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

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.

We study the following boundary value problem.Problem .Find a solution to (1) in Ω  which satisfies the boundary conditions: The adjoint problem to PK is as follows.

The Main Results
Problem PK is not well-posed.Actually, the adjoint homogeneous problem PK * has infinitely many classical solutions.
Further, let us denote by    (),  ∈ N ∪ {0},  = 1, 2, . . ., 2 + 1 the three-dimensional spherical functions.They are usually defined on the unit sphere  2 fl { ∈ R 3 : || = 1}, but for convenience of our discussions we extend them out of  2 radially, keeping the same notation for the extended functions: In this paper, we prove the following lemma.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 function (, ) to all these functions V , , (, ).Indeed, This means that an infinite number of orthogonality conditions  , , = 0 with V , , (, )  (, )   (11) must be fulfilled.
To avoid this we consider solutions to this problem in a generalized sense.In the present paper we study the case 0 < m < 4/3 and we use the following definition of a generalized solution of problem PK.
Definition 2 (see [1]).We call a function (, ) a generalized solution of problem PK in Ω  , 0 <  < 4/3, for (1), if ) for each  ∈ (0, 1) there exists a constant () > 0, such that (4) the identity holds for all V from 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 Definition 2 allows the generalized solution to have some singularity at the point .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 4 (see [1]).Let  ∈ (0, 4/3).Suppose that the righthand side function (, ) is fixed as a "harmonic polynomial" of order  with  ∈ N ∪ {0}: and  ∈  1 (Ω  ).Then there exists an unique generalized solution (, ) of problem PK in Ω  and it has the following form: In this paper, we derive an asymptotic formula concerning the behavior of the singularities of the generalized solution.
Theorem 5. Let  ∈ (0, 4/3) and the right-hand side function  ∈  1 (Ω  ) has the form (15). Then the unique generalized solution (, ) of problem PK on the characteristic surface has the following expansion at point : where (i) the function  ∈  1 (0 < || < 1/2) and satisfies the a priori estimate with a constant  > 0 independent of ; (ii) the functions   () have the following structure: where  +2, , ̸ = 0 are constants independent of (, ).Corollary 6. Suppose that at least one of the constants  +2, , in (20) is different from zero.Then for the corresponding function   () there exists a vector  ∈ R 3 , || = 1, such that   () →  , = const ̸ = 0 as  → +0.This means that the order of singularity of (, ) will be no smaller than  + 1.

Corollary 7.
Let the conditions of Theorem 5 be fulfilled and in addition (, ) satisfies the orthogonality conditions: where  is a positive constant independent of (, ).
Actually, Theorem 5 gives the asymptotic behavior of the singular solutions of problem PK on Σ  2 .It clarifies the significance of the orthogonality conditions (21): for fixed indexes (, , ), the corresponding condition ( 21) "controls" one power-type singularity.We mention here that some of the orthogonality conditions (21) involve functions V , , (, ), which are not classical solutions of problem PK * (see the proof of Lemma 1).

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: where () > 0 for  ̸ = 0.The Chaplygin equation ( 23) is elliptic in the subsonic half-plane  < 0 and hyperbolic in the supersonic half-plane  > 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  and in the hyperbolic halfplane by four characteristic segments that start from the three points  1 ,  2 , and  on the sonic line; see Figure 2(a).The values of the function are prescribed along  and along the characteristics  1  1 and  2  2 .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.
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: where  = ( 1 , . . .,   ),  ≥ 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 Σ1 .On the characteristic surface Σ2 , 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 () =   ,  ∈ R,  ≥ 0 in the domain Ω ∩ { > 0} with one of the following boundary conditions: These BVPs are multidimensional analogues of the Darboux-Goursat plane problems for the Gellerstedt equation ( > 0) or for the wave equation ( = 0).Garabedian [11] proved the uniqueness of a classical solution to problem 1 for the wave equation in R 4 .Popivanov and Schneider [12] showed that both problems 1 and 2 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][15][16][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][22][23][24][25][26][27].The Keldysh-type equations are another kind of mixedtype equations that also are known to play an important role in fluid mechanics, for example, near the line  = 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  = 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 ).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   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 1, 2: 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 function (, ) 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 (, ) is still allowed to have a singularity of order  ∈ (0, 1).
(2) To study the general case of problem PK when the right-hand side function (, ) is a smooth function not only of the form of "harmonic polynomial" is an open problem: (i) Find appropriate conditions for the function (, ) 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 (, ) 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 function (, ) 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 when 0 <  < 2. Let us mention that the presentation of the generalized solution (, ) from [1], which we are studying in the present paper, is valid only in the case when  ∈ (0, 4/3).Find appropriate techniques that work for 4/3 ≤  < 2.

The Two-Dimensional Darboux-Goursat Problems Corresponding to Problem PK
Problem PK in the case when the right-side function (, ) is of the form ( 15) can be reduced to a two-dimensional problem.
Problem  2 .Find a solution of satisfying the following boundary conditions: where (, ) fl 1 8 (2 −  − )    ( (, ) ,  (, )) , As far as we consider problem PK in the case when  ∈ (0, 4/3) for the parameter  we have In conformity with Definition 2, a generalized solution of problem PK 2 is defined as follows.
Theorem 9 (see [1]).Let 0 <  < 1 and  ∈  1 ().Then there exists one and only one generalized solution of problem  2 in , which has the following integral representation at a point ( 0 ,  0 ) ∈ : and it satisfies the following estimates: where  is a positive constant and The Riemann-Hadamard function Φ(, ;  0 ,  0 ), ( 0 ,  0 ) ∈ , which we have found in [1], can be represented as follows: where Here  3 ( 1 ,  2 ,  1 ,  2 , ; , ) is the Appell series: and  2 ( 1 ,  2 ,  1 ,  2 , ; , ) is the Horn series: (for basic information on the Appell and the Horn series, see [41], p. 222-228.)According to Theorem 9, the generalized solution is allowed to have a singularity of order no greater than  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 (, ), given by (40), or, more precisely, its restriction on the segment  = 1, 0 ≤  < 1.

The Asymptotic Expansion of the Solution of Problem PK 2
Introduce the following functions: where   , (||, ) are functions ( 6), closely connected with the solutions of the homogeneous adjoint problem PK * .Then we prove the following lemma.Lemma 10.For  = 0, . . ., [/2] the following equalities hold where  , , are the coefficients (11) and the relation between (, ) and the Fourier coefficient    from the expansion of (, ) is given by (33).
Denote also by Y   the spherical functions expressed in the spherical coordinates; that is,    () = Y   ((), ()).Then, using the orthonormality of the spherical functions on the unit sphere  2 , a direct calculation gives The proof is complete.
Further, we set  0 = 1 in (40).Essential for the following calculations is the decomposition of Φ − (, ;  0 , 1) given in Theorem A. 3  (55) According to (A.35) and Lemma 10 we have For  2 , using the estimate (A.36) from Theorem A.3, we obtain Making a substitution  =  + ( 0 − ) we compute with Formula (A.11) gives and with (A.13) we estimate Applying the results (60), (62), and (63) into (59) we obtain According to the results from [1] (Lemmas A.1, A.2, and A.3 therein), we have an estimate Then for  3 we have Therefore (54) holds with () fl From this theorem, we see that the generalized solution of problem PK 2 may have a singularity of order  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  , , are equal to zero.This result exactly corresponds to the estimate prescribed in Theorem 9.

Proof of the Main Results
Now, we are ready to prove the main results stated in Section 2.
The proof is complete.