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

A (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. MSC: 35D30; 35M12; 35A20


Introduction
In the present paper we consider an analogue of the Protter problems for (+)-D Keldyshtype equations. For m ∈ R,  < m < , we study some boundary value problems (BVPs) for the weakly hyperbolic equation (.) expressed in Cartesian coordinates (x, t) = (x  , x  , x  , t) ∈ R  in a simply connected region The adjoint problem to PK is as follows.
Problem PK * Find a solution to the self-adjoint equation (.) in m that satisfies the boundary conditions u| m  = ; t m u t → , as t → +.
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,  []. Let us also mention some results in the thermodynamic theory of porous elastic bodies given in [, ]. 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 [] and degenerated hyperbolic (or weakly hyperbolic) equations of Tricomi type []. In that case the Protter problems are multidimensional analogues of the plane Darboux or Cauchy-Goursat problems (see Kalmenov [] and Nakhushev []). The equations are considered in ( + )-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 [, ] Tricomi-type equations or the wave equation (m ∈ R, m ≥ ) bounded by  and two characteristics surfaces of (.) He proposed four problems, known now as Protter problems.
Protter problems Find a solution of equation (.) in the domain˜ m with one of the following boundary conditions: The boundary conditions in problem P * (respectively P * ) are the adjoint boundary conditions to problem P (respectively P) for (.) in˜ m .
It turns out that instead of both boundary conditions given in problems P on˜ m  ,  and in P on˜ m  ,  for the Tricomi-type equation (.), in the case of Keldysh-type equation (.), they are reduced to only one boundary condition on the characteristic m  and a condition on the growth of possible singularity of the derivative u t as t → +.
We mention some known results for Protter problems in the Tricomi case that make the investigation of such problems interesting and reasonable. Garabedian  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 PK * and to try to find out new effects that appear. In [] ill-posedness of -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 [], and some singular generalized solutions are announced in [].
In [, ] we study Protter problems for Tricomi-type equations. For -D Keldysh-type equation in [], we formulate a new Protter problem and announce some results for the existence and uniqueness of a generalized solution in the case  < m < . In [] we announce analogical results for ( + )-D equations of Keldysh type in a more general case  < m < / and claim the existence of infinitely many classical smooth solutions of the ( + )-D homogeneous problem PK * . Now, in the present paper we work in the case  < m < /. Using an appropriate Riemann-Hadamard function, we find an exact integral representation of the generalized solution and prove the results announced in []. 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 ε ∈ (, ), we introduce the region We give the following definition of a generalized solution of problem PK in the case  < m < /. Definition . We call a function u(x, t) a generalized solution of problem PK in m ,  < m <   , for equation (.) if: holds for all v from Remark . We mention that all the first derivatives of the generalized solutions of -D Protter problems in the Tricomi case can have singularity on the boundary of the domain (see [, ]). Actually, this fact corresponds to the analogical situation in a -D case of the Darboux problem (see []). While in the Keldysh case, according to Definition ., the derivative u t can be unbounded when t → +, but u x  , u x  and u x  are bounded in each¯ m,ε , ε > .
In this paper, first, we prove results for the uniqueness of a generalized solution to problem PK .
and Y s n form a complete orthonormal system in L  (S  ) (see []). For convenience of discussions that follow, we extend the spherical functions out of S  radially, keeping the same notation for the extended functions Y s n (x) := Y s n (x/|x|) for x ∈ R  \ {}. Let the right-hand side function f (x, t) of equation (.) be fixed as a "harmonic polynomial" of order l with l ∈ N ∪ {}, and it has the following representation: with some coefficients f s n (|x|, t). In this special case we give an existence result as well.
Theorem . Let m ∈ (,   ). Suppose that the right-hand side function f (x, t) has the form (.) and f ∈ C  (¯ m ). Then the unique generalized solution u(x, t) of problem PK in m exists and has the form Remark . Actually, when the right-hand side function f (x, t) has the form (.) in Theorem ., we find explicit representations for the functions u s n (|x|, t) in (.). These representations involve appropriate hypergeometric functions.
In the case when the right-hand side function f (x, t) has the form (.), we give an a priori estimate for the generalized solution of problem PK in m as well.
Theorem . Let the conditions in Theorem . be fulfilled. Then the unique generalized solution of problem PK in m has the form (.) and satisfies the a priori estimate with a constant c >  independent of f .
Estimate (.) shows the maximal order of possible singularity at point O, when the righthand 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 -D Protter problem P in the Tricomi case is obtained in [], while an estimate from below in this case is given in [].
The present paper contains the introduction and five more sections. In Section , the Protter problem PK is considered in a model case when the right-hand side function f (x, t) of equation (.) is fixed as a "harmonic polynomial" (.) of order l. In that case we formulate the -D boundary value problems PK  and PK  , corresponding to the ( + )-D problem PK . We give a notion for a generalized solution of Cauchy-Goursat problem PK  , and in Section , 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 PK  . Actually, this is the essential result in this paper and has the most difficult proof. Using the results of the previous section, in Section  we prove the main results in this paper, i.e., Theorem ., Theorem . and Theorem .. In Appendix A we give the main properties of the Riemann-Hadamard function associated to the Cauchy-Goursat problem PK  . In Appendix B some auxiliary results, needed for the study of the generalized solution to problem PK  , are proven.

Two-dimensional Cauchy-Goursat problems corresponding to problem PK
problem PK can suitably be treated. Written in the new coordinates, equation (.) becomes We consider equation (.) in the region bounded by the following surfaces: Problem PK becomes the following one: find a solution to equation (.) with the boundary conditions The two-dimensional spherical functions, expressed in terms of θ and ϕ in the traditional definition (see []), are Y s n (θ , ϕ) := Y s n (x), x ∈ S  , n ∈ N ∪ {}, s = , , . . . , n + , and satisfy the differential equation In the special case when the right-hand side function f (x, t) of equation (.) has the form we may look for a solution of the form with unknown coefficient u s n (r, t). For the coefficients u s n (r, t) which correspond to the right-hand sides f s n (r, t), we obtain the -D equation which is bounded by the segment S  = {(r, t) :  < r < , t = } and the characteristics In this case, for u(r, t), the -D problem corresponding to PK is the problem The generalized solution of problem PK  is defined as follows.
Definition . We call a function u(r, t) a generalized solution of problem PK  in G m ( < m <   ) if: Substituting the new characteristic coordinates and the new functions from problem PK  , we get the -D Cauchy-Goursat problem PK  : The generalized solution of problem PK  is defined as follows.

Existence and uniqueness of a generalized solution to the Cauchy-Goursat plane problem PK 2
In this section we prove the existence and uniqueness of a generalized solution to problem PK  . In order to do this, we use the Riemann-Hadamard function associated to problem PK  to find an integral representation for a generalized solution of this problem in D. According to Gellerstedt [] and the results of Nakhushev mentioned in the book of Smirnov [], this function has the form The Riemann-Hadamard function (ξ , η; ξ  , η  ) should have the following main properties (see [, ]): (i) The function as a function of (ξ  , η  ) satisfies and with respect to the first pair of variables (ξ , η) (vi)vanishes on the line {η = ξ } of power β. Actually, the function + is the Riemann function for equation (.).
Theorem . Let  < β <  and F ∈ C(D). Then each generalized solution of problem PK  in D has the following integral representation: Proof Let U(ξ , η) be a generalized solution of problem PK  in D. For any arbitrary function ψ(ξ , η) from C ∞  (D), we have ψ ∈ V () , and from (.) we obtain the identity where U ξη is the weak derivative of U. Therefore . From this it follows that U ξη is a classical derivative of U and U(ξ , η) satisfies the differential equation (.) in D in a classical sense. Now, using the properties of the Riemann-Hadamard function (ξ , η; ξ  , η  ), we obtain the integral representation (.) for a generalized solution of problem PK  by integrating the identity and then over the rectangle δ := (ξ , η) :  < ξ < ξ  -δ, ξ  + δ < η < η  with δ >  small enough, and finally letting δ → .
Theorem . claims the uniqueness of a generalized solution to problem PK  . Next, we prove that if F ∈ C  (D) and U(ξ , η) is a function defined by (.) in D, then U(ξ , η) is a generalized solution to problem PK  in D. In order to do this, we introduce the notation and we mention that, according to Lemma A. (see Appendix A below), the Riemann-Hadamard function (ξ , η; ξ  , η  ) can be decomposed in the following way: where H(ξ , η; ξ  , η  ) is the Riemann-Hadamard function (A.) associated to problem PK  in the case n =  and G(ξ , η; ξ  , η  ) is an additional term. Therefore we can rewrite representation (.) in the form Firstly, we will study the function U H (ξ  , η  ). To do this, we use the estimates for some integrals involving function H(ξ , η; ξ  , η  ) obtained in Appendix B.

Proof Step . From (.) and (B.) from Lemma B. (see Appendix B) we obtain
Step . Differentiating (.) with respect to η  and using (B.) from Lemma B., we obtain Step .
We definẽ

from (A.) we obtaiñ
Then we have Now the inverse transform of (.) gives )), and the following estimates hold inD \ (, ):

Theorem . Let the conditions in Theorem . be fulfilled. Then for the function
Proof Using estimates (A.) and (A.), from (.) we obtain estimate (.): Here where we used that Y =  on the line η = η  . Analogously, applying estimates (A.) and (A.), which are even better than (A.) and (A.), to the last integral for the derivative G η  , we obtain estimate (.).
As a direct consequence of Theorem . and Theorem ., in view of U = U H + U G , we have the following theorem.
Theorem . Let  < β <  and F ∈ C  (D). Then there exists one and only one generalized solution to problem PK  in D, which has integral representation (.), and it satisfies estimates (.).
To prove that U(ξ , η) satisfies identity (.) in Definition ., we need several steps as follows.

y) is the hypergeometric function (A.) of two variables (see Appendix A).
In [] the case  < β < / is considered, but here we find that in a more general case  < β <  the function R  (ξ , η; ξ  , η  ) solves where (ξ  , η  ) ∈ D and H(ξ , η; ξ  , η  ) is function (A.). Using (.), integration by parts and ii) Differentiating (.) we obtain that U H satisfies the differential equation where all derivatives are in a classical sense and they are continuous in D.
(.iii) Since H(ξ , η; ξ  , η  ) satisfies the differential equation (.) with n =  and = H +G satisfies (.) with n ≥  for the difference G = -H, we obtain where all derivatives are in a classical sense and they are continuous in D.
(.i) Let V (ξ , η) ∈ V () and in addition V (ξ , η) ≡  in a neighborhood of {η = ξ } and in a neighborhood of {η = }. From Step  we know that U(ξ , η) satisfies the differential equation (.), where all derivatives are in a classical sense, continuous in D. Let us consider Now we integrate by parts in I V in the following way: -in the term U ξ V η , we move the derivative from V η to U ξ and obtain the term (U ξ ) η V : -in the term U η V ξ , we move the derivative from U η to V ξ and obtain the term U(V ξ ) η : There are not integrals on the boundary of D because U(, η) = , V (ξ , η) ≡  in a neighborhood of {η = ξ } and in a neighborhood of {η = }.
we move the derivatives from (V η ) ξ to U and obtain the term Again there are not integrals on the boundary of D, and putting (.) and (.) into (.), we get identity (.) holds. Therefore we have Obviously, I ,kl → I V , as k, l → ∞.
We know that V ≡  in a neighborhood of (, ) and supp (l[ηξ ]) is contained in { ≤ l[ηξ ] ≤ }. Using estimate (.) we find that on supp (l[ηξ ]) the functions 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  < β <  (see (.)). Thus, according to the Lebesgue dominated convergence theorem, I ,kl →  as k, l → ∞.

Proof of the main results
In this section we give the proofs of Theorem ., Theorem . and Theorem . formulated in Section .
Proof of Theorem . Let and u  and u  be two generalized solutions of problem PK in m . Then the function u := u u  solves the homogeneous problem PK . We will show that the Fourier expansion has zero Fourier-coefficients holds for all test functions v(x, t) = w(r, t)Y s n (x) described in Remark .. Therefore from (.) we derive G m u s n,r w rt m u s n,t w t + n(n + ) r  u s n w r  dr dt =  (.) for all w(r, t) ∈ V () m (see Definition .), n ∈ N ∪ {}, s = , , . . . , n + . Since u(x, t) satisfies conditions (), () and () in Definition ., the functions u s n (r, t) satisfy conditions (), () and () in Definition ., and therefore they are generalized solutions of problem PK  .
Using (.) we see that the functions V (ξ , η) := r(ξ , η)w(r(ξ , η), t(ξ , η)) ∈ V () . Now from (.) we obtain that for the functions U s n (ξ , η) := r(ξ , η)u s n (r(ξ , η), t(ξ , η)) the identity To find such a solution means to find functions u s n (r, t) that satisfy the identities G m,ε u s n,r v rt m u s n,t v t + n(n + ) r  u s n v + f s n v r  dr dt =  for all v ∈ V () m and satisfy the corresponding conditions (), () and () in Definition .. In view of (.) to find such functions means to find functions U s n (ξ , η) = r(ξ , η)u s n r(ξ , η), t(ξ , η) , such that for F s n (ξ , η) :=   r(ξ , η)f s n (r(ξ , η), t(ξ , η)) the identity , η), t(ξ , η)) ∈ V () and satisfies the corresponding conditions (), () and () in Definition .. Theorem . gives the existence of such functions U s n (ξ , η) which are generalized solutions of problem PK  in D. In that way we found functions u r n (r, t) = r - U s n (ξ (r, t), η(r, t)) which are generalized solutions of problem PK  in G m . Therefore the function u(x, t), given by (.), is a generalized solution of problem PK in m .
Here F  (a  , a  , b  , b  , c; x, y) is the Appell series which converges absolutely for x, y ∈ C with |x| < , |y| <  (see [], pp.-) and H  (a  , a  , b  , b  , c; x, y) is the Horn series which converges absolutely for x, y ∈ C with |x| < , |y|( + |x|) <  (see [], pp.-). We mention that for (ξ  , η  ) ∈ D we have |X| <  in¯ and /|X| <  inT, while |Y | <  in but |Y | could be greater than  in T. However, the function is well defined because n ∈ N ∪ {}, since b  = -n, and we have a finite sum with respect to i in the function H  (ii) In view of (A.) we have