Open Access

On the existence and uniqueness of a generalized solution of the Protter problem for \((3+1)\)-D Keldysh-type equations

  • Nedyu Popivanov1Email author,
  • Tsvetan Hristov1,
  • Aleksey Nikolov2 and
  • Manfred Schneider3
Boundary Value Problems20172017:26

https://doi.org/10.1186/s13661-017-0757-1

Received: 19 November 2016

Accepted: 3 February 2017

Published: 13 February 2017

Abstract

\((3+1)\)-dimensional boundary value problem for equations of Keldysh type (the second kind) is studied. Such problems for equations of Tricomi type (the first kind) or for the wave equation were formulated by M.H. Protter (1954) as multidimensional analogues of Darboux or Cauchy-Goursat plane problems. Now, it is well known that Protter problems are not correctly set, and they have singular generalized solutions, even for smooth right-hand sides. In this paper an analogue of the Protter problem for equations of Keldysh type is given. An appropriate generalized solution with possible singularity is defined. Results for uniqueness and existence of such a generalized solution are obtained. Some a priori estimates are stated.

Keywords

weakly hyperbolic equations boundary value problems generalized solutions uniqueness behavior of solution

MSC

35D30 35M12 35A20

1 Introduction

In the present paper we consider an analogue of the Protter problems for \((3+1)\)-D Keldysh-type equations. For \(m \in \mathbf{R}\), \(0< m<2\), we study some boundary value problems (BVPs) for the weakly hyperbolic equation
$$ L_{m}[u]\equiv u_{x_{1}x_{1}}+u_{x_{2}x_{2}}+u_{x_{3}x_{3}} - \bigl(t^{m}u_{t} \bigr)_{t}=f(x,t), $$
(1.1)
expressed in Cartesian coordinates \((x,t)=(x_{1},x_{2},x_{3},t) \in \mathbf{R}^{4}\) in a simply connected region
$$\Omega_{m}:= \biggl\{ ( x,t ) : 0< t< t_{0}, \frac{2}{2-m} t^{\frac{2-m}{2}}< \sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}< 1- \frac{2}{2-m} t^{\frac{2-m}{2}} \biggr\} , $$
bounded by the ball \(\Sigma_{0}:=\{(x,t): t=0, \sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}<1\}\), centered at the origin \(O=(0,0,0,0)\) and by two characteristic surfaces of equation (1.1)
$$\begin{aligned}& \Sigma_{1}^{m}:= \biggl\{ (x,t):0< t< t_{0}, \sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}=1- \frac {2}{2-m} t^{\frac{2-m}{2}} \biggr\} , \\& \Sigma_{2}^{m}:= \biggl\{ (x,t): 0< t< t_{0}, \sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}= \frac{2}{2-m} t^{\frac{2-m}{2}} \biggr\} , \end{aligned}$$
where \(t_{0}:= ( \frac{2-m}{4} ) ^{\frac{2}{2-m}}\).

In this work we are interested in finding sufficient conditions for the existence and uniqueness of a generalized solution of the following problem.

Problem PK

Find a solution to equation (1.1) in \(\Omega_{m}\) that satisfies the boundary conditions
$$ u|_{\Sigma_{1}^{m}}=0; \quad\quad t^{m}u_{t} \rightarrow0, \quad \text{as } t\to+0. $$

The adjoint problem to PK is as follows.

Problem \(\mathbf {PK}^{\ast }\)

Find a solution to the self-adjoint equation (1.1) in \(\Omega_{m}\) that satisfies the boundary conditions
$$u|_{\Sigma_{2}^{m}}=0;\quad \quad t^{m}u_{t}\rightarrow0,\quad \text{as } t\to+0. $$

First, we present a brief historical overview here and provide an extensive list of references.

Protter arrived at similar problems, but for Tricomi-type equations, while studying BVPs which describe transonic flows in fluid dynamics. It is well known that most important boundary value problems that, in the case of linear mixed-type equations, appear in hodograph plane for two-dimensional transonic potential flows are the classical Tricomi, Frankl’, and Guderley-Morawetz problems. The first two for flows in nozzles and jets and the third one as an approximation in flows about airfoils. For such connections, see the paper of Morawetz [1]. About sixty years ago Murray Protter [2] stated a multidimensional variant of the famous Guderley-Morawetz plane problem for hyperbolic-elliptic equations that had been studied by Morawetz [3], Lax and Phillips [4]. This problem now is known as the Protter-Morawetz problem. A result for uniqueness was obtained by Aziz and Schneider [5] in the case of Frankl-Morawetz problem. However, the multidimensional case is rather different, and there is no general understanding of the situation. Even the question of well posedness is not completely resolved. For different statements of multidimensional Darboux-type problems or some related Protter-Morawetz problems for mixed-type equations, see [1, 613]. Some Tricomi problems for the Lavrent’ev-Bitsadze equation are studied in [1416]. On the other hand, different problems for elliptic-hyperbolic equations of Keldysh type have specific applications in plasma physics, optics, and analysis on projective spaces (see Otway [17, 18] and Otway and Marini [19]). Various statements of problems for mixed equations of Tricomi or Keldysh type can be found in Oleı̌nik and Radkevič [20], Nakhushev [21], and several applications of such problems in the study of transonic flows are described in Chen [22], Čanić and Keyfitz [23]. Let us also mention some results in the thermodynamic theory of porous elastic bodies given in [24, 25]. In order to analyze the spatial behavior of solutions, some appropriate estimates and similar procedures are used there.

In relation to the mixed-type problems, Protter also formulated and studied some BVPs in the hyperbolic part of the domain for the wave equation [26] and degenerated hyperbolic (or weakly hyperbolic) equations of Tricomi type [2]. In that case the Protter problems are multidimensional analogues of the plane Darboux or Cauchy-Goursat problems (see Kalmenov [27] and Nakhushev [28]). The equations are considered in \((3+1)\)-D domain, bounded by two characteristic surfaces and noncharacteristic plane region. The data are prescribed on one characteristic and on a noncharacteristic boundary part. Protter considered [2, 26] Tricomi-type equations or the wave equation (\(m \in\mathbf{R}\), \(m \geq0\))
$$ \tilde{L}_{m}[u]:= t^{m}[u_{x_{1}x_{1}}+u_{x_{2}x_{2}}+u_{x_{3}x_{3}}]- u_{tt}=f(x,t) $$
(1.2)
in the domain
$$ \tilde{\Omega}_{m}:= \biggl\{ ( x_{1},x_{2},x_{3},t ) : t>0, \frac{2}{m+2} t^{\frac{m+2}{2}}< \sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}< 1- \frac{2}{m+2} t^{\frac{m+2}{2}} \biggr\} , $$
bounded by \(\Sigma_{0}\) and two characteristics surfaces of (1.2)
$$\begin{aligned}& \tilde{\Sigma}_{1}^{m}= \biggl\{ t>0, \sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}=1- \frac{2}{m+2} t^{\frac{m+2}{2}} \biggr\} , \\& \tilde{\Sigma}_{2}^{m}= \biggl\{ t>0, \sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}= \frac{2}{m+2} t^{\frac{m+2}{2}} \biggr\} . \end{aligned}$$
He proposed four problems, known now as Protter problems.

Protter problems

Find a solution of equation (1.2) in the domain \(\tilde{\Omega}_{m}\) with one of the following boundary conditions:
$$\begin{aligned}& P1:\quad u|_{\Sigma_{0}\cup\tilde{\Sigma}_{1}^{m}}=0, \quad \quad P1^{\ast}:\quad u|_{\Sigma_{0}\cup\tilde{\Sigma}_{2}^{m}}=0; \\& P2:\quad u|_{\tilde{\Sigma}_{1}^{m}}=0,\quad\quad u_{t}|_{\Sigma_{0}}=0,\quad\quad P2^{\ast}:\quad u|_{\tilde{\Sigma}_{2}^{m}}=0,\quad\quad u_{t}|_{\Sigma_{0}}=0. \end{aligned}$$
The boundary conditions in problem P1 (respectively P2) are the adjoint boundary conditions to problem P1 (respectively P2) for (1.2) in \(\tilde{\Omega}_{m} \).

It turns out that instead of both boundary conditions given in problems P1 on \(\tilde{\Sigma}_{1}^{m}\), \(\Sigma_{0}\) and in P2 on \(\tilde{\Sigma}_{2}^{m}\), \(\Sigma_{0}\) for the Tricomi-type equation (1.2), in the case of Keldysh-type equation (1.1), they are reduced to only one boundary condition on the characteristic \(\Sigma_{1}^{m}\) and a condition on the growth of possible singularity of the derivative \(u_{t}\) as \(t \to+0\).

We mention some known results for Protter problems in the Tricomi case that make the investigation of such problems interesting and reasonable. Garabedian [29] obtained a result for the uniqueness of classical solution to problem P1 for the wave equation (i.e., equation (1.2) with \(m=0\)). It is interesting that contrary to their plane analogues, the 3-D Protter problems are not well posed (see [30, 31] and the monograph of Bitsadze [32]). The reason is that the adjoint homogeneous problems P1 and P2 have an infinite number of linearly independent nontrivial classical solutions. On the other hand, the unique generalized solutions of 3-D problems P1 and P2 could have strong power-type singularity on the \(\tilde{\Sigma}^{m} _{2}\) even for smooth right-hand sides (see [31, 33, 34]). Behavior of the singular solutions to 3-D problems P1 and P2 is studied in [35, 36]. Such results are announced for the 4-D case as well [37]. Didenko [38] studied problems P1 and P1 for the Tricomi equation (\(m = 1\)) in the symmetric case. Aldashev [39] studied some multidimensional analogues of Protter problems for equation (1.2), but he did not mention any possible singular solutions.

These known results for Protter problems for Tricomi-type equations and many interesting applications of different boundary value problems for equations of Keldysh type motivate us to study problems PK and \(\mathit {PK}^{\ast}\) and to try to find out new effects that appear. In [40] ill-posedness of 3-D Protter problems for Keldysh-type equations in the frame of classical solvability is discussed, and the results for uniqueness of quasi-regular solutions are obtained. Existence and uniqueness of generalized solutions to problem PK in that case are obtained in [41], and some singular generalized solutions are announced in [42].

In [31, 33] we study Protter problems for Tricomi-type equations. For 3-D Keldysh-type equation in [43], we formulate a new Protter problem and announce some results for the existence and uniqueness of a generalized solution in the case \(0< m<1\). In [44] we announce analogical results for \((3+1)\)-D equations of Keldysh type in a more general case \(0< m<4/3\) and claim the existence of infinitely many classical smooth solutions of the \((3+1)\)-D homogeneous problem \(\mathit {PK}^{\ast}\). Now, in the present paper we work in the case \(0< m<4/3\). Using an appropriate Riemann-Hadamard function, we find an exact integral representation of the generalized solution and prove the results announced in [44]. To avoid an infinite number of necessary conditions in the frame of classical solvability, we give a notion of a generalized solution to problem PK which can have some singularity at the point O. In order to deal successfully with the encountered difficulties for \(\varepsilon\in(0,1)\), we introduce the region
$$\Omega_{m,\varepsilon}:=\Omega_{m} \cap \bigl\{ \vert x\vert > \varepsilon \bigr\} , $$
where \(\vert x\vert =\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}\).

We give the following definition of a generalized solution of problem PK in the case \(0< m<4/3\).

Definition 1.1

We call a function \(u(x,t)\) a generalized solution of problem PK in \(\Omega_{m}\), \(0< m<\frac{4}{3}\), for equation (1.1) if:
  1. 1.

    \(u,u_{x_{j}} \in C(\bar{\Omega}_{m} \setminus O)\), \(j=1,2,3\), \(u_{t} \in C(\bar{\Omega}_{m} \setminus\bar{\Sigma}_{0})\);

     
  2. 2.

    \(u|_{\Sigma_{1}^{m}}=0\);

     
  3. 3.
    For each \(\varepsilon\in(0,1)\) there exists a constant \(C(\varepsilon)>0\) such that in \(\Omega_{m,\varepsilon}\)
    $$ \bigl\vert u_{t}(x,t) \bigr\vert \leq C( \varepsilon)t^{-\frac{3m}{4}}; $$
    (1.3)
     
  4. 4.
    The identity
    $$\begin{aligned} \int_{\Omega_{m}} \bigl\{ t^{m} u_{t}v_{t}-u_{x_{1}}v_{x_{1}}- u_{x_{2}}v_{x_{2}}-u_{x_{3}}v_{x_{3}}-fv \bigr\} \,dx_{1}\,dx_{2}\,dx_{3}\,dt =0 \end{aligned}$$
    (1.4)
    holds for all v from
    $$V_{m}:= \bigl\{ v(x,t):v \in C^{2}(\bar{\Omega}_{m}), v|_{\Sigma_{2}^{m}} =0, v \equiv0 \text{ in a neighborhood of }O \bigr\} . $$
     

Remark 1.1

We mention that all the first derivatives of the generalized solutions of 3-D Protter problems in the Tricomi case can have singularity on the boundary of the domain (see [31, 33]). Actually, this fact corresponds to the analogical situation in a 2-D case of the Darboux problem (see [27]). While in the Keldysh case, according to Definition 1.1, the derivative \(u_{t}\) can be unbounded when \(t\to+0\), but \(u_{x_{1}}\), \(u_{x_{2}}\) and \(u_{x_{3}}\) are bounded in each \(\bar{\Omega}_{m,\varepsilon}\), \(\varepsilon >0\).

In this paper, first, we prove results for the uniqueness of a generalized solution to problem PK.

Theorem 1.1

If \(m \in(0,\frac{4}{3})\), then there exists at most one generalized solution of problem PK in \(\Omega_{m}\).

Further, we use the three-dimensional spherical functions \(Y_{n}^{s}(x)\) with \(n \in\mathbf{N} \cup\{0\}\); \(s=1,2, \ldots,2n+1\). The functions \(Y_{n}^{s}(x)\) are defined usually on the unit sphere \(S^{2}:=\{(x_{1},x_{2},x_{3}):x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\}\), and \(Y_{n}^{s}\) form a complete orthonormal system in \(L_{2}(S^{2})\) (see [45]). For convenience of discussions that follow, we extend the spherical functions out of \(S^{2}\) radially, keeping the same notation for the extended functions \(Y_{n}^{s}(x):=Y_{n}^{s}(x/\vert x\vert )\) for \(x \in\mathbf {R}^{3}\setminus\{0\}\).

Let the right-hand side function \(f(x,t)\) of equation (1.1) be fixed as a “harmonic polynomial” of order l with \(l \in\mathbf{N} \cup\{0\}\), and it has the following representation:
$$ f(x,t)=\sum_{n=0}^{l}\sum _{s=1}^{2n+1}f_{n}^{s} \bigl( \vert x\vert ,t \bigr)Y_{n}^{s}(x) $$
(1.5)
with some coefficients \(f_{n}^{s}(\vert x\vert ,t)\).

In this special case we give an existence result as well.

Theorem 1.2

Let \(m \in(0,\frac{4}{3})\). Suppose that the right-hand side function \(f(x,t)\) has the form (1.5) and \(f \in C^{1}(\bar{\Omega}_{m})\). Then the unique generalized solution \(u(x,t)\) of problem PK in \(\Omega_{m}\) exists and has the form
$$ u(x,t)=\sum_{n=0}^{l}\sum _{s=1}^{2n+1}u_{n}^{s} \bigl( \vert x\vert ,t \bigr)Y_{n}^{s}(x). $$
(1.6)

Remark 1.2

Actually, when the right-hand side function \(f(x,t)\) has the form (1.5) in Theorem 1.2, we find explicit representations for the functions \(u_{n}^{s}(\vert x\vert ,t)\) in (1.6). These representations involve appropriate hypergeometric functions.

In the case when the right-hand side function \(f(x,t)\) has the form (1.5), we give an a priori estimate for the generalized solution of problem PK in \(\Omega_{m}\) as well.

Theorem 1.3

Let the conditions in Theorem  1.2 be fulfilled. Then the unique generalized solution of problem PK in \(\Omega_{m}\) has the form (1.6) and satisfies the a priori estimate
$$ \bigl\vert u(x ,t) \bigr\vert \leq c \Bigl( \max _{\bar{\Omega}_{m}} \vert f\vert \Bigr) \vert x\vert ^{-l-1}, $$
(1.7)
with a constant \(c>0\) independent of f.

Estimate (1.7) shows the maximal order of possible singularity at point O, when the right-hand side function \(f(x,t)\) is a “harmonic polynomial” of fixed order l. We will point out that a similar a priori estimate for generalized solutions to 3-D Protter problem P1 in the Tricomi case is obtained in [36], while an estimate from below in this case is given in [31].

The present paper contains the introduction and five more sections. In Section 2, the Protter problem PK is considered in a model case when the right-hand side function \(f(x,t)\) of equation (1.1) is fixed as a “harmonic polynomial” (1.5) of order l. In that case we formulate the 2-D boundary value problems \(\mathit {PK}_{1}\) and \(\mathit {PK}_{2}\), corresponding to the \((3+1)\)-D problem PK. We give a notion for a generalized solution of Cauchy-Goursat problem \(\mathit {PK}_{2}\), and in Section 3, using the Riemann-Hadamard function associated to this problem, we find an integral representation for a generalized solution. Further, we obtain existence and uniqueness results for a generalized solution of problem \(\mathit {PK}_{2}\). Actually, this is the essential result in this paper and has the most difficult proof. Using the results of the previous section, in Section 4 we prove the main results in this paper, i.e., Theorem 1.1, Theorem 1.2 and Theorem 1.3. In Appendix A we give the main properties of the Riemann-Hadamard function associated to the Cauchy-Goursat problem \(\mathit {PK}_{2}\). In Appendix B some auxiliary results, needed for the study of the generalized solution to problem \(\mathit {PK}_{2}\), are proven.

2 Two-dimensional Cauchy-Goursat problems corresponding to problem PK

Using spherical coordinates \((r,\theta, \varphi, t)\in \mathbf{R}^{4}\), \(0 \leq\theta< \pi\), \(0 \leq\varphi< 2 \pi\), \(r>0 \) with
$$x_{1}=r \sin\theta\cos\varphi, \quad\quad x_{2}=r \sin\theta\sin \varphi, \quad\quad x_{3}=r \cos\theta $$
problem PK can suitably be treated. Written in the new coordinates, equation (1.1) becomes
$$ L_{m}u=\frac{1}{r^{2}} \bigl(r^{2} u_{r } \bigr)_{r }+ \frac{1}{r^{2} \sin\theta} (\sin\theta u_{\theta})_{\theta} +\frac{1}{r ^{2} \sin^{2} \theta}u_{\varphi\varphi}- \bigl(t^{m}u_{t} \bigr)_{t}=f. $$
(2.1)
We consider equation (2.1) in the region
$$\Omega_{m}= \biggl\{ (r,\theta, \varphi, t):0< t< t_{0}, 0 \leq\theta< \pi, 0 \leq\varphi< 2 \pi, \frac {2}{2-m}t^{\frac{2-m}{2}} < r < 1-\frac{2}{2-m}t^{\frac{2-m}{2}} \biggr\} , $$
bounded by the following surfaces:
$$\begin{aligned}& \Sigma_{0}= \bigl\{ (r,\theta, \varphi, t):t=0, 0 \leq\theta< \pi, 0 \leq\varphi< 2 \pi, r< 1 \bigr\} , \\& \Sigma_{1}^{m}= \biggl\{ (r,\theta, \varphi, t):t>0, 0 \leq \theta< \pi, 0 \leq\varphi< 2 \pi, r=1-\frac{2}{2-m}t^{\frac{2-m}{2}} \biggr\} , \\& \Sigma_{2}^{m}= \biggl\{ (r,\theta, \varphi, t):t>0, 0 \leq\theta< \pi, 0 \leq\varphi< 2 \pi, r=\frac{2}{2-m}t^{\frac{2-m}{2}} \biggr\} . \end{aligned}$$
Problem PK becomes the following one: find a solution to equation (2.1) with the boundary conditions
$$u|_{\Sigma_{1}^{m} } =0; \quad\quad t^{m} u_{t} \to0,\quad \text{as }t \to+0. $$
The two-dimensional spherical functions, expressed in terms of θ and φ in the traditional definition (see [45]), are \(Y_{n}^{s}(\theta,\varphi):=Y_{n}^{s}(x)\), \(x \in S^{2}\), \(n \in\mathbf{N} \cup\{0\}\), \(s=1,2,\ldots,2n+1\), and satisfy the differential equation
$$ \frac{1}{\sin\theta}\frac{\partial}{\partial \theta} \biggl(\sin\theta \frac{\partial}{\partial\theta}Y_{n}^{s} \biggr)+\frac{1}{\sin^{2} \theta} \frac{\partial^{2}}{\partial\varphi^{2}}Y_{n}^{s}+n(n+1)Y_{n}^{s}=0. $$
In the special case when the right-hand side function \(f(x,t)\) of equation (2.1) has the form
$$ f(r ,\theta, \varphi,t)=f_{n}^{s}(r,t)Y_{n}^{s}( \theta, \varphi), $$
we may look for a solution of the form
$$ u(r ,\theta, \varphi,t)=u_{n}^{s}(r,t)Y_{n}^{s}( \theta, \varphi) $$
with unknown coefficient \(u_{n}^{s}(r,t)\).
For the coefficients \(u_{n}^{s}(r ,t)\) which correspond to the right-hand sides \(f_{n}^{s}(r ,t)\), we obtain the 2-D equation
$$\begin{aligned} u_{rr}+\frac{2}{r}u_{r}- \bigl(t^{m}u_{t } \bigr)_{t}-\frac{n(n+1)}{r ^{2}}u=f \end{aligned}$$
in the domain
$$G_{m }= \biggl\{ (r ,t):0< t< t_{0}, \frac{2}{2-m}t^{\frac{2-m}{2}}< r < 1-\frac{2}{2-m}t^{\frac{2-m}{2}} \biggr\} , $$
which is bounded by the segment \(S_{0} =\{( r ,t): 0 < r <1, t=0\}\) and the characteristics
$$\begin{aligned}& S_{1}^{m}:= \biggl\{ (r ,t):0< t< t_{0}, r =1-\frac{2}{2-m}t^{\frac{2-m}{2}} \biggr\} , \\& S^{m}_{2}:= \biggl\{ (r ,t):0< t< t_{0}, r = \frac{2}{2-m}t^{\frac{2-m}{2}} \biggr\} . \end{aligned}$$
In this case, for \(u(r ,t)\), the 2-D problem corresponding to PK is the problem
$$ \mathit {PK}_{1} \quad \textstyle\begin{cases} u_{rr}+\frac{2}{r}u_{r}-(t^{m}u_{t })_{t}-\frac{n(n+1)}{r ^{2}}u=f(r,t) \quad\text{in } G_{m }, \\ u|_{S_{1}^{m}}=0; \quad\quad t^{m} u_{t} \to0,\quad \text{as }t \to+0. \end{cases} $$

Remark 2.1

When the right-hand side function \(f(x,t)\) has the form (1.5), it is enough to take test functions \(v \in V_{m}\) in the identity (1.4) to have the form \(v=w(\vert x\vert ,t)Y_{n}^{s}(x)\), \(n \in\mathbf{N} \cup\{0\}\), \(s=1,2,\ldots,2n+1\) and
$$\begin{aligned} w \in V_{m}^{(1)}:= \bigl\{ w(r,t): w \in C^{2}( \bar{G}_{m}), w|_{S_{2}^{m}}=0, w \equiv0 \text{ in a neighborhood of }(0,0) \bigr\} . \end{aligned}$$

The generalized solution of problem \(\mathit {PK}_{1}\) is defined as follows.

Definition 2.1

We call a function \(u(r,t)\) a generalized solution of problem \(\mathit {PK}_{1}\) in \(G_{m}\) (\(0< m<\frac{4}{3}\)) if:
  1. 1.

    \(u, u_{r} \in C(\bar{G}_{m}\setminus(0,0))\), \(u_{t}\in C(\bar {G}_{m}\setminus\bar{S}_{0})\);

     
  2. 2.

    \(u|_{S_{1}^{m}}=0\);

     
  3. 3.
    For each \(\varepsilon\in(0,1)\) there exists a constant \(C(\varepsilon)>0\) such that the estimates
    $$\bigl\vert u_{t}(r,t) \bigr\vert \leq C(\varepsilon)t^{-\frac{3m}{4}} $$
    hold in \(G_{m,\varepsilon}:=G_{m} \cap\{r> \varepsilon\}\);
     
  4. 4.
    The identity
    $$ \int_{G_{m}} \biggl\{ u_{r}v_{r}-t^{m} u_{t}v_{t}+\frac{n(n+1)}{r^{2}}uv+fv \biggr\} r^{2} \,dr \,dt =0 $$
    (2.2)
    holds for all \(v \in V_{m}^{(1)}\).
     
Substituting the new characteristic coordinates
$$ \xi=1-r -\frac{2}{2-m}t^{\frac{2-m}{2}}, \quad \quad\eta=1-r + \frac{2}{2-m}t^{\frac{2-m}{2}} $$
(2.3)
and the new functions
$$ \begin{aligned} &U(\xi,\eta)=r(\xi,\eta) u \bigl(r (\xi,\eta ),t(\xi, \eta ) \bigr), \\ &V(\xi,\eta)=r(\xi,\eta) v \bigl(r (\xi,\eta),t(\xi,\eta) \bigr), \\ &F(\xi,\eta)=\frac{1}{8}(2-\xi-\eta)f \bigl(r (\xi,\eta),t(\xi,\eta ) \bigr), \end{aligned} $$
(2.4)
from problem \(\mathit {PK}_{1}\), we get the 2-D Cauchy-Goursat problem
where
$$D:= \bigl\{ (\xi,\eta):0< \xi< \eta< 1 \bigr\} \subset\mathbf{R}^{2}, $$
and the parameter \(\beta=\frac{ m}{ 2(2-m)}\in(0,1)\) since \(0< m<\frac{4}{3}\).

The generalized solution of problem \(\mathit {PK}_{2}\) is defined as follows.

Definition 2.2

We call a function \(U(\xi,\eta )\) a generalized solution of problem \(\mathit {PK}_{2}\) in D (\(0<\beta<1\)) if:
  1. 1.

    \(U, U_{\xi}+U_{\eta} \in C(\bar{D} \setminus(1,1))\), \(U_{\xi}-U_{\eta} \in C(\bar{D}\setminus \{\eta=\xi\})\);

     
  2. 2.
    $$ U(0,\eta)=0; $$
    (2.7)
     
  3. 3.
    For each \(\varepsilon\in(0,1)\) there exists a constant \(C(\varepsilon)>0\) such that
    $$ \bigl\vert (U_{\xi}-U_{\eta}) (\xi,\eta) \bigr\vert \leq C(\varepsilon) (\eta-\xi)^{-\beta}\quad \text{in }\bar{D}_{\varepsilon} \setminus\{\eta=\xi\}, $$
    (2.8)
    where \(D_{\varepsilon}:=D \cap\{\xi< 1-\varepsilon\}\);
     
  4. 4.
    The identity
    $$ \int_{D}(\eta-\xi)^{2 \beta} \biggl\{ U_{\xi}V_{\eta}+U_{\eta}V_{\xi} +\frac{2n(n+1)}{(2-\xi-\eta)^{2}}UV+2FV \biggr\} \,d\xi \,d\eta=0 $$
    (2.9)
    holds for all
    $$V \in V^{(2)}:= \bigl\{ V(\xi,\eta): V \in C^{2}(\bar{D}), V (\xi,1)= 0, V \equiv0 \text{ in a neighborhood of }(1,1) \bigr\} . $$
     

3 Existence and uniqueness of a generalized solution to the Cauchy-Goursat plane problem \(\mathit {PK}_{2}\)

In this section we prove the existence and uniqueness of a generalized solution to problem \(\mathit {PK}_{2}\). In order to do this, we use the Riemann-Hadamard function associated to problem \(\mathit {PK}_{2}\) to find an integral representation for a generalized solution of this problem in D. According to Gellerstedt [46] and the results of Nakhushev mentioned in the book of Smirnov [47], this function has the form
$$ \Phi(\xi,\eta; \xi_{0} ,\eta_{0} ) = \textstyle\begin{cases} \Phi^{+}(\xi,\eta; \xi_{0},\eta_{0} ), & \eta> \xi_{0}, \\ \Phi^{-}(\xi,\eta; \xi_{0},\eta_{0} ), & \eta< \xi_{0}, \end{cases} $$
(3.1)
for \((\xi_{0},\eta_{0}) \in D\) and \((\xi,\eta) \in \bar{T} \cup\bar{\Pi} \setminus\{\eta=\xi_{0}\}\), where
$$\begin{aligned} T:= \bigl\{ (\xi,\eta): 0 < \xi< \eta< \xi_{0} \bigr\} , \quad\quad \Pi:= \bigl\{ ( \xi,\eta): 0 < \xi< \xi_{0}, \xi_{0} < \eta< \eta_{0} \bigr\} . \end{aligned}$$
The Riemann-Hadamard function \(\Phi(\xi,\eta;\xi_{0},\eta_{0})\) should have the following main properties (see [46, 47]):
  1. (i)
    The function Φ as a function of \((\xi_{0},\eta_{0})\) satisfies
    $$ \begin{aligned}[b] E_{\xi_{0},\eta_{0}}[\Phi]&:= \frac{\partial^{2} \Phi }{\partial\xi_{0}\, \partial \eta_{0}}+\frac{\beta}{\eta_{0}-\xi_{0}} \biggl( \frac{\partial \Phi}{\partial\xi_{0}}-\frac{\partial\Phi}{\partial \eta_{0}} \biggr)-\frac{n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}}\Phi \\ &=0 \quad\text{in } D, \eta\neq\xi_{0} \end{aligned} $$
    (3.2)
    and with respect to the first pair of variables \((\xi,\eta)\)
    $$ \begin{aligned}[b] E^{\ast}_{\xi,\eta}[\Phi]&:= \frac{\partial^{2} \Phi }{\partial\xi\,\partial \eta}- \frac{\partial}{\partial\xi} \biggl(\frac{\beta \Phi}{\eta-\xi} \biggr)+ \frac{\partial}{\partial\eta} \biggl( \frac{\beta \Phi}{\eta-\xi} \biggr)-\frac{n(n+1)}{(2-\xi-\eta)^{2}}\Phi \\ &=0 \quad\text{in } D, \eta\neq \xi_{0}; \end{aligned} $$
    (3.3)
     
  2. (ii)

    \(\Phi^{+}(\xi_{0},\eta_{0};\xi_{0},\eta_{0})=1\);

     
  3. (iii)

    \(\Phi^{+}(\xi,\eta_{0};\xi_{0},\eta_{0})= ( \frac{\eta_{0}-\xi}{\eta_{0}-\xi_{0}} )^{\beta}\);

     
  4. (iv)

    \(\Phi^{+}(\xi_{0},\eta;\xi_{0},\eta_{0})= ( \frac{\eta-\xi_{0}}{\eta_{0}-\xi_{0}} )^{\beta}\);

     
  5. (v)
    The jump of the function Φ on the line \(\{\eta=\xi_{0}\}\) is
    $$\begin{aligned}{} [ [\Phi] ] :=& \lim_{\delta\to +0} \bigl\{ \Phi^{-}(\xi,\xi_{0}-\delta;\xi_{0}, \eta_{0})- \Phi^{+}(\xi,\xi_{0}+\delta; \xi_{0},\eta_{0}) \bigr\} \\ =& \cos(\pi\beta) \lim_{\delta\to +0} \bigl\{ \Phi^{+}( \xi,\xi_{0}+\delta;\xi_{0},\xi_{0}+\delta) \Phi^{+}(\xi_{0},\xi_{0}+\delta; \xi_{0},\eta_{0}) \bigr\} \\ =& \cos(\pi\beta) \biggl(\frac{\xi_{0}-\xi}{\eta_{0}-\xi_{0}} \biggr)^{\beta}; \end{aligned}$$
     
  6. (vi)

    \(\Phi^{-}\) vanishes on the line \(\{\eta=\xi\}\) of power 2β.

     

Actually, the function \(\Phi^{+}\) is the Riemann function for equation (2.5).

Remark 3.1

In the case \(\boldsymbol{0<\beta<1/2} \) and \(\boldsymbol{F(\xi,\eta)=(\eta-\xi)^{-4 \beta}f(\xi,\eta)}\), where \(f \in C(\bar{D})\), a generalized solution of problem \(\mathit {PK}_{2}\) has an explicit integral representation (see [46] and [47]). We find an integral representation in the case \(\boldsymbol{0<\beta<1}\) and \(F\in C(\bar{D})\) using the properties of the Riemann-Hadamard function \(\Phi(\xi,\eta;\xi_{0},\eta_{0})\). The existence of a function \(\Phi(\xi,\eta;\xi_{0},\eta_{0})\) with properties \(\mbox{(i)} \div\mbox{(vi)}\) is shown in Appendix A (see also [44]).

Theorem 3.1

Let \(0<\beta<1\) and \(F\in C(\bar{D})\). Then each generalized solution of problem \(\mathit {PK}_{2}\) in D has the following integral representation:
$$ U (\xi_{0},\eta_{0}) = \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta) \Phi(\xi,\eta; \xi_{0},\eta_{0}) \,d\eta \,d\xi. $$
(3.4)

Proof

Let \(U(\xi,\eta)\) be a generalized solution of problem \(\mathit {PK}_{2}\) in D. For any arbitrary function \(\psi(\xi,\eta)\) from \(C_{0}^{\infty}(D)\), we have \(\psi\in V^{(2)}\), and from (2.9) we obtain the identity
$$ \int_{D}(\eta-\xi)^{2 \beta} \biggl\{ U_{\xi\eta}+ \frac{\beta}{\eta-\xi}(U_{\xi}-U_{\eta}) -\frac{n(n+1)}{(2-\xi-\eta)^{2}}U-F \biggr\} \psi \,d\xi \,d\eta=0, $$
where \(U_{\xi\eta}\) is the weak derivative of U. Therefore
$$U_{\xi\eta}=F +\frac{n(n+1)}{(2-\xi-\eta)^{2}}U-\frac{\beta}{\eta-\xi}(U_{\xi}-U_{\eta }) \in C(D) $$
since \(F, U, U_{\xi}-U_{\eta} \in C(D)\). From this it follows that \(U_{\xi\eta}\) is a classical derivative of U and \(U(\xi,\eta)\) satisfies the differential equation (2.5) in D in a classical sense.
Now, using the properties of the Riemann-Hadamard function \(\Phi(\xi,\eta;\xi_{0},\eta_{0})\), we obtain the integral representation (3.4) for a generalized solution of problem \(\mathit {PK}_{2}\) by integrating the identity
$$\Phi(\xi,\eta;\xi_{0},\eta_{0}) E_{\xi,\eta} \bigl[U( \xi,\eta) \bigr] -U(\xi,\eta) E^{\ast}_{\xi,\eta} \bigl[\Phi(\xi, \eta; \xi_{0},\eta_{0}) \bigr] = F(\xi,\eta) \Phi (\xi,\eta; \xi_{0},\eta_{0}) $$
over a triangle
$$T_{\delta}:= \bigl\{ (\xi,\eta):0 < \xi< \xi_{0}-2 \delta, \xi+ \delta< \eta< \xi_{0} - \delta \bigr\} $$
and then over the rectangle
$$\Pi_{\delta}:= \bigl\{ (\xi,\eta): 0 < \xi< \xi_{0}-2 \delta, \xi_{0} + \delta< \eta< \eta_{0} \bigr\} $$
with \(\delta>0\) small enough, and finally letting \(\delta\to0\). □
Theorem 3.1 claims the uniqueness of a generalized solution to problem \(\mathit {PK}_{2}\). Next, we prove that if \(F \in C^{1}(\bar{D})\) and \(U(\xi,\eta)\) is a function defined by (3.4) in D, then \(U(\xi,\eta)\) is a generalized solution to problem \(\mathit {PK}_{2}\) in D. In order to do this, we introduce the notation
$$M_{F}:=\max \Bigl\{ \max_{\bar{D}_{0}}\vert F\vert , \max_{\bar {D}_{0}}\vert F_{\xi}+F_{\eta} \vert \Bigr\} , $$
and we mention that, according to Lemma A.1 (see Appendix A below), the Riemann-Hadamard function \(\Phi(\xi,\eta;\xi_{0},\eta_{0})\) can be decomposed in the following way:
$$\Phi(\xi,\eta;\xi_{0},\eta_{0})= H(\xi,\eta; \xi_{0},\eta_{0})+G(\xi,\eta;\xi_{0}, \eta_{0}), $$
where \(H(\xi,\eta;\xi_{0},\eta_{0})\) is the Riemann-Hadamard function (A.12) associated to problem \(\mathit {PK}_{2}\) in the case \(n=0\) and \(G(\xi,\eta;\xi_{0},\eta_{0})\) is an additional term. Therefore we can rewrite representation (3.4) in the form
$$ U(\xi_{0},\eta_{0})=U^{H}( \xi_{0},\eta_{0})+U^{G}(\xi_{0}, \eta_{0}), $$
(3.5)
where
$$ U^{H}(\xi_{0},\eta_{0}):= \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta) H (\xi,\eta; \xi_{0},\eta_{0}) \,d\eta \,d\xi $$
(3.6)
and
$$ U^{G}(\xi_{0},\eta_{0}):= \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta) G (\xi,\eta; \xi_{0},\eta_{0}) \,d\eta \,d\xi. $$
(3.7)
Firstly, we will study the function \(U^{H}(\xi_{0},\eta_{0})\). To do this, we use the estimates for some integrals involving function \(H(\xi,\eta;\xi_{0},\eta_{0})\) obtained in Appendix B.

Theorem 3.2

Let \(0 < \beta<1\) and \(F \in C^{1}(\bar{D})\). Then, for the function \(U^{H}(\xi_{0},\eta_{0})\), we have \(U^{H}, U^{H}_{\xi_{0}}+U^{H}_{\eta_{0}}\in C(\bar{D}\setminus(1,1))\), \(U^{H}_{\eta_{0}}\in C(\bar{D}\setminus\{\eta_{0}=\xi_{0}\})\) and the following estimates hold:
$$\begin{aligned}& \bigl\vert U^{H}(\xi_{0}, \eta_{0}) \bigr\vert \leq K_{1} M_{F} \xi_{0}\quad \textit{in } \bar{D}\setminus(1,1), \\& \bigl\vert U^{H}_{\xi_{0}}+U^{H}_{\eta_{0}} \bigr\vert (\xi_{0},\eta_{0}) \leq K_{1} M_{F} \eta_{0} \quad \textit{in } \bar{D}\setminus(1,1), \\& \bigl\vert U^{H}_{\eta_{0}}(\xi_{0}, \eta_{0}) \bigr\vert \leq K_{1} M_{F} \xi_{0}(\eta_{0}-\xi_{0})^{-\beta} \quad \textit{in } \bar{D}\setminus\{\eta_{0}=\xi_{0}\}, \end{aligned}$$
where \(K_{1}>0\) is a constant independent of F.

Proof

Step 1. From (3.6) and (B.1) from Lemma B.1 (see Appendix B) we obtain
$$ \bigl\vert U^{H}(\xi_{0}, \eta_{0}) \bigr\vert \leq M_{F} \int_{0}^{\xi_{0}} \int_{\xi }^{\eta_{0}} H (\xi,\eta;\xi_{0}, \eta_{0}) \,d\eta \,d\xi =M_{F}I^{1}( \xi_{0},\eta_{0}) \leq k_{1} M_{F} \xi_{0}. $$
Step 2. Differentiating (3.6) with respect to \(\eta_{0}\) and using (B.4) from Lemma B.2, we obtain
$$ \begin{aligned} \bigl\vert U^{H}_{\eta_{0}}( \xi_{0},\eta_{0}) \bigr\vert &= \biggl\vert \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta) H_{\eta_{0}} (\xi, \eta;\xi_{0},\eta_{0}) \,d\eta \,d\xi+ \int_{0}^{\xi_{0}}F(\xi,\eta_{0}) \frac{(\eta_{0}-\xi)^{\beta}}{(\eta_{0}-\xi_{0})^{\beta}} \,d\xi \biggr\vert \\ & \leq M_{F} \biggl\{ \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}} \bigl\vert H_{\eta_{0}} (\xi, \eta;\xi_{0},\eta_{0}) \bigr\vert \,d\eta \,d\xi+ \int_{0}^{\xi_{0}} \biggl(\frac{\eta_{0}-\xi}{\eta_{0}-\xi_{0}} \biggr)^{\beta} \,d\xi \biggr\} \\ &= M_{F} \biggl\{ I^{2}(\xi_{0}, \eta_{0})+ \int_{0}^{\xi_{0}} \biggl(\frac{\eta_{0}-\xi}{\eta_{0}-\xi_{0}} \biggr)^{\beta} \,d\xi \biggr\} \\ &\leq M_{F}(k_{2}+1) \xi_{0}( \eta_{0}-\xi_{0})^{-\beta}. \end{aligned} $$
Step 3. According to Remark A.1, the derivatives \(H_{\xi_{0}}^{+}\), \(H_{\xi_{0}}^{-}\) have singularities of order \(\vert \eta-\xi_{0}\vert ^{-1}\) on the line \(\{\eta=\xi_{0}\}\). Gellerstedt [46] and Moiseev [48] consider the case \(n= 0\) and suggest differentiating (3.6) after appropriate substitutions of variables. In that way one can find integral representations for the first derivatives of the solution which do not involve the first derivatives of function H. In order to do this, following Moiseev [48], we introduce new variables
$$ \tilde{\xi}:=\frac{\xi_{0}-\xi}{\eta_{0}-\xi_{0}}, \quad\quad \tilde{\eta}:= \frac{\eta_{0}-\eta}{\eta_{0}-\xi_{0}}. $$
(3.8)
We define
$$\tilde{H}^{+}(\tilde{\xi},\tilde{\eta}):=H^{+}(\xi,\eta; \xi_{0},\eta_{0}), \quad\quad \tilde{H}^{-}(\tilde{\xi}, \tilde{\eta}):=H^{-}(\xi,\eta;\xi_{0},\eta_{0}), $$
from (A.12) we obtain
$$\begin{aligned}& \tilde{H}^{+} (\tilde{\xi},\tilde{\eta})= (1-\tilde{ \eta}+\tilde{\xi})^{\beta}F \biggl(\beta,1-\beta,1;\frac{\tilde{\xi} \tilde{\eta}}{1-\tilde{\eta}+\tilde{\xi}} \biggr),\quad\tilde{\eta}< 1, \\& \tilde{H}^{-}(\tilde{\xi},\tilde{\eta}) = \frac{k (1-\tilde{\eta}+\tilde{\xi}) ^{2\beta}}{\tilde{\xi}^{\beta}\tilde{\eta}^{\beta}} F \biggl(\beta,\beta,2\beta;\frac{1-\tilde{\eta}+\tilde{\xi}}{\tilde{\xi} \tilde{\eta}} \biggr),\quad\tilde{\eta}>1. \end{aligned}$$
Then we have
$$\begin{aligned} &U^{H}(\xi_{0}, \eta_{0}) \\ &\quad =(\eta_{0}-\xi_{0})^{2} \int_{0}^{\frac{\xi_{0}}{\eta_{0}-\xi_{0}}} \biggl\{ \int_{0}^{1+\tilde{\xi}}F \bigl(\xi_{0}-( \eta_{0}-\xi_{0})\tilde{\xi},\eta _{0}-( \eta_{0}-\xi_{0})\tilde{\eta} \bigr) \tilde{H }(\tilde{\xi}, \tilde{\eta}) \,d\tilde{\eta} \biggr\} \,d\tilde{\xi}, \end{aligned} $$
and
$$ \begin{aligned} & \bigl(U^{H}_{\xi_{0}}+U^{H}_{\eta_{0}} \bigr) (\xi_{0},\eta_{0}) \\ &\quad = (\eta_{0}-\xi_{0})^{2} \int_{0}^{\frac{\xi_{0}}{\eta_{0}-\xi_{0}}} \int_{0}^{1+\tilde{\xi}}(F_{\xi}+F_{\eta}) \bigl(\xi_{0}-(\eta_{0}-\xi_{0})\tilde{\xi}, \eta_{0}-(\eta_{0}-\xi_{0})\tilde{\eta} \bigr) \tilde{H} (\tilde{\xi},\tilde{\eta}) \,d\tilde{\eta} \,d\tilde{\xi} \\ &\quad\quad{} +(\eta_{0}-\xi_{0}) \int_{0}^{\frac{\eta_{0}}{\eta_{0}-\xi_{0}}}F \bigl(0,\eta_{0}-( \eta_{0}-\xi_{0})\tilde{\eta} \bigr) \tilde{H} \biggl( \frac{\xi_{0}}{\eta_{0}-\xi_{0}},\tilde{\eta} \biggr) \,d\tilde{\eta}. \end{aligned} $$
Now the inverse transform of (3.8) gives
$$ \begin{aligned} \bigl(U^{H}_{\xi_{0}}+U^{H}_{\eta_{0}} \bigr) (\xi_{0},\eta_{0})&= \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}(F_{\xi}+F_{\eta}) (\xi,\eta) H (\xi,\eta;\xi_{0},\eta_{0}) \,d\eta \,d\xi \\ &\quad{} + \int_{0}^{\eta_{0}}F(0,\eta)H(0,\eta;\xi_{0}, \eta_{0}) \,d\eta. \end{aligned} $$
Now (B.1) from Lemma B.1 and (B.6) from Lemma B.3 give
$$\bigl\vert \bigl(U^{H}_{\xi_{0}}+U^{H}_{\eta_{0}} \bigr) (\xi_{0},\eta_{0}) \bigr\vert \leq M_{F} \bigl\{ I^{1}(\xi_{0},\eta_{0})+I^{3}( \xi_{0},\eta_{0}) \bigr\} \leq M_{F}(k_{1}+k_{3}) \eta_{0}. $$
 □

Theorem 3.3

Let the conditions in Theorem  3.2 be fulfilled. Then for the function \(U^{G}(\xi_{0},\eta_{0})\) we have \(U^{G}, U^{G}_{\xi_{0}}, U^{G}_{\eta_{0}} \in C(\bar{D}\setminus(1,1))\), and the following estimates hold in \(\bar{D}\setminus(1,1)\):
$$\begin{aligned}& \bigl\vert U^{G}(\xi_{0}, \eta_{0}) \bigr\vert \leq K_{2}M_{F} \xi_{0}(2-\xi_{0}-\eta_{0})^{-n}, \end{aligned}$$
(3.9)
$$\begin{aligned}& \bigl\vert U^{G}_{\xi_{0}}(\xi_{0}, \eta_{0}) \bigr\vert \leq K_{2}M_{F} \xi_{0}(2-\xi_{0}-\eta_{0})^{-n-1}, \end{aligned}$$
(3.10)
$$\begin{aligned}& \bigl\vert U^{G}_{\eta_{0}}(\xi_{0}, \eta_{0}) \bigr\vert \leq K_{2}M_{F} \xi_{0}(2-\xi_{0}-\eta_{0})^{-n-1}, \end{aligned}$$
(3.11)
where \(K_{2}>0\) is a constant independent of F.

Proof

Using estimates (A.26) and (A.27), from (3.7) we obtain estimate (3.9):
$$ \begin{aligned} \bigl\vert U^{G}(\xi_{0}, \eta_{0}) \bigr\vert &= \biggl\vert \int_{0}^{\xi_{0}} \int_{\xi}^{\xi_{0}}F(\xi,\eta) G^{-}(\xi,\eta; \xi_{0},\eta_{0}) \,d\eta \,d\xi \\ &\quad{} + \int_{0}^{\xi_{0}} \int_{\xi_{0}}^{\eta_{0}}F(\xi,\eta) G^{+}(\xi,\eta; \xi_{0},\eta_{0}) \,d\eta \,d\xi \biggr\vert \\ &\leq C_{G}M_{F}\xi_{0} \biggl\{ \frac{1}{2}\xi_{0}(2-\xi_{0}-\eta_{0})^{-n}+( \eta_{0}-\xi_{0})^{1-\beta} \biggr\} \\ &\leq K_{2}M_{F}\xi_{0}(2-\xi_{0}- \eta_{0})^{-n}. \end{aligned} $$
Now we calculate
$$U^{G}_{\xi_{0}}(\xi_{0},\eta_{0})= \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta ) G_{\xi_{0}}(\xi, \eta,\xi_{0},\eta_{0}) \,d\eta \,d\xi. $$
Here we do not have integrals on the boundaries because \(Y=0\) on the line \(\{\xi=\xi_{0}\}\), and the function \(G(\xi,\eta,\xi_{0},\eta_{0})\) has no jump on the line \(\{\eta=\xi_{0}\}\) (see Appendix A). Applying estimates (A.30) and (A.31) to this integral, we have
$$ \begin{aligned} \bigl\vert U^{G}_{\xi_{0}}( \xi_{0},\eta_{0}) \bigr\vert &\leq\frac{M_{F}C_{G}}{(2-\xi_{0}-\eta_{0})^{n+1}} \int_{0}^{\xi_{0}} \int_{\xi}^{\xi_{0}} (\xi_{0}- \eta)^{-\beta} \,d\eta \,d\xi \\ &\quad{} +\frac{M_{F}C_{G}}{2-\xi_{0}-\eta_{0}} \int_{0}^{\xi_{0}} \int_{\xi_{0}}^{\eta_{0}}(\eta-\xi_{0})^{-\beta} \,d\eta \,d\xi \\ &\leq\frac{M_{F} C_{G}}{(2-\xi_{0}-\eta_{0})^{n+1}} \bigl(I_{1}^{1}+2^{n}I_{2}^{1} \bigr). \end{aligned} $$
Now (B.2) and (B.3) from Lemma B.1 give estimate (3.10). Further, we calculate
$$U^{G}_{\eta_{0}}(\xi_{0},\eta_{0})= \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta ) G_{\eta_{0}}(\xi, \eta;\xi_{0},\eta_{0}) \,d\eta \,d\xi, $$
where we used that \(Y=0\) on the line \(\eta=\eta_{0}\). Analogously, applying estimates (A.28) and (A.29), which are even better than (A.30) and (A.31), to the last integral for the derivative \(G_{\eta_{0}}\), we obtain estimate (3.11). □

As a direct consequence of Theorem 3.2 and Theorem 3.3, in view of \(U=U^{H}+U^{G}\), we have the following theorem.

Theorem 3.4

Let \(0 < \beta<1\) and \(F \in C^{1}(\bar{D})\). Then, for the function \(U(\xi,\eta)\) defined by (3.4), we have \(U, U_{\xi}+U_{\eta} \in C(\bar{D} \setminus(1,1))\), \(U_{\eta} \in C(\bar{D}\setminus \{\eta=\xi\})\) and for some constant \(K_{3}>0\) the estimates below hold
$$ \begin{gathered} \bigl\vert U(\xi,\eta) \bigr\vert \leq K_{3}M_{F}\xi(2-\xi-\eta)^{-n} \quad\textit{in }\bar{D} \setminus(1,1), \\ \bigl\vert (U_{\xi}+U_{\eta}) (\xi,\eta) \bigr\vert \leq K_{3}M_{F}(2-\xi-\eta)^{-n-1} \quad\textit{in } \bar{D} \setminus (1,1), \\ \bigl\vert U_{\eta}(\xi,\eta) \bigr\vert \leq K_{3}M_{F} \xi(\eta-\xi)^{-\beta}(2-\xi- \eta)^{-n-1} \quad\textit{in } \bar{D}\setminus\{ \eta=\xi\}. \end{gathered} $$
(3.12)

Now, we are able to prove the following existence result.

Theorem 3.5

Let \(0 < \beta<1\) and \(F \in C^{1}(\bar{D})\). Then there exists one and only one generalized solution to problem \(\mathit {PK}_{2}\) in D, which has integral representation (3.4), and it satisfies estimates (3.12).

Proof

Let \(U(\xi,\eta)\) be the function known from Theorem 3.4. Therefore \(U, U_{\xi}+U_{\eta} \in C(\bar{D} \setminus(1,1))\), \(U_{\eta} \in C(\bar{D}\setminus \{\eta=\xi\})\), and it satisfies estimates (3.12) in Definition 2.2. But in view of (3.12) it is obvious that condition (2.7) and estimate (2.8) hold.

To prove that \(U(\xi,\eta)\) satisfies identity (2.9) in Definition 2.2, we need several steps as follows.

Step 1. We prove that \(U(\xi,\eta)\) satisfies the differential equation (2.5) in a classical sense and \(\frac{\partial}{\partial\eta} (U_{\xi} ) \in C(D)\).

(1.i) Following Smirnov [47], we find another representation formula for the function \(U^{H}(\xi,\eta)\). Let us introduce the function
$$ R_{0}(\xi,\eta;\xi_{0},\eta_{0}):= \textstyle\begin{cases} R_{0}^{+}(\xi,\eta;\xi_{0},\eta_{0}), &\eta>\xi_{0},\\ R_{0}^{-}(\xi,\eta;\xi_{0},\eta_{0}), &\eta< \xi_{0}, \end{cases} $$
where
$$ \begin{gathered} R_{0}^{+}(\xi,\eta;\xi_{0}, \eta_{0}) := \biggl(\frac{\eta_{0}-\eta}{\eta_{0}-\xi_{0}} \biggr)^{\beta} \biggl( \frac{\eta_{0}-\eta}{\eta_{0}-\xi} \biggr)^{1-\beta}F_{1} \biggl(1-\beta,\beta,1- \beta,2;\frac{\eta_{0}-\eta}{\eta_{0}-\xi_{0}},\frac{\eta_{0}-\eta}{\eta_{0}-\xi} \biggr), \\ R_{0}^{-}(\xi,\eta;\xi_{0},\eta_{0}) :=\gamma \biggl(\frac{\eta-\xi}{\xi_{0}-\xi} \biggr)^{\beta} \biggl(\frac{\eta-\xi }{\eta_{0}-\xi} \biggr)^{\beta}F_{1} \biggl(\beta,\beta,\beta,1+2 \beta; \frac{\eta-\xi}{\xi_{0}-\xi},\frac{\eta-\xi}{\eta_{0}-\xi} \biggr). \end{gathered} $$
Here \(\gamma=- \frac{\Gamma(\beta)}{\Gamma(1-\beta) \Gamma(1+2 \beta )}\) and \(F_{1}(a,b_{1},b_{2},c;x,y)\) is the hypergeometric function (A.8) of two variables (see Appendix A).
In [47] the case \(0<\beta<1/2\) is considered, but here we find that in a more general case \(0<\beta<1\) the function \(R_{0}(\xi,\eta;\xi_{0},\eta_{0})\) solves
$$ \begin{gathered} \frac{\partial R_{0}}{\partial\eta}=-(\eta- \xi)^{-1}H( \xi,\eta;\xi_{0},\eta_{0})\quad\text{for }(\xi, \eta) \in\Pi\cup T, \\ R_{0}|_{\eta=\eta_{0}}=0,\quad\quad R_{0}|_{\eta=\xi}=0, \end{gathered} $$
(3.13)
where \((\xi_{0},\eta_{0}) \in D\) and \(H(\xi,\eta;\xi_{0},\eta_{0})\) is function (A.12).
Using (3.13), integration by parts and
$$\bigl[R_{0}^{+} - R_{0}^{-} \bigr]\big|_{\eta=\xi_{0}}= \frac{1}{\beta} $$
leads to the integral representation
$$ \begin{aligned}[b] U^{H}(\xi_{0}, \eta_{0}) &:= \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}\frac{\partial}{\partial\eta} \bigl[(\eta-\xi)F( \xi,\eta) \bigr] R_{0} (\xi,\eta;\xi_{0}, \eta_{0}) \,d\eta \,d\xi \\ &\quad{} +\frac{1}{\beta} \int_{0}^{\xi_{0}}(\xi_{0}-\xi)F(\xi, \xi_{0}) \,d\xi. \end{aligned} $$
(3.14)
(1.ii) Differentiating (3.14) we obtain that \(U^{H}\) satisfies the differential equation
$$ \bigl(U^{H}_{\xi_{0}} \bigr)_{\eta_{0}}+ \frac{\beta}{\eta_{0}-\xi_{0}} \bigl( U^{H}_{ \xi_{0}}- U^{H}_{\eta_{0}} \bigr)=F(\xi_{0},\eta_{0}), $$
(3.15)
where all derivatives are in a classical sense and they are continuous in D.
(1.iii) Since \(H(\xi,\eta;\xi_{0},\eta_{0})\) satisfies the differential equation (3.2) with \(n=0\) and \(\Phi =H+G\) satisfies (3.2) with \(n \geq0\) for the difference \(G=\Phi- H\), we obtain
$$G_{\xi_{0}\eta_{0}}+\frac{\beta}{\eta_{0}-\xi_{0}} ( G_{ \xi_{0}}- G_{\eta_{0}} )- \frac{n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}}G=\frac{n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}}H. $$
Now, using integral representation (3.7) for \(U^{G}(\xi_{0},\eta _{0})\), we calculate
$$ \begin{aligned}[b] & \bigl(U^{G}_{\xi_{0}} \bigr)_{\eta_{0}}+\frac{\beta}{\eta_{0}-\xi_{0}} \bigl( U^{G}_{ \xi_{0}}- U^{G}_{\eta_{0}} \bigr)-\frac{n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}}U^{G} \\ &\quad = \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta) \biggl[G_{\xi_{0}\eta_{0}}+ \frac{\beta}{\eta_{0}-\xi_{0}} ( G_{ \xi_{0}}- G_{\eta_{0}} ) \\ &\quad\quad{} -\frac {n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}}G \biggr](\xi,\eta;\xi_{0},\eta_{0}) \,d\eta \,d\xi \\ &\quad =\frac{n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}} \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta) H(\xi,\eta; \xi_{0},\eta_{0}) \,d\eta \,d\xi \\ &\quad =\frac{n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}}U^{H}, \end{aligned} $$
(3.16)
where all derivatives are in a classical sense and they are continuous in D.
(1.iv) Since \(U=U^{H}+U^{G}\), summing up equations (3.15) and (3.16), we obtain the differential equation
$$(U_{\xi_{0}} )_{\eta_{0}}+\frac{\beta}{\eta_{0}-\xi_{0}} ( U_{ \xi_{0}}- U_{\eta_{0}} )-\frac{n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}}U=F(\xi_{0},\eta_{0}) $$
in a classical sense. But, since \(F, U, U_{\xi_{0}}-U_{\eta_{0}} \in C(D)\), it follows that \((U_{\xi_{0}} )_{\eta _{0}}\in C(D)\).

Step 2. We will prove that identity (2.9) holds for all \(V(\xi,\eta) \in V^{(2)}\).

(2.i) Let \(V(\xi,\eta) \in V^{(2)}\) and in addition \(V(\xi,\eta)\equiv0\) in a neighborhood of \(\{\eta=\xi\}\) and in a neighborhood of \(\{\eta=1\}\). From Step 1 we know that \(U(\xi,\eta)\) satisfies the differential equation (2.5), where all derivatives are in a classical sense, continuous in D. Let us consider
$$ I_{V}:= \int_{D}(\eta-\xi)^{2 \beta} \biggl\{ U_{\xi}V_{\eta}+U_{\eta}V_{\xi} +\frac{2n(n+1)}{(2-\xi-\eta)^{2}}UV+2FV \biggr\} \,d\xi \,d\eta. $$
(3.17)
Now we integrate by parts in \(I_{V}\) in the following way:
  • in the term \(U_{\xi}V_{\eta}\), we move the derivative from \(V_{\eta}\) to \(U_{\xi}\) and obtain the term \((U_{\xi} )_{\eta}V\):
    $$ \int_{D}(\eta-\xi)^{2 \beta}U_{\xi}V_{\eta} \,d\xi \,d\eta= - \int_{D}(\eta-\xi)^{2 \beta} \biggl[ (U_{\xi} )_{\eta}+\frac{2\beta}{\eta-\xi}U_{\xi} \biggr]V \,d\xi \,d\eta; $$
    (3.18)
  • in the term \(U_{\eta}V_{\xi}\), we move the derivative from \(U_{\eta}\) to \(V_{\xi}\) and obtain the term \(U (V_{\xi} )_{\eta}\):
    $$\int_{D}(\eta-\xi)^{2 \beta}U_{\eta}V_{\xi} \,d\xi \,d\eta= - \int_{D}(\eta-\xi)^{2 \beta} \biggl[ (V_{\xi} )_{\eta}+\frac{2\beta}{\eta-\xi}V_{\xi} \biggr]U \,d\xi \,d\eta. $$
There are not integrals on the boundary of D because \(U(0,\eta)=0\), \(V(\xi,\eta)\equiv0\) in a neighborhood of \(\{\eta=\xi\}\) and in a neighborhood of \(\{\eta=1\}\).
  • since \(V \in C^{2}(\bar{D})\), we have \((V_{\xi} )_{\eta}= (V_{\eta } )_{\xi}\);

  • in the term \((V_{\eta} )_{\xi}U\), we move the derivatives from \((V_{\eta} )_{\xi}\) to U and obtain the term \((U_{\xi} )_{\eta}V\):
    $$ \begin{aligned}[b] \int_{D}(\eta-\xi)^{2 \beta}U_{\eta}V_{\xi} \,d\xi \,d\eta&= - \int_{D}(\eta-\xi)^{2 \beta} \biggl[ (V_{\eta} )_{\xi}+\frac{2\beta}{\eta-\xi}V_{\xi} \biggr]U \,d\xi \,d\eta \\ &= \int_{D}(\eta-\xi)^{2 \beta} \biggl[U_{\xi}V_{\eta}- \frac{2\beta}{\eta-\xi}(V_{\xi}+V_{\eta})U \biggr] \,d\xi \,d\eta \\ &=- \int_{D}(\eta-\xi)^{2 \beta} \biggl[ (U_{\xi} )_{\eta}-\frac{2\beta}{\eta-\xi}U_{\eta} \biggr]V \,d\xi \,d\eta. \end{aligned} $$
    (3.19)
Again there are not integrals on the boundary of D, and putting (3.18) and (3.19) into (3.17), we get
$$ I_{V}= -2 \int_{D}(\eta-\xi)^{2 \beta} \biggl\{ (U_{\xi} )_{\eta}+\frac{\beta}{\eta-\xi}(U_{\xi}-U_{\eta}) - \frac{n(n+1)}{(2-\xi-\eta)^{2}}U-F \biggr\} V \,d\xi \,d\eta=0. $$
(3.20)
(2.ii) Let \(V(\xi,\eta) \in V^{(2)}\) and \(\Psi(s)\) be a function having the properties \(\Psi(s) \in C^{\infty}(\mathbf{R}^{1})\), \(\Psi(s)=1\) for \(s\geq2\), \(\Psi(s)=0\) for \(s\leq1\). If \(k, l \in\mathbf{N}\), then according to (2.i) (see (3.17) and (3.20)) for the functions
$$V_{k,l}(\xi,\eta):=V(\xi,\eta)\Psi \bigl(k [1-\eta] \bigr)\Psi \bigl(l [ \eta-\xi] \bigr) $$
identity (2.9) holds. Therefore we have
$$ \begin{aligned}[b] 0&= \int_{D}(\eta-\xi)^{2 \beta} \biggl\{ U_{\xi}V_{\eta}+U_{\eta}V_{\xi} +\frac{2n(n+1)}{(2-\xi-\eta)^{2}}UV+2FV \biggr\} \\ &\quad{}\times \Psi \bigl(k [1-\eta] \bigr) \Psi \bigl(l [ \eta-\xi] \bigr) \,d\xi \,d\eta \\ &\quad{} + \int_{D}l(\eta-\xi)^{2 \beta} \{U_{\xi}-U_{\eta} \}\Psi \bigl(k [1-\eta] \bigr)\Psi' \bigl(l [\eta-\xi] \bigr) V \,d \xi \,d\eta \\ &\quad{} - \int_{D}k(\eta-\xi)^{2 \beta}U_{\xi} \Psi' \bigl(k [1-\eta] \bigr) \Psi \bigl(l [\eta-\xi] \bigr)V \,d\xi \,d\eta \\ &=: I_{1,kl}+I_{2,kl}+I_{3,kl}. \end{aligned} $$
(3.21)
Obviously, \(I_{1,kl} \to I_{V}\), as \(k, l \to\infty\).
We know that \(V \equiv0\) in a neighborhood of \((0,0)\) and \(\operatorname {supp}\Psi'(l[\eta-\xi])\) is contained in \(\{1\leq l[\eta-\xi]\leq2 \}\). Using estimate (3.12) we find that on \(\operatorname {supp}\Psi'(l[\eta-\xi])\) the functions
$$W_{k,l}(\xi,\eta):=l(\eta-\xi)^{2 \beta} \{U_{\xi}-U_{\eta} \}\Psi \bigl(k [1-\eta] \bigr)\Psi' \bigl(l [\eta-\xi] \bigr) V $$
satisfy the estimates
$$ \bigl\vert W_{k,l}(\xi,\eta) \bigr\vert \leq \mathit{const.} ( \eta-\xi)^{\beta-1}. $$
(3.22)
Since, obviously, the sequence \({W_{k,l}}\) converges pointwise almost everywhere to zero and it is dominated by a Lebesgue integrable function in D for \(0<\beta<1\) (see (3.22)). Thus, according to the Lebesgue dominated convergence theorem, \(I_{2,kl} \to0\) as \(k, l \to \infty\).
Since \(V(\xi,1)=0\), we have
$$k \bigl\vert V(\xi,\eta) \bigr\vert \bigl\vert \Psi' \bigl(k [1-\eta] \bigr) \bigr\vert = k(1-\eta) \bigl\vert V_{\eta}(\xi, \tilde{ \eta}) \bigr\vert \bigl\vert \Psi' \bigl(k [1-\eta] \bigr) \bigr\vert \leq c_{v}, $$
where \(c_{v}\) is a constant and \(\eta<\tilde{\eta}<1\). Therefore \(I_{3,kl} \to0\) as \(k, l \to\infty\).

Thus, letting \(k, l \to\infty\) in (3.21), we obtain that identity (2.9) holds for \(V \in V^{(2)}\). Consequently, the function \(U(\xi,\eta)\) is a generalized solution to problem \(\mathit {PK}_{2}\). □

4 Proof of the main results

In this section we give the proofs of Theorem 1.1, Theorem 1.2 and Theorem 1.3 formulated in Section 1.

Proof of Theorem 1.1

Let and \(u_{1}\) and \(u_{2}\) be two generalized solutions of problem PK in \(\Omega_{m}\). Then the function \(u:=u_{1}-u_{2}\) solves the homogeneous problem PK. We will show that the Fourier expansion
$$u(r ,\theta, \varphi ,t)=\sum_{n=0}^{\infty} \sum_{s=1}^{2n+1}u_{n}^{s}(r,t)Y_{n}^{s}( \theta, \varphi) $$
has zero Fourier-coefficients
$$\begin{aligned} u_{n}^{s}(r ,t):= \int_{0}^{\pi} \int_{0}^{2\pi} u(r,\theta,\varphi,t)Y_{n}^{s}( \theta,\varphi)\sin\theta \,d\varphi \,d\theta \end{aligned}$$
in \(G_{m}\), i.e., \(u \equiv0\) in \(\Omega_{m }\).
For u we know that the identity
$$\begin{aligned} \int_{\Omega_{m}} \bigl\{ t^{m} u_{t}v_{t}-u_{x_{1}}v_{x_{1}}-u_{x_{2}}v_{x_{2}}-u_{x_{3}}v_{x_{3}} \bigr\} \,dx_{1}\,dx_{2}\,dx_{3}\,dt =0 \end{aligned}$$
(4.1)
holds for all test functions \(v(x,t)=w(r,t)Y_{n}^{s}(x)\) described in Remark 2.1. Therefore from (4.1) we derive
$$ \int_{G_{m}} \biggl\{ u_{n,r}^{s}w_{r}-t^{m} u_{n,t}^{s}w_{t}+\frac{n(n+1)}{r^{2}}u_{n}^{s}w \biggr\} r^{2} \,dr \,dt =0 $$
(4.2)
for all \(w(r,t)\in V_{m}^{(1)}\) (see Definition 2.1), \(n \in\mathbf{N}\cup\{0\}\), \(s=1,2,\ldots, 2n+1\). Since \(u(x,t)\) satisfies conditions (1), (2) and (3) in Definition 1.1, the functions \(u_{n}^{s}(r,t)\) satisfy conditions (1), (2) and (3) in Definition 2.1, and therefore they are generalized solutions of problem \(\mathit {PK}_{1}\).
Using (2.3) we see that the functions \(V(\xi,\eta):=r(\xi,\eta)w(r(\xi,\eta),t(\xi,\eta)) \in V^{(2)}\). Now from (4.2) we obtain that for the functions \(U_{n}^{s}(\xi,\eta):=r(\xi,\eta)u_{n}^{s}(r(\xi,\eta),t(\xi,\eta))\) the identity
$$ \int_{D}(\eta-\xi)^{2 \beta} \biggl\{ U_{n,\xi}^{s}V_{\eta}+U_{n,\eta }^{s}V_{\xi} +\frac{2n(n+1)}{(2-\xi-\eta)^{2}}U_{n}^{s}V \biggr\} \,d\xi \,d\eta=0 $$
holds for all \(V(r,t)\in V^{(2)}\) (see Definition 2.2), \(n \in\mathbf{N}\cup\{0\}\), \(s=1,2,\ldots, 2n+1\). The functions \(U_{n}^{s}(\xi,\eta)\) satisfy conditions (1), (2) and (3) in Definition 2.2 and, consequently, \(U_{n}^{s}(\xi,\eta)\) are generalized solutions of the 2-D homogeneous problem \(\mathit {PK}_{2}\). Theorem 3.1 gives \(U_{n}^{s}(\xi,\eta) \equiv0\) in D. Therefore \(u_{n}^{s}(r,t)\equiv0\) in \(G_{m}\) and thus \(u=u_{1}-u_{2} \equiv0\) in \(\Omega_{m}\). □

Proof of Theorem 1.2

From Theorem 1.1 it follows that there exists at most one generalized solution of problem PK in \(\Omega_{m}\). Since \(f(x,t)\) has the form (1.5), we look for a generalized solution of the form (1.6), i.e.,
$$u(x,t)=\sum_{n=0}^{l}\sum _{s=1}^{2n+1}u_{n}^{s} \bigl( \vert x\vert ,t \bigr)Y_{n}^{s}(x). $$
To find such a solution means to find functions \(u_{n}^{s}(r,t)\) that satisfy the identities
$$\begin{aligned} \int_{G_{m,\varepsilon}} \biggl[u_{n,r }^{s}v_{r }-t^{m} u_{n,t}^{s}v_{t} +\frac{n(n+1)}{r^{2}}u_{n}^{s}v+f_{n}^{s}v \biggr]r^{2} \,dr \,dt =0 \end{aligned}$$
for all \(v \in V_{m}^{(1)}\) and satisfy the corresponding conditions (1), (2) and (3) in Definition 2.1. In view of (2.3) to find such functions means to find functions
$$U_{n}^{s}(\xi,\eta)=r(\xi,\eta)u_{n}^{s} \bigl(r(\xi,\eta),t(\xi,\eta) \bigr), $$
such that for \(F_{n}^{s}(\xi,\eta):=\frac{1}{4}r(\xi,\eta)f_{n}^{s}(r(\xi,\eta),t(\xi,\eta))\) the identity
$$ \int_{D}(\eta-\xi)^{2 \beta} \biggl\{ U_{n,\xi}^{s}V_{\eta}+U_{n,\eta }^{s}V_{\xi} +\frac{2n(n+1)}{(2-\xi-\eta)^{2}}U_{n}^{s}V+2F_{n}^{s}V \biggr\} \,d\xi \,d\eta=0 $$
holds for all \(V(\xi,\eta)=r(\xi,\eta)v(r(\xi,\eta),t(\xi,\eta)) \in V^{(2)}\) and satisfies the corresponding conditions (1), (2) and (3) in Definition 2.2. Theorem 3.5 gives the existence of such functions \(U^{s}_{n}(\xi,\eta)\) which are generalized solutions of problem \(\mathit {PK}_{2}\) in D. In that way we found functions \(u_{n}^{r}(r,t)=r^{-1}U_{n}^{s}(\xi(r,t),\eta(r,t))\) which are generalized solutions of problem \(\mathit {PK}_{1}\) in \(G_{m}\). Therefore the function \(u(x ,t)\), given by (1.6), is a generalized solution of problem PK in \(\Omega_{m}\). □

Proof of Theorem 1.3

Theorem 1.1 and Theorem 1.2 claim the existence and uniqueness of generalized solutions \(u(x ,t)\) of problem PK in \(\Omega_{m}\), which has the form (1.6). Using (2.3) for functions \(U_{n}^{s}(\xi,\eta)=r(\xi,\eta)u_{n}^{s}(r(\xi,\eta),t(\xi,\eta))\) and \(F_{n}^{s}(\xi,\eta)=\frac{1}{4}r(\xi,\eta)f_{n}^{s}(r (\xi,\eta),t(\xi ,\eta))\), we obtain the 2-D problem \(\mathit {PK}_{2}\). According to Theorem 3.5, estimates (3.12) hold, and in view of (3.6) and (3.7) we see that the estimate for \(\vert U_{n}^{s}(\xi,\eta)\vert \) holds with \({ ( \max_{\bar{G}_{m}} \vert f_{n}^{s}\vert )}\) instead of \(M_{F}\):
$$\bigl\vert U_{n}^{s}(\xi,\eta) \bigr\vert \leq K \Bigl( \max_{\bar{G}_{m}} \bigl\vert f_{n}^{s} \bigr\vert \Bigr) (2-\xi-\eta)^{-n} $$
with a constant \(K>0\) independent of \(f_{n}^{s}\). That implies
$$\bigl\vert u_{n}^{s}(r,t) \bigr\vert \leq2^{-n}K \Bigl(\max_{\bar{G}_{m}} \bigl\vert f_{n}^{s} \bigr\vert \Bigr) r^{-n-1}. $$
Therefore in view of (1.6), summing up over n and s, we get the desired estimate (1.7). □

Remark 4.1

It is interesting that in the case \(0< m<1\) problem PK for the Keldysh-type equation (1.1) can be formally reduced to problem P2 for the Tricomi-type equation (1.2) with power of degeneration \(m_{1}:=m/(1-m)>0\) and the right-hand side function, which vanishes on \(\Sigma_{0}\) like \(t^{m_{1}}\). That implies many differences between the investigation and behavior of the solution to the obtained problem and the usual Protter problem P2. However, in the present paper we study the \((3+1)\)-D problem PK in a more general case when \(\boldsymbol{0< m<4/3}\).

Declarations

Acknowledgements

The authors thank the anonymous referees for making several helpful suggestions. The research of Tsvetan Hristov, Aleksey Nikolov and Nedyu Popivanov was partially supported by the Sofia University Grant 152/2016.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Faculty of Mathematics and Informatics, University of Sofia
(2)
Faculty of Applied Mathematics and Informatics, Technical University of Sofia
(3)
Faculty of Mathematics, Karlsruhe Institute of Technology

References

  1. Morawetz, C: Mixed equations and transonic flow. J. Hyperbolic Differ. Equ. 1(1), 1-26 (2004) MathSciNetView ArticleMATHGoogle Scholar
  2. Protter, M: New boundary value problem for the wave equation and equations of mixed type. J. Ration. Mech. Anal. 3, 435-446 (1954) MathSciNetMATHGoogle Scholar
  3. Morawetz, C: A weak solution for a system of equations of elliptic-hyperbolic type. Commun. Pure Appl. Math. 11, 315-331 (1958) MathSciNetView ArticleMATHGoogle Scholar
  4. Lax, P, Phillips, R: Local boundary conditions for dissipative symmetric linear differential operators. Commun. Pure Appl. Math. 13, 427-455 (1960) MathSciNetView ArticleMATHGoogle Scholar
  5. Aziz, A, Schneider, M: Frankl-Morawetz problems in \({R}^{3}\). SIAM J. Math. Anal. 10, 913-921 (1979) MathSciNetView ArticleMATHGoogle Scholar
  6. Barros-Neto, J, Gelfand, I: Fundamental solutions for the Tricomi operator. Duke Math. J. 128(1), 119-140 (2005) MathSciNetView ArticleMATHGoogle Scholar
  7. Dechevski, L, Popivanov, N: Morawetz-Protter 3D problem for quasilinear equations of elliptic-hyperbolic type. Critical and supercritical cases. C. R. Acad. Bulgare Sci. 61(12), 1501-1508 (2008) MathSciNetMATHGoogle Scholar
  8. Keyfitz, B, Tesdall, A, Payne, K, Popivanov, N: The sonic line as a free boundary. Q. Appl. Math. 71(1), 119-133 (2013) MathSciNetView ArticleMATHGoogle Scholar
  9. Lupo, D, Monticelli, D, Payne, K: On the Dirichlet problem of mixed type for lower hybrid waves in axisymmetric cold plasmas. Arch. Ration. Mech. Anal. 217(1), 37-69 (2015) MathSciNetView ArticleMATHGoogle Scholar
  10. Lupo, D, Morawetz, C, Payne, K: On closed boundary value problems for equations of mixed elliptic-hyperbolic type. Commun. Pure Appl. Math. 60(9), 1319-1348 (2007) MathSciNetView ArticleMATHGoogle Scholar
  11. Lupo, D, Payne, K, Popivanov, N: Nonexistence of nontrivial solutions for supercritical equations of mixed elliptic-hyperbolic type. Prog. Nonlinear Differ. Equ. Appl. 66, 371-390 (2006) MathSciNetMATHGoogle Scholar
  12. Lupo, D, Payne, K, Popivanov, N: On the degenerate hyperbolic Goursat problem for linear and nonlinear equations of Tricomi type. Nonlinear Anal. 108, 29-56 (2014) MathSciNetView ArticleMATHGoogle Scholar
  13. Rassias, J: Tricomi-Protter problem of nD mixed type equations. Int. J. Appl. Math. Stat. 8(M07), 76-86 (2007) MathSciNetMATHGoogle Scholar
  14. Kalmenov, T, Koshanov, B, Iskakova, U: Completeness of the root vectors to the Tricomi problem for Lavrent’ev-Bitsadze equation. In: Nonclassical Equations of Mathematical Physics. Proceedings of the Seminar Dedicated to the 60-th Birthday of Prof. V. N. Vragov, Novosibirsk, Russia, pp. 125-129. Sobolev Institute of Mathematics RAN, Novosibirsk (2005) Google Scholar
  15. Moiseev, T: Gellerstedt problem with a generalized Frankl matching condition on the type change line with data on external characteristics. Differ. Equ. 52, 240-247 (2016) MathSciNetView ArticleMATHGoogle Scholar
  16. Sadybekov, M, Yessirkegenov, N: Properties of solutions to Neumann-Tricomi problems for Lavrent’ev-Bitsadze equations at corner points. Electron. J. Differ. Equ. 2014, 193 (2014) MathSciNetView ArticleMATHGoogle Scholar
  17. Otway, T: The Dirichlet Problem for Elliptic-Hyperbolic Equations of Keldysh Type. Lecture Notes in Mathematics, vol. 2043. Springer, Berlin (2012) MATHGoogle Scholar
  18. Otway, T: Elliptic-Hyperbolic Partial Differential Equations. A Mini-Course in Geometric and Quasilinear Methods. Springer Briefs in Mathematics. Springer, Cham (2015) View ArticleMATHGoogle Scholar
  19. Otway, T, Marini, A: Strong solutions to a class of boundary value problems on a mixed Riemannian-Lorentzian metric. Discrete Contin. Dyn. Syst. 2015(Suppl.), 801-808 (2015) MathSciNetMATHGoogle Scholar
  20. Oleı̌nik, O, Radkevič, E: Second Order Equations with Nonnegative Characteristic Form. Am. Math. Soc., Providence (1973) View ArticleGoogle Scholar
  21. Nakhushev, A: Problems with Shifts for Partial Differential Equations. Nauka, Moscow (2006) [in Russian] MATHGoogle Scholar
  22. Chen, S: A mixed equation of Tricomi-Keldysh type. J. Hyperbolic Differ. Equ. 9(3), 545-553 (2012) MathSciNetView ArticleMATHGoogle Scholar
  23. Čanić, S, Keyfitz, B: A smooth solution for a Keldysh type equation. Commun. Partial Differ. Equ. 21(1-2), 319-340 (1996) MathSciNetView ArticleMATHGoogle Scholar
  24. Marin, M, Lupu, M: On harmonic vibrations in thermoelasticity of micropolar bodies. J. Vib. Control 4(5), 507-518 (1998) MathSciNetView ArticleMATHGoogle Scholar
  25. Marin, M, Craciun, E-M, Pop, N: Considerations on mixed initial-boundary value problems for micropolar porous bodies. Dyn. Syst. Appl. 25(1-2), 175-196 (2016) MATHGoogle Scholar
  26. Protter, M: A boundary value problem for the wave equation and mean value problems. Ann. Math. Stud. 33, 247-257 (1954) Google Scholar
  27. Kalmenov, T: On the strong solutions of the Darboux and Tricomi problems. Sov. Math. Dokl. 28, 672-674 (1983) Google Scholar
  28. Nakhushev, A: Criteria for continuity of the gradient of the solution to the Darboux problem for the Gellerstedt equation. Differ. Equ. 28(10), 1445-1457 (1992) MathSciNetMATHGoogle Scholar
  29. Garabedian, P: Partial differential equations with more than two variables in the complex domain. J. Math. Mech. 9, 241-271 (1960) MathSciNetMATHGoogle Scholar
  30. Khe, KC: On nontrivial solutions of some homogeneous boundary value problems for the multidimensional hyperbolic Euler-Poisson-Darboux equation in an unbounded domain. Differ. Equ. 34(1), 139-142 (1998) MathSciNetMATHGoogle Scholar
  31. Popivanov, N, Schneider, M: The Darboux problems in \({R}^{3}\) for a class of degenerating hyperbolic equations. J. Math. Anal. Appl. 175(2), 537-579 (1993) MathSciNetView ArticleMATHGoogle Scholar
  32. Bitsadze, A: Some Classes of Partial Differential Equations. Gordon & Breach, New York (1988) MATHGoogle Scholar
  33. Hristov, T, Popivanov, N: Singular solutions to Protter’s problem for a class of 3-D weakly hyperbolic equations. C. R. Acad. Bulgare Sci. 60(7), 719-724 (2007) MathSciNetMATHGoogle Scholar
  34. Popivanov, N, Popov, T: Asymptotic expansions of singular solutions for \((3+1)\)-D Protter problems. J. Math. Anal. Appl. 331, 1093-1112 (2007) MathSciNetView ArticleMATHGoogle Scholar
  35. Nikolov, A, Popivanov, N: Asymptotic expansion of singular solutions to Protter problem for \((2+1)\)-D degenerate wave equation. In: Venkov, G, Pasheva, V (eds.) Applications of Mathematics in Engineering and Economics: AMEE’13, Sozopol, Bulgaria, June 2013. AIP Conf. Proc., vol. 1570, pp. 249-256. AIP, New York (2013) Google Scholar
  36. Popivanov, N, Popov, T: Behaviour of singular solutions to 3-D Protter problem for a degenerate hyperbolic equation. C. R. Acad. Bulgare Sci. 63(6), 829-834 (2010) MathSciNetMATHGoogle Scholar
  37. Nikolov, A, Popivanov, N: Singular solutions to Protter’s problem for \((3+1)\)-D degenerate wave equation. In: Venkov, G, Pasheva, V (eds.) Applications of Mathematics in Engineering and Economics: AMEE’12, Sozopol, Bulgaria, June 2012. AIP Conf. Proc., vol. 1497, pp. 233-238. AIP, New York (2012) Google Scholar
  38. Didenko, V: On boundary-value problems for multidimensional hyperbolic equations with degeneration. Sov. Math. Dokl. 13, 998-1002 (1972) MATHGoogle Scholar
  39. Aldashev, S: A criterion for the existence of eigenfunctions of the Darboux-Protter spectral problem for degenerating multidimentional hyperbolic equations. Differ. Equ. 41, 833-839 (2005) MathSciNetView ArticleMATHGoogle Scholar
  40. Hristov, T, Popivanov, N, Schneider, M: On the uniqueness of generalized and quasi-regular solutions for equations of mixed type in \({R}^{3}\). Sib. Adv. Math. 21(4), 262-273 (2011) MathSciNetView ArticleMATHGoogle Scholar
  41. Hristov, T, Popivanov, N, Schneider, M: Generalized solutions to Protter problems for 3-D Keldysh type equations. In: Sivasundaram, S (ed.) 10th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences: ICNPAA 2014, Narvik, Norway, July 2014. AIP Conf. Proc., vol. 1637, pp. 422-430. AIP, New York (2014) Google Scholar
  42. Hristov, T: Singular solutions to Protter problem for Keldysh type equations. In: Venkov, G, Pasheva, V (eds.) Applications of Mathematics in Engineering and Economics: AMEE’14, Sozopol, Bulgaria, June 2014. AIP Conf. Proc., vol. 1631, pp. 255-262. AIP, New York (2014) Google Scholar
  43. Hristov, T, Popivanov, N, Schneider, M: Protter problem for 3-D Keldysh type equations involving lower order terms. In: Applications of Mathematics in Engineering and Economics: AMEE’15, Sozopol, Bulgaria, June 2015. AIP Conf. Proc., vol. 1690, pp. 04002001-04002012. AIP, New York (2015) Google Scholar
  44. Hristov, T, Nikolov, A, Popivanov, N, Schneider, M: Generalized solutions of Protter problem for \((3+1)\)-D Keldysh type equations. In: Pasheva, V, Venkov, G, Popivanov, N (eds.) Applications of Mathematics in Engineering and Economics: AMEE’16, Sozopol, Bulgaria, June 2016. AIP Conf. Proc., vol. 1789, pp. 04000701-04000713. AIP, New York (2016) Google Scholar
  45. Jones, M: Spherical Harmonics and Tensors for Classical Field Theory. Research Studies Press, Letchworth (1986) Google Scholar
  46. Gellerstedt, S: Sur une équation linéaire aux dérivées partielles de type mixte. Ark. Mat. Astron. Fys. A 25(29), 1-23 (1937) MATHGoogle Scholar
  47. Smirnov, M: Degenerating Hyperbolic Equations. Vysheishaia shkola Publ. House, Minsk (1977) [in Russian] MATHGoogle Scholar
  48. Moiseev, E: Approximation of the classical solution of a Darboux problem by smooth solutions. Differ. Equ. 20, 59-74 (1984) MathSciNetMATHGoogle Scholar
  49. Bateman, H, Erdelyi, A: Higher Transcendental Functions, vol. 1. McGraw-Hill, New York (1953) MATHGoogle Scholar
  50. Smirnov, M: Mixed Type Equations. Vishaya shkola Publ. House, Moskow (1985) [in Russian] Google Scholar
  51. Volkodavov, V, Zaharov, V: Tables of Riemann and Riemann-Hadamard Functions for Some Differential Equations in n-Dimensional Euclidean Spaces. Samara State Teacher’s Training University, Samara (1994) [in Russian] Google Scholar
  52. Meredov, M: The unique solvability of Darboux’s problem for a degenerating system. Differ. Equ. 10, 63-70 (1975) MATHGoogle Scholar

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