# Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations

Authors: Nedyu Popivanov; Tsvetan Hristov; Aleksey Nikolov; Manfred Schneider
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

We study a boundary value problem for (3 + 1)-D weakly hyperbolic equations of Keldysh type (problem PK). The Keldysh-type equations are known in some specific applications in plasma physics, optics, and analysis on projective spaces. Problem PK is not well-posed since it has infinite-dimensional cokernel. Actually, this problem is analogous to a similar one proposed by M. Protter in 1952, but for Tricomi-type equations which, in part, are closely connected with transonic fluid dynamics. We consider a properly defined, in a special function space, generalized solution to problem PK for which existence and uniqueness theorems hold. It is known that it may have a strong power-type singularity at one boundary point even for very smooth right-hand sides of the equation. In the present paper we study the asymptotic behavior of the generalized solutions of problem PK at the singular point. There are given orthogonality conditions on the right-hand side of the equation, which are necessary and sufficient for the existence of a generalized solution with fixed order of singularity.


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## Body

## 1. Statement of the Problem

Form∈R,0<m<2, we consider the equation(1)Lmu≡ux1x1+ux2x2+ux3x3-tmutt=fx,tin the domain(2)Ωm≔x,t:0<t<t0,22-mt2-m/2<x<1-22-mt2-m/2,where (x,t)≔(x1,x2,x3,t)∈R4,x=x12+x22+x32 and t0=(2-m)/42/(2-m).The regionΩm (see Figure 1) is bounded by the ball Σ0≔x,t:t=0,x<1 centered at the origin O=(0,0,0,0) and by two characteristic surfaces of (1): (3)Σ1m≔x,t:0<t<t0,x=1-22-mt2-m/2,Σ2m≔x,t:0<t<t0,x=22-mt2-m/2.Figure 1
RegionΩm.In our case (0<m<2), the hyperplane t=0 is tangential to the characteristics Σ1m and Σ2m.In the given domain, (1) is hyperbolic, with parabolic power-type degeneration at Σ0⊂t=0; that is, we have a weakly hyperbolic equation of Keldysh type.We study the following boundary value problem.Problem
P
K. Find a solution to (1) in Ωm which satisfies the boundary conditions:(4)uΣ1m=0,tmut⟶0ast⟶+0.The adjoint problem to PK is as follows.Problem
P
K
∗. Find a solution to the self-adjoint equation (1) in Ωm which satisfies the boundary conditions:(5)uΣ2m=0,tmut⟶0ast⟶+0.


## 2. The Main Results

Problem PK is not well-posed. Actually, the adjoint homogeneous problemPK∗ has infinitely many classical solutions.In order to give their exact representation, fork,n∈N∪0, let us introduce the following functions:(6)Ek,mnx,t≔∑i=0kAikx-n+2i-1x2-42-m2t2-mn-k-i-m/4-2m,where (7)Aik≔-1ik-i+1in-k-i+4-3m/4-2mii!n+1/2-iiwith (a)i≔Γ(a+i)/Γ(a), which gives (a)i=aa+1⋯(a+i-1), for i∈N, and (a)0=1.Further, let us denote byYns(x),n∈N∪0,s=1,2,…,2n+1 the three-dimensional spherical functions. They are usually defined on the unit sphere S2≔x∈R3:x=1, but for convenience of our discussions we extend them out of S2 radially, keeping the same notation for the extended functions: (8)Ynsx≔Ynsxx,x∈R3∖0.In this paper, we prove the following lemma.Lemma 1.
For allm∈R,0<m<2,k,n∈N∪0,n≥N(m,k) and s=1,2,…,2n+1, the functions(9)vk,mn,sx,t≔Ek,mnx,tYnsx,x,t≠O,0,x,t=Oare classical solutions of the homogeneous problem PK∗.It is easy to see that a necessary condition for the existence of a classical solution of problem PK is the orthogonality of the right-hand side functionf(x,t) to all these functions vk,mn,s(x,t). Indeed, (10)∫Ωmvk,mn,sx,tfx,tdxdt=∫Ωmvk,mn,sx,tLmux,tdxdt=∫ΩmLmvk,mn,sx,tux,tdxdt=0.This means that an infinite number of orthogonality conditions μk,mn,s=0 with(11)μk,mn,s≔∫Ωmvk,mn,sx,tfx,tdxdtmust be fulfilled.To avoid this we consider solutions to this problem in a generalized sense. In the present paper we study the case0<m<4/3 and we use the following definition of a generalized solution of problem PK.Definition 2 (see [1]).
We call a functionu(x,t) a generalized solution of problem PK in Ωm,0<m<4/3, for (1), if(1)
u,uxj∈C(Ω¯m∖O),j=1,2,3,ut∈C(Ω¯m∖Σ¯0);(2)
uΣ1m=0;(3)
for eachε∈(0,1) there exists a constant C(ε)>0, such that(12)utx,t≤Cεt-3m/4inΩm∩x>ε;(4)
the identity(13)∫Ωmtmutvt-ux1vx1-ux2vx2-ux3vx3-fvdx1dx2dx3dt=0holds for allv from(14)Vm≔vx,t:v∈C2Ω¯m,vΣ2m=0,v≡0in  a  neighborhood  ofO.We mention that the inequality (12) restricts the generalized solution’s function space to a class which is smaller than it is allowed by the second boundary condition in (4).Note that Definition2 allows the generalized solution to have some singularity at the point O. The results of the present paper show that indeed there exist such singular solutions to this problem.In our recent paper [1], we proved the following results on the existence and uniqueness of a generalized solution of problem PK.Theorem 3 (see [1]).
Ifm∈(0,4/3), then there exists at most one generalized solution of problem PK in Ωm.Theorem 4 (see [1]).
Letm∈(0,4/3). Suppose that the right-hand side function f(x,t) is fixed as a “harmonic polynomial” of order l with l∈N∪0:(15)fx,t=∑n=0l∑s=12n+1fnsx,tYnsxand f∈C1(Ω¯m). Then there exists an unique generalized solution u(x,t) of problem PK in Ωm and it has the following form:(16)ux,t=∑n=0l∑s=12n+1unsx,tYnsx.In this paper, we derive an asymptotic formula concerning the behavior of the singularities of the generalized solution.Theorem 5.
Letm∈(0,4/3) and the right-hand side function f∈C1(Ω¯m) has the form (15). Then the unique generalized solution u(x,t) of problem PK on the characteristic surface (17)Σ2m=x,t:0<x<12,t=τx≔2-12-mx2/2-mhas the following expansion at point O:(18)ux,τx=∑p=0lGpxx-p-1+Gxx-m/4-2m,where(i)
the functionG∈C1(0<x<1/2) and satisfies the a priori estimate (19)Gx≤CmaxΩ¯m⁡fwith a constantC>0 independent of f;(ii)
the functionsGp(x) have the following structure:(20)Gpx=∑k=0l-p/2∑s=12p+4k+1ck,mp+2k,sμk,mp+2k,sYp+2ksx,p=0,1,…,l,whereck,mp+2k,s≠0 are constants independent of f(x,t).Corollary 6.
Suppose that at least one of the constantsμk,mp+2k,s in (20) is different from zero. Then for the corresponding function Gp(x) there exists a vector α∈R3,α=1, such that Gpαϱ→cp,α=const≠0asϱ→+0. This means that the order of singularity of u(x,t) will be no smaller than p+1.Corollary 7.
Let the conditions of Theorem5 be fulfilled and in addition f(x,t) satisfies the orthogonality conditions:(21)∫Ωmvk,mn,sx,tfx,tdxdt=0for all n=0,1,…,l;k=0,1,…,[n/2] and s=1,2,…,2n+1. Then the unique generalized solution u(x,t) of problem PK fulfills the a priori estimate on Σ2m: (22)ux,τx≤CmaxΩ¯m⁡fx-m/4-2m,where C is a positive constant independent of f(x,t).Actually, Theorem5 gives the asymptotic behavior of the singular solutions of problem PK on Σ2m. It clarifies the significance of the orthogonality conditions (21): for fixed indexes (n,k,s), the corresponding condition (21) “controls” one power-type singularity. We mention here that some of the orthogonality conditions (21) involve functions vk,mn,s(x,t), which are not classical solutions of problem PK∗ (see the proof of Lemma 1).


## 3. History of the Problem and Motivation

It is well-known that different boundary value problems (BVPs) for mixed-type equations have important applications in transonic gas dynamics (see Bers [2], Morawetz [3], and Kuz’min [4]). After a space symmetry assumption, the transonic potential flows in fluid dynamics are described in the hodograph plane by two-dimensional BVPs for the Chaplygin equation:(23)Ktuxx-utt=0,where tK(t)>0 for t≠0. The Chaplygin equation (23) is elliptic in the subsonic half-plane t<0 and hyperbolic in the supersonic half-plane t>0.In particular, certain flows around airfoils are modeled by the Guderley-Morawetz plane problem for (23) (see the monograph of Bers [2]). The domain is bounded in the elliptic half-plane by a smooth arc σ and in the hyperbolic half-plane by four characteristic segments that start from the three points A1, A2, and B on the sonic line; see Figure 2(a). The values of the function are prescribed along σ and along the characteristics A1C1 and A2C2. The Guderley-Morawetz problem is well studied. The existence of weak solutions and the uniqueness of a strong solution in weighted Sobolev spaces were obtained by Morawetz [5]. Lax and Phillips [6] proved that the weak solutions are strong.Figure 2
(a) Guderley-Morawetz domain. (b) Protter-Morawetz domainΩ~.
(a)
(b)Results on another BVP for the Chaplygin equation in mixed-type domain can be found in the recent paper of Liu et al. [7].An interesting multidimensional generalization of Guderley-Morawetz problem was proposed by Protter [8, 9] for the multidimensional Chaplygin equation:(24)KtΔxu-utt=Kt∑j=1Nuxjxj-utt=fx,t,where x=(x1,…,xN),N≥2. Protter considered (24) in the Protter-Morawetz domain Ω~, which could be obtained by rotating of a symmetrical Guderley-Morawetz domain around the axis of symmetry (see Figure 2). The boundary data are prescribed on Γ in the elliptic part and on the outer characteristic surface Σ~1. On the characteristic surface Σ~2, data are not imposed. Aziz and Schneider [10] obtained uniqueness result for this problem, but even now there is not a single example of a nontrivial solution to the multidimensional problem (as, for example, in Lemma 1 above), neither a general existence result is known. Many difficulties and differences in comparison with the planar problems can be illustrated as well by the related problems in the hyperbolic part of the domain, also formulated by Protter.Protter Problems. Find a solution of (24) with K(t)=tm,m∈R,m≥0 in the domain Ω~∩t>0 with one of the following boundary conditions: (25)P1:uΣ0∪Σ~1=0,P2:uΣ~1=0,utΣ0=0.These BVPs are multidimensional analogues of the Darboux-Goursat plane problems for the Gellerstedt equation (m>0) or for the wave equation (m=0). Garabedian [11] proved the uniqueness of a classical solution to problem P1 for the wave equation in R4. Popivanov and Schneider [12] showed that both problems P1 and P2 are not well-posed in the frame of classical solvability, since they have infinite-dimensional cokernels (see also Khe [13]). In [12], they suggested to study the Protter problems in the frame of generalized solutions with possible big singularities. Today it is well-known that the Protter problems have singular generalized solutions, even for smooth right-hand sides [12, 14–17]. Different aspects of Protter problems and several their generalizations (including some applications in the industrial explosion process) are studied by many authors (see Aldashev and Kim [18], Choi and Park [19], Aldashev [20], and references therein). For different statements of other related problems for mixed-type equations of the first kind, including nonlinear equations, see [21–27].The Keldysh-type equations are another kind of mixed-type equations that also are known to play an important role in fluid mechanics, for example,(26)uxx+tmutt+aux+but+cu=0near the line t=0.Otway [28, 29] and Lupo et al. [30] gave a statement of some 2D BVPs for elliptic-hyperbolic Keldysh-type equations with specific applications in plasma physics, including a model for analyzing the possible heating in axisymmetric cold plasmas. Čanić and Keyfitz [31] studied some plane problems for a nonlinear degenerate elliptic equation, whose solutions behave like those of a Keldysh-type equation. Such an equation arises in the modeling of a weak shock reflection at a wedge. A 2D mixed-type equation analogous in part to the Tricomi-type and the Keldysh-type equations has also been studied recently by Shuxing [32].Keldysh [33] studied the regularity of the solutions of 2D elliptic equations of second order near the boundary, in the case when the boundary contains a segment of the line t=0. He showed that for degenerating elliptic equation (26) the formulation of the Dirichlet problem may depend on the lower order terms (the dependence is different for different values of m). Fichera [34] generalized Keldysh’s results for multidimensional linear second-order equations with nonnegative characteristic form and now BVPs for them are well understood in the sense that boundary conditions should not be imposed on the whole boundary. A summary of Fichera’s theory can be found in Radkevich [35, 36]. Keyfitz [37] examined whether the Fichera’s classification could be extended to quasilinear equations and mentioned that contrasting behavior of the characteristics of the Tricomi and Keldysh equation (see Figures 1 and 2) may have implications, unexplored yet, for the solution of some free boundary problems arising in the fluid dynamics models.All these results and the fact that the solutions of the Keldysh-type equation are not differentiable at the degenerate boundary (see [38]) make it interesting to formulate and study the multidimensional Protter-Morawetz problem for Keldysh-type equations. In [39], using the exact Hardy-Sobolev inequality, we proved the uniqueness of a quasiregular solution to problem PK for equations involving lower order terms. Let us mention here that, in problem PK, unlike Tricomi case, a data on the degenerate boundary is not prescribed (similar to the elliptic case) and derivative ut can have singularity on it, but up to the prescribed level. On the other hand, the results in [1] and the results of the present paper show some similarities between problem PK and problems P1, P2: the infinite-dimensional cokernel of the problem and the existence of generalized solutions with isolated singularities.There are still some open questions in this area that naturally arise.Open Problems
(1)
In the case when the right-hand side functionf(x,t) is a “harmonic polynomial” to find additional conditions under which problem PK has a bounded solution. According to Corollary 7, when all the orthogonality conditions, which we prescribe in the present paper, are fulfilled, the generalized solution u(x,t) is still allowed to have a singularity of order β∈(0,1).
(2)
To study the general case of problem PK when the right-hand side functionf(x,t) is a smooth function not only of the form of “harmonic polynomial” is an open problem:
(i)
Find appropriate conditions for the functionf(x,t) under which there exists a generalized solution.
(ii)
What kind of singularity may have the generalized solution in this case? The a priori estimate, obtained in [1], shows that when the function f(x,t) is a “harmonic polynomial” the generalized solution may have at most a polynomial growth. Are there exist singular solutions with an exponential growth, as it is in the case of the Protter problems for the usual wave equation?
(iii)
To find some appropriate conditions for the functionf(x,t) under which problem PK has only regular, bounded, or even classical solution. Up to now such conditions for the existence of a bounded solution to Protter problems are obtained only in the case of the wave equation.
(3)
To study problem PK in the more general case when0<m<2. Let us mention that the presentation of the generalized solution u(x,t) from [1], which we are studying in the present paper, is valid only in the case when m∈(0,4/3). Find appropriate techniques that work for 4/3≤m<2.


## 4. The Two-Dimensional Darboux-Goursat Problems Corresponding to Problem PK

Problem PK in the case when the right-side functionf(x,t) is of the form (15) can be reduced to a two-dimensional problem.More precisely, let us look for a solution to problem PK of the form (16). Using the spherical coordinates r,θ,φ,t∈R4,0≤θ<π,0≤φ<2π,r>0 with(27)x1=rsin⁡θcos⁡φ,x2=rsin⁡θsin⁡φ,x3=rcos⁡θ,and later in the characteristic coordinates(28)ξ=1-r-22-mt2-m/2,η=1-r+22-mt2-m/2,for the functions (29)Uξ,η≔rξ,ηunsrξ,η,tξ,ηwe obtain (see [1]) the following Darboux-Goursat problem.Problem
P
K
2. Find a solution of(30)Uξη+βη-ξUξ-Uη-nn+12-ξ-η2U=Fξ,ηinD,satisfying the following boundary conditions:(31)U0,η=0,limη-ξ→+0⁡η-ξ2βUξ-Uη=0,where (32)D≔ξ,η:0<ξ<η<1,(33)Fξ,η≔182-ξ-ηfnsrξ,η,tξ,η,(34)β≔m22-m.As far as we consider problem PK in the case when m∈(0,4/3) for the parameter β we have (35)0<β<1.In conformity with Definition2, a generalized solution of problem PK2 is defined as follows.Definition 8 (see [1]).
We call a functionU(ξ,η) a generalized solution of problem PK2 in D, (0<β<1), if(1)
U,Uξ+Uη∈C(D¯∖(1,1)),Uξ-Uη∈C(D¯∖η=ξ);(2)
(36)U0,η=0;(3)
for eachε∈(0,1) there exists a constant C(ε)>0, such that(37)Uξ-Uηξ,η≤Cεη-ξ-βinD∩ξ<1-ε;(4)
the identity(38)∫Dη-ξ2βUξVη+UηVξ+2nn+12-ξ-η2UV+2FVdξdη=0holds for all(39)V∈V2≔Vξ,η:V∈C2D¯,Vξ,1=0,V≡0in  a  neighborhood  of1,1.Further, using the Riemann-Hadamard functionΦ(ξ,η;ξ0,η0) associated with problem PK2, an explicit integral representation of the generalized solution U(ξ,η) was found. A survey of the Riemann method can be found in [40].Theorem 9 (see [1]).
Let0<β<1 and F∈C1(D¯). Then there exists one and only one generalized solution of problem PK2 in D, which has the following integral representation at a point (ξ0,η0)∈D:(40)Uξ0,η0=∫0ξ0∫ξη0Fξ,ηΦξ,η;ξ0,η0dηdξ,and it satisfies the following estimates:(41)Uξ,η≤KMFξ2-ξ-η-ninD¯∖1,1,Uξ+Uηξ,η≤KMF2-ξ-η-n-1inD¯∖1,1,Uηξ,η≤KMFξη-ξ-β2-ξ-η-n-1inD¯∖η=ξ,where K is a positive constant and (42)MF≔max⁡maxD¯⁡F,maxD¯⁡Fξ+Fη.The Riemann-Hadamard functionΦ(ξ,η;ξ0,η0),(ξ0,η0)∈D, which we have found in [1], can be represented as follows:(43)Φξ,η;ξ0,η0=Φ+ξ,η;ξ0,η0,η>ξ0,Φ-ξ,η;ξ0,η0,η<ξ0,where(44)Φ+ξ,η;ξ0,η0≔η-ξη0-ξ0βF3β,n+1,1-β,-n,1;X,Y,Φ-ξ,η;ξ0,η0≔γη-ξη0-ξ0βX-βH2β,β,-n,n+1,2β;1X,-Y,(45)X=Xξ,η,ξ0,η0≔ξ0-ξη0-ηη-ξη0-ξ0,(46)Y=Yξ,η,ξ0,η0≔-ξ0-ξη0-η2-ξ-η2-ξ0-η0,(47)γ=ΓβΓ1-βΓ2β.Here F3(a1,a2,b1,b2,c;x,y) is the Appell series:(48)F3a1,a2,b1,b2,c;x,y≔∑i=0∞∑j=0∞a1ja2ib1jb2ici+ji!j!xjyi,and H2(a1,a2,b1,b2,c;x,y) is the Horn series:(49)H2a1,a2,b1,b2,c;x,y≔∑i=0∞∑j=0∞a1j-ia2jb1ib2icji!j!xjyi(for basic information on the Appell and the Horn series, see [41], p. 222–228.)According to Theorem9, the generalized solution is allowed to have a singularity of order no greater than n at point (1,1). But it is still not clear if such a singularity really exists and how it depends on the right-hand side of the equation. In the next section, we study more deeply the function U(ξ,η), given by (40), or, more precisely, its restriction on the segment η=1, 0≤ξ<1.


## 5. The Asymptotic Expansion of the Solution of Problem PK2

Introduce the following functions:(50)E~knξ,η≔rξ,ηEk,mnrξ,η,tξ,η=∑i=0kAik1-ξn-k-i-β1-ηn-k-i-β22i-n2-ξ-ηn-2i,where Ek,mn(x,t) are functions (6), closely connected with the solutions of the homogeneous adjoint problem PK∗. Then we prove the following lemma.Lemma 10.
Fork=0,…,[n/2] the following equalities hold (51)∫Dη-ξ2βE~knξ,ηFξ,ηdξdη=Cmμk,mn,s,Cm=23m-2/2-m2-m-m/2-m,where μk,mn,s are the coefficients (11) and the relation between F(ξ,η) and the Fourier coefficient fns from the expansion of f(x,t) is given by (33).Proof.
Denote(52)Gm≔r,t:0<t<t0,m2-mt2-m/m<r<1-m2-mt2-m/m.Denote also by Yns the spherical functions expressed in the spherical coordinates; that is, Yns(x)=Yns(θ(x),φ(x)). Then, using the orthonormality of the spherical functions on the unit sphere S2, a direct calculation gives (53)μk,mn,s=∫Ωmvk,mn,sx,tfx,tdxdt=∫0π∫02π∫GmEk,mnr,tYnsθ,φ∑p=0l∑q=12p+1fpqr,tYpqθ,φsin⁡θr2drdtdφdθ=∫S2Yns2θ,φdS∫GmEk,mnfnsr,tr2drdt=23m-2/m-22-mm/2-m∫Dη-ξ2βE~knξ,ηFξ,ηdξdη.
The proof is complete.Theorem 11.
Suppose thatF∈C1(D¯). Then the restriction U(ξ,1) of the generalized solution of problem PK2 has the following expansion on the segment 0≤ξ<1:(54)Uξ,1=∑k=0n/2bknμk,mn,s1-ξ-n+2k+1-ξ1-βgξ,where g(ξ)∈C1([0,1)) and gξ≤CFC(D), with constants C>0 and bkn≠0 independent of F.Proof.
According to Theorem9 the condition F∈C1(D¯) assures that there exists an unique generalized solution U(ξ,η) of problem PK2, given by (40). According to Definition 8, we see that the restriction U(ξ,1) should belong to C1([0,1)).
Further, we setη0=1 in (40). Essential for the following calculations is the decomposition of Φ-(ξ,η;ξ0,1) given in Theorem A.3 which we prove in Appendix. Using (A.34) we obtain(55)Uξ0,1=∫0ξ0∫ξ1Fξ,ηΦξ,η;ξ0,1dηdξ=∫0ξ0∫ξξ0Fξ,ηΦ1-ξ,η;ξ0dηdξ+∫0ξ0∫ξξ0Fξ,ηΦ2-ξ,η;ξ0dηdξ+∫0ξ0∫ξ01Fξ,ηΦ+ξ,η;ξ0,1dηdξ≕J1+J2+J3.According to (A.35) and Lemma 10 we have(56)J1=∑k=0n/2ckn1-ξ0-n+2k∫0ξ0∫ξξ0η-ξ2βFξ,ηE~k,mnξ,ηdηdξ=∑k=0n/2ckn1-ξ0-n+2k∫Dη-ξ2βFξ,ηE~k,mnξ,ηdηdξ-∫ξ01∫0ηη-ξ2βFξ,ηE~k,mnξ,ηdξdη=∑k=0n/2ckn1-ξ0-n+2kCmμ~k,mn-∫ξ01∫0ηη-ξ2βFξ,ηE~k,mnξ,ηdξdη=∑k=0n/2bknμ~k,mn1-ξ0-n+2k+J1,0,with bkn=Cmckn≠0.
The functionsE~kn, given by (50), can be estimated in D as follows: (57)E~k,mnξ,η≤akn1-ξ-β1-ηn-2k-β,akn=const>0.Then we have (58)J1,0≤∑k=0n/2ckn1-ξ0-n+2k∫ξ01∫0ηη-ξ2βFξ,ηE~k,mnξ,ηdξdη≤FCD∑k=0n/2aknckn∫ξ01∫0ηη-ξ2β1-ξ-β1-η-βdξdη≤k2FCD1-ξ01-β,k2=const>0.
ForJ2, using the estimate (A.36) from Theorem A.3, we obtain(59)J2≤k1FCD1-ξ0∫0ξ0∫ξξ0ξ0-η-β1-η-1dηdξ.Making a substitution η=ξ+(ξ0-ξ)σ we compute(60)∫ξξ0ξ0-η-β1-η-1dη=ξ0-ξ1-β1-ξ∫011-σ-β1-zσ-1dσwith (61)z≔ξ0-ξ1-ξ.Formula (A.11) gives(62)∫011-σ-β1-zσ-1dσ=Γ1-βΓ2-βF121,1,2-β;zand with (A.13) we estimate(63)F121,1,2-β;z≤k31-ξβ1-ξ0β,k3=const>0.Applying the results (60), (62), and (63) into (59) we obtain (64)J2≤k4FCD1-ξ01-β,k4=const>0.
According to the results from [1] (Lemmas A.1, A.2, and A.3 therein), we have an estimate (65)Φ+ξ,η;ξ0,1≤k5η-ξ0-β,k5=const>0.
Then forJ3 we have (66)J3≤k6FCD1-ξ01-β,k6=const>0.
Therefore (54) holds with g(ξ)≔(1-ξ)β-1(J1,0+J2+J3). Obviously, g(ξ)∈C1([0,1)), because J1,0+J2+J3=U(ξ,1)-∑k=0[n/2]‍bknμk,mn,s(1-ξ)-n+2k.
The proof is complete.From this theorem, we see that the generalized solution of problemPK2 may have a singularity of order n and this happens in the general case: a bounded solution, or a solution with a smaller order of singularity, is possible only if some of the coefficients μk,mn,s are equal to zero. This result exactly corresponds to the estimate prescribed in Theorem 9.


## 6. Proof of the Main Results

Now, we are ready to prove the main results stated in Section2.Proof of Lemma1.
First, we have obviouslyvk,mn,s(x,t)∈C∞(Ωm).
Forn>2k+m/(4-2m) it is easy to check that vk,mn,s∈C(Ω¯m∖O) and vk,mn,sΣ2m=0.
Forn>N(m,k)≔2k+1+m/(2-m) we see that Ek,mn(x,t)→0 as (x,t)→O.
Thereforevk,mn,s(x,t)∈C∞(Ωm)∩C(Ω¯m).
It is easy to check that forn>N(m,k) the boundary conditions (5) are also satisfied.
Now, let us look for solutions of the homogeneous problemPK∗ of the form (9). Passing to the spherical coordinates (27) in the homogeneous equation (1) and using that, the spherical functions satisfy the differential equation: (67)1sin⁡θ∂∂θsin⁡θ∂∂θYns+1sin2⁡θ∂2∂φ2Yns+nn+1Yns=0and we see that the functions Ek,mn(r,t) should be solutions of(68)vrr+2rvr-tmvtt-nn+1r2v=0in Gm. A direct calculation of the derivatives of Ek,mn(r,t) shows that these functions indeed satisfy (68).
The proof is complete.Proof of Theorem5.
Letu(x,t) be the unique generalized solution of problem PK. Then it has the form (16) (see Theorem 4)
As we saw, all the functions(69)Unsξ,η≔rξ,ηunsrξ,η,tξ,η,n=0,…,l,s=1,…,2n+1, are generalized solutions of problem PK2 with right-hand sides Fns(ξ,η) given by (70)Fnsξ,η≔182-ξ-ηfnsrξ,η,tξ,η.Now, Theorem 11 states that (71)Unsξ,1=∑k=0n/2bkn,sμk,mn,s1-ξ-n+2k+1-ξ1-βgnsξ,where bkn,s are nonzero constant independent of Fns,gns(ξ)∈C1([0,1)) and gnsξ≤CFnsC(D).
Now, using the relations (27)-(28), we make the inverse transformation from problem PK2 to problem PK. In this way we obtain the expansion (18), (20) with (72)Gx=21-β∑p=0l∑k=0l-p/2∑s=12p+4k+1Yp+2ksxgp+2ks1-2xand ck,mn,s=22k-nbkn,s≠0.
The assertion (i) in Theorem5 follows from the properties of the functions gns(ξ) and the fact that the functions Yns(x) are bounded and belong to C1(0<x<1/2).
The proof is complete.Proof of Corollary6.
Let at least one of the constantsμk,mp+2k,s in (20) (73)Gpx=∑k=0l-p/2∑s=12p+4k+1ck,mp+2k,sμk,mp+2k,sYp+2ksx,p=0,1,…,lbe different from zero. Then by the linear independence of the spherical functions Yns it follows that, for the corresponding function Gp(x), there exists a vector α∈S2, such that Gp(α)≠0. But, recalling that we extend functions Yns out of S2 radially, we have that Gp(α)=Gp(αϱ)=const,ϱ>0. Therefore Gp(αϱ)→cp,α=const≠0asϱ→+0.
The proof is complete.



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*Source: 1571959-2017-10-16.xml*