₀^1/2𝑀^𝜈_𝑚𝜉₁,0𝐺𝜉₁𝑑𝜉₁≤_𝜉^1/2𝑀^𝜈_𝑚𝜉₁,𝜉−𝑀^𝜈_𝑚𝜉₁,0𝐺𝜉₁𝑑𝜉₁+^𝜉₀𝑀^𝜈_𝑚𝜉₁,0𝐺𝜉₁𝑑𝜉₁. (4.30) For the first integral, using the estimate (4.10), we calculate 𝑀^𝜈_𝑚𝜉₁,𝜉−𝑀^𝜈_𝑚𝜉₁,0=^𝜉₀𝜕𝜕𝜉₂𝑀^𝜈_𝑚𝜉₁,𝜉₂𝑑𝜉₂=^𝜉₀𝜉₂𝑀^𝜈_𝑚+1𝜉₁,𝜉₂𝑑𝜉₂≤𝑐^𝜈_𝑚+1𝜉²𝜉₁^{𝜈−2𝑚−2}=𝑐^𝜈_𝑚+1𝜉²𝜉₁^−3/2, (4.31) and, therefore, _𝜉^1/2𝑀^𝜈_𝑚𝜉₁,𝜉−𝑀^𝜈_𝑚𝜉₁,0𝐺𝜉₁𝑑𝜉₁≤𝐶₁𝐴𝜉^3/2. (4.32) For the second integral ^𝜉₀𝑀^𝜈_𝑚𝜉₁,0𝐺𝜉₁𝑑𝜉₁≤𝑐^𝜈_𝑚𝐴^𝜉₀𝜉₁^𝜈−2𝑚𝑑𝜉₁=𝐶𝐴^𝜉₀𝜉₁^1/2𝑑𝜉₁=𝐶₂𝐴𝜉^3/2. (4.33) From the last two inequalities we get the estimate _𝜉^1/2𝑀^𝜈_𝑚𝜉₁,𝜉𝐺𝜉₁𝑑𝜉₁−𝛾^′_𝑚≤𝐶𝐴𝜉^3/2. (4.34) Therefore, in the case [𝜈]=2𝑚, 𝑚∈ℕ, 𝐹_[𝜈](𝜉)=𝛾^′_𝑚+𝜓_[𝜈](𝜉), (4.35) where |𝜓_[𝜈](𝜉)|≤𝐶𝐴𝜉^3/2.(2b) When [𝜈]=2𝑚−1,thatis,𝜈=2𝑚−1/2, we have to study the integral (4.28). Obviously, _𝜉^1/2𝜉𝑀^𝜈_𝑚𝜉₁,𝜉𝐺𝜉₁𝑑𝜉₁−𝛾^′_𝑚𝜉≤𝜉_𝜉^1/2𝑀^𝜈_𝑚𝜉₁,𝜉−𝑀^𝜈_𝑚𝜉₁,0𝐺𝜉₁𝑑𝜉₁+𝜉^𝜉₀𝑀^𝜈_𝑚𝜉₁,0𝐺𝜉₁𝑑𝜉₁. (4.36) For the last integral we have ^𝜉₀𝑀^𝜈_𝑚𝜉₁,0𝐺𝜉₁𝑑𝜉₁≤𝑐^𝜈_𝑚𝐴^𝜉₀𝜉₁^𝜈−2𝑚𝑑𝜉₁=2𝐶𝐴𝜉^1/2. (4.37) Now, to estimate the first term in the right-hand side of (4.36), there are two cases:(i) when 𝑚≥2, we have 𝜈>[𝜈]=2𝑚−1≥𝑚+1 and similarly to the previous case (2a) we can apply inequality (4.31). Thus, we estimate the difference: 𝑀^𝜈_𝑚𝜉₁,𝜉−𝑀^𝜈_𝑚𝜉₁,0=^𝜉₀𝜕𝜕𝜉₂𝑀^𝜈_𝑚𝜉₁,𝜉₂𝑑𝜉₂=^𝜉₀𝜉₂𝑀^𝜈_𝑚+1𝜉₁,𝜉₂𝑑𝜉₂≤^𝜉₀𝑐^𝜈_𝑚+1𝜉₂𝜉₁^{𝜈−2𝑚−2}𝑑𝜉₂≤𝑐^𝜈_𝑚+1𝜉²𝜉₁^−5/2, (4.38)(ii) when 𝑚=1, denote for simplicity 𝑝∶=1−𝜉²/𝜉²₁, then 𝑝∈[0,1] and directly from the definition (4.6) of the functions 𝑀^𝜈_𝑚, we get 𝑀₁^3/2𝜉₁,𝜉−𝑀₁^3/2𝜉₁,0=𝐶𝜉₁^−1/21₋₁1−𝑝𝑧^−1/2−(1−𝑧)^−1/2√1−𝑧²𝑑𝑧≤𝐶𝜉₁^−1/21₋₁1−𝑝|𝑧|√1+𝑧1−𝑝𝑧1−𝑝𝑧+√1−𝑧𝑑𝑧≤𝐶₁𝜉₁^−1/21−𝑝+𝐶₁𝜉₁^−1/21−𝑝¹₀11−𝑝𝑧(1−𝑧)𝑑𝑧≤𝐶₁𝜉₁^−1/21−𝑝1+¹₀11−𝑝+𝑝𝜏𝜏𝑑𝜏≤𝐶₂𝜉₁^−1/21−𝑝≤𝐶₂𝜉₁^−1/21−𝑝²=𝐶𝜉𝜉₁^−3/2. (4.39)Thus for 𝑚≥1, both cases lead to _𝜉^1/2𝑀^𝜈_𝑚𝜉₁,𝜉−𝑀^𝜈_𝑚𝜉₁,0𝐺𝜉₁𝑑𝜉₁≤𝐶𝐴𝜉^1/2. (4.40)
Finally, substituting (4.37) and (4.40) in (4.36), we find the estimate _𝜉^1/2𝑀^𝜈_𝑚𝜉₁,𝜉𝐺𝜉₁𝑑𝜉₁−𝛾^′_𝑚𝜉≤𝐶𝐴𝜉^3/2. (4.41)
Summarizing, in the case (2b) of odd [𝜈]=2𝑚−1, we have 𝐹_[𝜈](𝜉)=𝛾^′_𝑚𝜉+𝜓_[𝜈](𝜉), (4.42) where |𝜓_𝜈](𝜉)|≤𝐶𝐴𝜉^3/2.
Now, we are ready to go backwards from 𝐹_[𝜈](𝜉) to 𝐹₀(𝜉). Integrating (4.20) we find 𝐹_𝑗(𝜉)=𝐹_𝑗(0)+^𝜉₀𝐹_𝑗+1𝜉₁𝑑𝜉₁+𝑑^𝜈_𝑗^𝜉₀𝜉₁^𝜈−𝑗𝐺𝜉₁𝑑𝜉₁, (4.43) for 𝑗=0,…,[𝜈]−1. Using (4.43) and (4.23), one can find the relation between the functions 𝐹(𝜉)≡𝐹₀(𝜉) and 𝐹_[𝜈](𝜉). Starting from (4.43) with 𝑗=0 one expresses 𝐹₁(𝜉) by applying again (4.43) in the right-hand side but for 𝑗=1: 𝐹(𝜉)≡𝐹₀(𝜉)=𝐹₀(0)+^𝜉₀𝐹₁𝜉₁𝑑𝜉₁+𝑑^𝜈₀^𝜉₀𝜉^𝜈₁𝐺𝜉₁𝑑𝜉₁=𝐹₀(0)+𝐹₁(0)𝜉+^𝜉₀^𝑡₀𝐹₂𝜉₁𝑑𝜉₁𝑑𝑡+^𝜉₀𝑑^𝜈₁^𝑡₀𝜉₁^𝜈−1𝐺𝜉₁𝑑𝜉₁+𝑑^𝜈₀𝑡^𝜈𝐺(𝑡)𝑑𝑡. (4.44) Similarly, (4.43) gives a representation of 𝐹₂(𝜉) through 𝐹₃(𝜉) and so on. Finally, we get the sum 𝐹(𝜉)=𝐹₀(0)+𝐹₁(0)𝐹1!𝜉+²(0)𝜉2!²+⋯+𝑑^𝜈₀^𝜉₀𝐺𝜉₁𝜉^𝜈₁𝑑𝜉₁+𝑑^𝜈₁^𝜉₀𝐺𝜉₁𝜉₁^𝜈−1𝜉−𝜉₁𝑑𝜉₁+𝑑^𝜈₂2^𝜉₀𝐺𝜉₁𝜉₁^𝜈−2𝜉−𝜉₁²𝑑𝜉₁+⋯, (4.45) by consequently substituting 𝐹_𝑘+1(𝜉) in the resulting expression at each step by applying the same formula (4.43) for 𝑗=𝑘+1. Thus, for 𝐹(𝜉) we find inductively 𝐹(𝜉)=_𝑗𝐹_𝑗(0)𝜉𝑗!^𝑗+⋯=_𝑘𝛾^′_𝑘()!𝜉2𝑘^2𝑘+⋯, (4.46) since 𝐹_2𝑖+1(0)=0 and 𝐹_2𝑖(0)=𝛾^′_𝑖, in view of (4.23). The process ends when 𝐹_[𝜈], integrated [𝜈] times, appears and we can apply (4.35) or (4.42) instead of (4.43). Therefore, according to (4.35), when [𝜈]=2𝑚 is an even number, the last 𝛾^′_𝑖 in this sum will be 𝛾^′_𝑚, and its coefficient will be (1/(2𝑚)!)𝜉^2𝑚, while when [𝜈]=2𝑚−1 is an odd number, formula (4.42) shows that the last term will be also (𝛾^′_𝑚/(2𝑚)!)𝜉^2𝑚. In both cases 𝑚=[(𝜈+1)/2] and the constant coefficients are independent of 𝐺(𝜉). Then, for the function 𝐹(𝜉) we have the representation 𝐹(𝜉)=^[(𝜈+1)/2]_𝑘=0𝛾^′_𝑘()!𝜉2𝑘^2𝑘+Ψ(𝜉), (4.47) where the function Ψ(𝜉) is defined by Ψ(𝜉)∶=^[𝜈]−1_𝑗=0𝑑^𝜈_𝑗𝑗!^𝜉₀𝐺𝜉₁𝜉₁^𝜈−𝑗𝜉−𝜉₁^𝑗𝑑𝜉₁+^𝜉₀𝜓_[𝜈]𝜉₁𝜉−𝜉₁^[𝜈]−1([𝜈]−1)!𝑑𝜉₁. (4.48) Therefore |Ψ(𝜉)|≤𝐶𝐴𝜉^𝜈+1, because |𝐺(𝜉)|≤𝐴 and |𝜓_[𝜈](𝜉₁)|≤𝐶𝐴𝜉₁^3/2.
Finally, recall that 𝛾^′_𝑘=𝑏^𝜈_𝑘𝛾_𝑘, with coefficients 𝑏^𝜈_𝑘≠0 from (4.9) and 𝜏(𝜉)=−𝜋⁻¹𝜉^−𝜈𝐹(𝜉) from (4.4), and, therefore, 𝜏(𝜉)=^[(𝜈+1)/2]_𝑘=0𝐶_𝑘𝛾_𝑘𝜉^{−(𝜈−2𝑘)}+𝜓(𝜉), (4.49) where 𝐶_𝑘=−𝜋⁻¹ 𝑏^𝜈_𝑘/(2𝑘)!≠0 and |𝜓(𝜉)|≤𝐶𝐴𝜉.

This lemma, due to formula (3.20), helps us to examine the solution 𝑣(𝜉,𝜂) of Problem (P2.2) and therefore due to (3.18), the solution 𝑢_𝑛(𝜌,𝑡)=𝜌^−1/2𝑣 of Problem (P2.1). First, for 𝑘=0,…,[𝑛/2], denote by 𝛼^𝑛_𝑘 the parameters 𝛼^𝑛_𝑘∶=₀^1/2_𝑡^1−𝑡𝐸^𝑛_𝑘𝜌,𝑡𝑓_𝑛𝜌,𝑡𝜌𝑑𝜌𝑑𝑡. (4.50)

Theorem 4.2. Let 𝑓_𝑛∈𝐶¹(𝐷₁). Then the generalized solution 𝑢_𝑛(𝜌,𝑡) of Problem (P2.1) belongs to 𝐶²(𝐷₁⧵(0,0)) and has the following asymptotic expansion at the origin (0,0): 𝑢_𝑛𝜌,𝑡=^[𝑛/2]_𝑘=0𝜌^−1/2𝜌+𝑡^{−(𝑛−2𝑘−1/2)}𝛼^𝑛_𝑘𝐹^𝑛_𝑘𝜌,𝑡+𝜌^1/2ln𝜌𝐹^𝑛𝜌,𝑡, (4.51) where 𝐹^𝑛_𝑘(𝜌,𝑡),𝐹^𝑛(𝜌,𝑡)∈𝐶²(𝐷₁⧵(0,0)), 𝐹^𝑛_𝑘𝜌,𝑡≤𝐶,𝐹^𝑛𝜌,𝑡≤𝐶max_𝐷1𝑓_𝑛𝜌,𝑡+max_𝐷1𝑓_𝑛𝑡𝜌,𝑡, (4.52) with functions 𝐹^𝑛_𝑘 and a constant 𝐶 independent of 𝑓_𝑛 and lim_𝑡→+0𝐹^𝑛_𝑘(𝑡,𝑡)=𝑐𝑜𝑛𝑠𝑡≠0.

First, let us shortly outline the proof. In order to find the behavior of 𝑢_𝑛 we apply the relation (3.18) and study the function 𝑣. Actually, Lemma 3.4 uses Lemmas 3.2 and 3.3 to describe 𝑣 by formulas (3.20)–(3.22) and the analysis passes to 𝜏, 𝐺, and the Legendre function 𝑃_𝜈. This way the base for the asymptotic expansion (4.51) is the expansion found in Lemma 4.1 for 𝜏 given by (3.21).

Proof of Theorem 4.2. Denote 𝐴∶=max_𝐷1𝑓_𝑛𝜌,𝑡+max_𝐷1𝑓_𝑛𝑡𝜌,𝑡 (4.53) and thus |𝑔(𝜉,𝜂)|≤𝐶𝐴,|𝐺(𝜉)|≤𝐶𝐴 with the constant 𝐶 independent of 𝑓_𝑛. Our goal is to apply Lemma 4.1.
The key of this will be the equality ₀^1/2𝜉^𝜈−2𝑘𝐺(𝜉)=𝑑𝜉_{01/2𝜉1/2𝜉0}𝜉^𝜈−2𝑘𝑃_𝜈𝜉₁𝜂₁+𝜉²𝜉𝜉₁+𝜂₁𝜕𝜕𝜉₁−𝜕𝜕𝜂₁𝑔𝜉₁,𝜂₁𝑑𝜂₁𝑑𝜉₁−𝑑𝜉_01/2𝜉0𝜉^𝜈−2𝑘𝑃_𝜈𝜂₁+2𝜉²𝜉1+2𝜂₁𝑔12,𝜂₁𝑑𝜂₁−𝑑𝜉_01/2𝜉1/2𝜉^𝜈−2𝑘𝑃_𝜈𝜉𝜉₁𝑔𝜉₁,0𝑑𝜉₁𝑑𝜉=₀^1/2𝜉1₀^𝜉1_𝜂1𝜉^𝜈−2𝑘𝑃_𝜈𝜉₁𝜂₁+𝜉²𝜉𝜉₁+𝜂₁𝜕𝑑𝜉𝜕𝜉₁−𝜕𝜕𝜂₁𝑔𝜉₁,𝜂₁𝑑𝜂₁𝑑𝜉₁−₀^1/2_𝜂^1/2₁𝜉^𝜈−2𝑘𝑃_𝜈𝜂₁+2𝜉²𝜉1+2𝜂₁𝑔1𝑑𝜉2,𝜂₁𝑑𝜂₁−₀^1/2𝜉1₀𝜉^𝜈−2𝑘𝑃_𝜈𝜉𝜉₁𝑑𝜉𝑔𝜉₁,0𝑑𝜉₁=−₀^1/2𝜉1₀𝜕𝜕𝜉₁−𝜕𝜕𝜂₁^𝜉1_𝜂1𝜉^𝜈−2𝑘𝑃_𝜈𝜉₁𝜂₁+𝜉²𝜉𝜉₁+𝜂₁𝑑𝜉𝑔𝜉₁,𝜂₁𝑑𝜂₁𝑑𝜉₁=−₀^1/2𝜉1₀𝜕𝜕𝜉₁−𝜕𝜕𝜂₁ℎ^𝜈_𝜈−2𝑘𝜉₁,𝜂₁𝑔𝜉₁,𝜂₁𝑑𝜂₁𝑑𝜉₁𝑛,𝑘=0,…,2, (4.54) according to Definition (3.2) of functions ℎ^𝜈_𝑘 in Lemma 3.1.
On the other hand, Lemma 3.1 gives 𝜕−𝜕𝜕𝜉𝜕𝜂𝜉+𝜂^−1/2ℎ^𝜈_𝜈−2𝑘𝜉,𝜂=𝑎^𝑛_𝑘𝐸^𝑛_𝑘𝜉+𝜂,𝜉−𝜂, (4.55) and by definition 𝑓=(𝜉+𝜂)^−1/2𝑔. Therefore, we have the following relation between 𝛼^𝑛_𝑘 defined in (4.50) and the constants 𝛾_𝑘 from Lemma 4.1: 𝛼^𝑛_𝑘=₀^1/2_𝑡^1−𝑡𝐸^𝑛_𝑘𝑓=2𝜌,𝑡𝜌,𝑡𝜌𝑑𝜌𝑑𝑡𝑎^𝑛_𝑘0^1/2𝜉1₀𝜕𝜕𝜉₁−𝜕𝜕𝜂₁𝜉₁+𝜂₁^−1/2ℎ^𝜈_𝜈−2𝑘𝜉₁,𝜂₁𝑔𝜉₁,𝜂₁𝜉₁+𝜂₁^1/2𝑑𝜂₁𝑑𝜉₁2=−𝑎^𝑛_𝑘0^1/2𝜉₁^𝜈−2𝑘𝐺𝜉₁𝑑𝜉₁2=−𝑎^𝑛_𝑘𝛾_𝑘, (4.56) with coefficients 𝑎^𝑛_𝑘≠0 from (3.3). Then Lemma 4.1 gives 𝜏(𝜉)=^[(𝜈+1)/2]_𝑘=0𝐶^𝑛_𝑘𝛼^𝑛_𝑘𝜉^{−(𝜈−2𝑘)}+𝜓(𝜉), (4.57) where the constants 𝐶^𝑛_𝑘≠0 are independent of 𝑓 and |𝜓(𝜉)|≤𝐶𝐴𝜉. Hence, for the solution 𝑣(𝜉,𝜂) of Problem (P2.1) we have 𝑣𝜉,𝜂=𝜏(𝜉)+_𝜉^1/2𝜏𝜉₁𝜕𝜕𝜉₁𝑃_𝜈𝜉−𝜂𝜉₁+2𝜉𝜂𝜉₁𝜉+𝜂𝑑𝜉₁−_𝜉^1/2𝜂₀𝑃_𝜈𝜉−𝜂𝜉₁−𝜂₁+2𝜉₁𝜂₁+2𝜉𝜂𝜉₁+𝜂₁𝑔𝜉𝜉+𝜂¹,𝜂₁𝑑𝜂₁𝑑𝜉₁=^[(𝜈+1)/2]_𝑘=0𝐶^𝑛_𝑘𝛽^𝑛_𝑘𝐺^𝑛_𝑘𝜉,𝜂𝜉^{−(𝜈−2𝑘)}+Ψ₁𝜉,𝜂, (4.58) where 𝐺^𝑛_𝑘𝜉,𝜂=1+𝜉^𝜈−2𝑘_𝜉^1/2𝜉₁^2𝑘−𝜈𝜕𝜕𝜉₁𝑃_𝜈𝜉−𝜂𝜉₁+2𝜉𝜂𝜉₁𝜉+𝜂𝑑𝜉₁,Ψ₁𝜉,𝜂=𝜓(𝜉)+_𝜉^1/2𝜓𝜉₁𝜕𝜕𝜉₁𝑃_𝜈𝜉−𝜂𝜉₁+2𝜉𝜂𝜉₁𝜉+𝜂𝑑𝜉₁−_𝜉^1/2𝜂₀𝑃_𝜈𝜉−𝜂𝜉₁−𝜂₁+2𝜉₁𝜂₁+2𝜉𝜂𝜉₁+𝜂₁𝑔𝜉𝜉+𝜂¹,𝜂₁𝑑𝜉₁𝑑𝜂₁. (4.59)(4.60) Notice that the arguments of the Legendre’s functions 𝑃_𝜈 in (4.58), (4.59) and (4.60) vary in the interval [0,1]. Thus, 𝐺^𝑛_𝑘𝜉,𝜂≤1+𝐶₁𝜂𝜉^{𝜈−2𝑘+1}𝜉+𝜂_𝜉^1/2𝜉₁^{2𝑘−𝜈−2}𝑑𝜉₁𝐶≤1+¹𝜈+1−2𝑘=𝐶. (4.61) Therefore, the functions 𝐹^𝑛_𝑘𝜌,𝑡∶=2^𝜈−2𝑘𝐶^𝑛_𝑘𝐺^𝑛_𝑘𝜌+𝑡2,𝜌−𝑡2 (4.62) are also bounded. On the other hand, 𝑣(𝜉,0)=𝜏(𝜉) and therefore 𝐹^𝑛_𝑘(𝑡,𝑡)=2^𝜈−2𝑘𝐶^𝑛_𝑘𝐺^𝑛_𝑘(𝑡,0)=2^𝜈−2𝑘𝐶^𝑛_𝑘 (4.63) with coefficients 𝐶^𝑛_𝑘≠0 from (4.57).
Let us now evaluate the function Ψ₁ defined in (4.60). We have |𝜓(𝜉)|≤𝐶𝐴𝜉, _𝜉^1/2𝜓𝜉₁𝜕𝜕𝜉₁𝑃_𝜈𝜉−𝜂𝜉₁+2𝜉𝜂𝜉₁𝜉+𝜂𝑑𝜉₁≤𝐶𝐴𝜉𝜂𝜉+𝜂_𝜉^1/2𝜉₁⁻¹𝑑𝜉₁≤𝐶𝐴𝜉ln𝜉_𝜉1/2𝜂0𝑃_𝜈𝜉−𝜂𝜉₁−𝜂₁+2𝜉₁𝜂₁+2𝜉𝜂𝜉₁+𝜂₁𝑔𝜉𝜉+𝜂¹,𝜂₁𝑑𝜉₁𝑑𝜂₁≤𝐶𝐴𝜉. (4.64) Finally, let us return to (𝜌,𝑡) coordinates using (3.17) and (3.18). The representation (4.58) gives (4.51), where the function 𝐹^𝑛𝜌,𝑡∶=𝜌^−1/2ln𝜌⁻¹Ψ₁𝜌+𝑡2,𝜌−𝑡2 (4.65) is continuous in 𝐷₁ and the estimate |𝐹^𝑛(𝜌,𝑡)|≤𝐶₁𝐴𝜌^1/2, holds with 𝐶₁=const.

Finally, we are ready to prove our main result.

Proof of Theorem 2.3. The uniqueness and the existence of the generalized solution when 𝑓∈𝐶¹(Ω) is a trigonometric polynomial, follows from the results in [6]. Now the right-hand side function satisfies (2.5), and thus it can be written in the form 𝑓𝑥₁,𝑥₂,𝑡=^𝑙_𝑛=2𝑓¹_𝑛(|𝑥|,𝑡)cos𝑛𝜑+𝑓²_𝑛(|𝑥|,𝑡)sin𝑛𝜑. (4.66) According to [6] the unique generalized solution 𝑢(𝑥₁,𝑥₂,𝑡) also has the form 𝑢𝑥₁,𝑥₂,𝑡=^𝑙_𝑛=2𝑢¹_𝑛𝜌,𝑡cos𝑛𝜑+𝑢²_𝑛𝜌,𝑡sin𝑛𝜑, (4.67) where the functions 𝑢^𝑖_𝑛(𝜌,𝑡) are solutions of Problem (P2.1) with right-hand side function 𝑓^𝑖_𝑛∈𝐶¹(𝐺) and are described in Theorem 4.2. Then, for the constants 𝛼^𝑛_𝑘 from Theorem 4.2 and 𝛽^𝑛_𝑘,𝑖 from (2.6), we have the following relation: 𝛼^𝑛_𝑘=₀^1/2_𝑡^1−𝑡𝐸^𝑛_𝑘𝜌,𝑡𝑓^𝑖_𝑛𝜌,𝑡𝜌𝑑𝜌𝑑𝑡=𝜋⁻¹_Ω𝑊^𝑛_𝑘,𝑖(𝑥,𝑡)𝑓(𝑥,𝑡)𝑑𝑥𝑑𝑡=𝜋⁻¹𝛽^𝑛_𝑘,𝑖. (4.68) Therefore, from Theorem 4.2 it follows that 𝑢^𝑖_𝑛𝜌,𝑡=𝜋^−1[𝑛/2]_𝑘=0𝜌^−1/2𝜌+𝑡^{−(𝑛−2𝑘−1/2)}𝛽^𝑛_𝑘,𝑖𝐹_𝑘^𝑛,𝑖𝜌,𝑡+𝜌^1/2ln𝜌𝐹^𝑛,𝑖𝜌,𝑡, (4.69) where the functions 𝐹_𝑘^𝑛,𝑖 are independent of 𝑓,  |𝐹_𝑘^𝑛,𝑖(𝜌,𝑡)|≤𝐶 and 𝐹^𝑛,𝑖𝜌,𝑡≤𝐶max_𝐷1𝑓^𝑖_𝑛+max_𝐷1𝑓^𝑖_𝑛𝑡≤𝐶₁max_Ω𝑓(𝑥,𝑡)+max_Ω𝑓_𝑡(𝑥,𝑡). (4.70) Summing over 𝑛 and 𝑖 in (4.67) we get the expansion 𝑢(𝑥,𝑡)=^𝑙2_𝑛=2𝑖=1^[𝑛/2]_𝑘=0|𝑥|²+𝑡^{2𝑘−𝑛/2}𝛽^𝑛_𝑘,𝑖𝐹^𝑛_𝑘,𝑖()+|𝑥|𝑥,𝑡₂+𝑡^21/4𝐹(𝑥,𝑡)ln|𝑥|²+𝑡², (4.71) where 𝐹^𝑛_𝑘,1(𝑥,𝑡)=𝜋⁻¹𝜌^−1/2𝜌²+𝑡^{2𝑘−𝑛/2}𝜌+𝑡^{2𝑘−𝑛+1/2}𝐹_𝑘^𝑛,1𝐹𝜌,𝑡cos𝑛𝜑,_𝑛𝑘,2(𝑥,𝑡)=𝜋⁻¹𝜌^−1/2𝜌²+𝑡^{2𝑘−𝑛/2}𝜌+𝑡^{2𝑘−𝑛+1/2}𝐹_𝑘^𝑛,2𝐹()=|𝑥|𝜌,𝑡sin𝑛𝜑,𝑥,𝑡^1/2ln|𝑥||𝑥|²+𝑡^21/4ln|𝑥|²+𝑡^2𝑙𝑛=2𝐹^𝑛,1𝜌,𝑡cos𝑛𝜑+𝐹^𝑛,2𝜌,𝑡sin𝑛𝜑. (4.72) Obviously the functions 𝐹^𝑛_𝑘,𝑖(𝑥,𝑡) are bounded and independent of 𝑓. Also, we have |𝐹(𝑥,𝑡)|≤𝐶max_Ω𝑓(𝑥,𝑡)+max_Ω𝑓_𝑡(𝑥,𝑡). (4.73)
Notice that the singularity (|𝑥|²+𝑡²)^−𝑚/2 for fixed 𝑚 appears in the sum for the solution 𝑢(𝑥,𝑡) when 𝑛=𝑚,𝑚+2,𝑚+4,…, and the corresponding coefficients are 𝛽^𝑚_0,𝑖𝐹^𝑚_0,𝑖; 𝛽^𝑚_1,𝑖𝐹^𝑚+2_1,𝑖; 𝛽^𝑚_2,𝑖𝐹^𝑚+4_2,𝑖; an so on, until 𝑛≤𝑙. Therefore, (4.71) is equivalent to 𝑢(𝑥,𝑡)=^𝑙_𝑚=0|𝑥|²+𝑡^{2−𝑚/2[(𝑙−𝑚)/2]2}_𝑘=0𝑖=1𝛽^𝑚+2𝑘_𝑘,𝑖𝐹^𝑚+2𝑘_𝑘,𝑖()+|𝑥|𝑥,𝑡₂+𝑡^21/4𝐹(𝑥,𝑡)ln|𝑥|²+𝑡². (4.74) Thus, the properties (i), (ii), and (v) are proved.
Finally, let us prove the properties (iii) and (iv). For a fixed direction (𝛼₁,𝛼₂,1)𝑡=(cos𝛾,sin𝛾,1)𝑡 ∈𝑆₂, 0<𝑡<1/2,𝛾∈[0,2𝜋) we have the expressions 𝐹^𝑛_𝑘,1𝛼₁𝑡,𝛼₂𝑡,𝑡=𝜋⁻¹2^{2𝑘−𝑛+1/2}𝐹_𝑘^𝑛,1()𝐹𝑡,𝑡cos𝑛𝛾,_𝑛𝑘,2𝛼₁𝑡,𝛼₂𝑡,𝑡=𝜋⁻¹2^{2𝑘−𝑛+1/2}𝐹_𝑘^𝑛,2(𝑡,𝑡)sin𝑛𝛾, (4.75) with the functions 𝐹_𝑘^𝑛,𝑖(𝜌,𝑡) from (4.69) and 𝐹^𝑛_𝑘,𝑖(𝑥₁,𝑥₂,𝑡) from (2.8). Therefore, according to (2.8) and (4.74), lim_𝑡→+0𝐹^𝑚𝛼₁𝑡,𝛼₂=𝑡,𝑡_{[(𝑙−𝑚)/2]𝑘=0}𝐶^𝑚_𝑘,1𝛽^𝑚+2𝑘_𝑘,1cos(𝑚+2𝑘)𝛾+𝐶^𝑚_𝑘,2𝛽^𝑚+2𝑘_𝑘,2sin(𝑚+2𝑘)𝛾 (4.76) with some constants 𝐶^𝑚_𝑘,𝑖≠0. Thus, this expression is zero for all 𝛾∈[0,2𝜋] if and only if all the constants 𝛽^𝑚+2𝑘_𝑘,𝑖 involved are zero, because the trigonometric functions are linearly independent in [0,2𝜋]. Thus, if at least one 𝛽^𝑚+2𝑘_𝑘,𝑖≠0, one could choose 𝛾, that is, a direction (𝛼₁,𝛼₂,1), such that lim_𝑡→+0𝐹^𝑚(𝛼₁𝑡,𝛼₂𝑡,𝑡)=𝐶_𝑚=const≠0, which proves (iii).
For (iv) in the case 𝑚=0 we have lim_𝑡→+0𝐹⁰𝛼₁𝑡,𝛼₂𝑡,𝑡=^[𝑙/2]_𝑘=1𝐶⁰_𝑘,1𝛽^2𝑘_𝑘,1cos2𝑘𝛾+𝐶⁰_𝑘,2𝛽^2𝑘_𝑘,2sin2𝑘𝛾, (4.77) and the sum starts at 𝑘=1 since 𝛽⁰_0,1=𝛽⁰_0,2=0 according to Definition (2.6) and the special form (2.5) of 𝑓. Now, 𝐹⁰ is continuous at (0,0,0) only when the expression in the right-hand side of (4.77) is a constant. However, the constant 1 and the trigonometric functions involved in (4.77) are linearly independent. Therefore, if at least one 𝛽^2𝑘_𝑘,𝑖 is not zero, then 𝐹⁰ is not continuous at the origin.

Acknowledgments

This research was supported in part by the research grants of the Priority R&D Group for Mathematical Modelling, Numerical Simulation and Computer Visualization at Narvik University College. The research of N. Popivanov and T. Popov was partially supported by the Bulgarian NSF under Grants DO 02-115/2008 and DO 02-75/2008. The authors would like to thank the anonymous referees for the useful comments and suggestions.