NEW SINGULAR SOLUTIONS OF PROTTER ’ S PROBLEM FOR THE 3 D WAVE EQUATION

In 1952, for the wave equation, Protter formulated some boundary value problems (BVPs), which are multidimensional analogues of Darboux problems on the plane. He studied these problems in a 3D domain Ω0, bounded by two characteristic cones Σ1 and Σ2,0 and a plane region Σ0. What is the situation around these BVPs now after 50 years? It is well known that, for the infinite number of smooth functions in the right-hand side of the equation, these problems do not have classical solutions. Popivanov and Schneider (1995) discovered the reason of this fact for the cases of Dirichlet’s or Neumann’s conditions on Σ0. In the present paper, we consider the case of third BVP on Σ0 and obtain the existence of many singular solutions for the wave equation. Especially, for Protter’s problems in R3, it is shown here that for any n∈N there exists a C(Ω̄0) right-hand side function, for which the corresponding unique generalized solution belongs to Cn(Ω̄0\O), but has a strong power-type singularity of order n at the point O. This singularity is isolated only at the vertex O of the characteristic cone Σ2,0 and does not propagate along the cone.


Introduction
In 1952, at a conference of the American Mathematical Society in New York, Protter introduced some boundary value problems (BVPs) for the 3D wave equation in a domain Ω 0 ⊂ R 3 .These problems are three-dimensional analogous of the Darboux problems (or Cauchy-Goursat problems) on the plane.The simply connected domain 316 New singular solutions of Protter's problem is bounded by the disk Σ 0 := x 1 ,x 2 ,t : t = 0, x 2 1 + x 2 2 < 1 , (1.3) centered at the origin O(0,0,0) and by the two characteristic cones of (1.1) (1.4) Similar to the plane problems, Protter formulated and studied [24] some 3D problems with data on the noncharacteristic disk Σ 0 and on one of the cones Σ 1 and Σ 2,0 .These problems are known now as Portter's problems, defined as follows.
Substituting the boundary condition on Σ 0 by the third-type condition [u t + αu]| Σ0 = 0, we arrive at the following problems.
The boundary conditions of problem (P1 * ) (resp., of (P2 * )) are the adjoined boundary conditions to such ones of (P1) (resp., of (P2)) for the wave equation (1.1) in Ω 0 .Note that Garabedian in [10] proved the uniqueness of a classical solution of problem (P1).For recent results concerning Protter's problems (P1) and (P1 * ), we refer to [23] and the references therein.For further publications in this area, see [1,2,8,14,17,18,19,21].For problems (P α ), we refer to [11] and the references therein.In the case of the hyperbolic equation with the wave operator in the main part, which involves either lower-order terms or other type perturbations, problem (P α ) in Ω 0 has been studied by Aldashev in [1,2,3] and by Grammatikopoulos et al. [12].On the other hand, Ar.B. Bazarbekov and Ak.B. Bazarbekov [5] give another analogue of the classical Darboux problem in the same domain Ω 0 .Some other statements of Darboux-type problems can be found in [4,6,16] in bounded or unbounded domains different from Ω 0 .
It is well known that, in contrast to the Darboux problem on the plane, the 3D problems (P1) and (P2) are not well posed.It is due to the fact that their adjoint homogeneous problems (P1 * ) and (P2 * ) have smooth solutions, whose span is infinite-dimensional (see, e.g., Tong [26], Popivanov and Schneider [22], and Khe [18]).M. K. Grammatikopoulos et al. 317 Now we formulate the following useful lemma, the proof of which is given in Section 2.
A necessary condition for the existence of a classical solution for problem (P2) is the orthogonality of the right-hand side function f to all solutions W n,i k of the homogeneous adjoined problem.In order to avoid an infinite number of necessary conditions in the frame of classical solvability, Popivanov and Schneider in [22,23] gave definitions of a generalized solution of problem (P2) with an eventual singularity on the characteristic cone Σ 2,0 , or only at its vertex O. On the other hand, Popivanov and Schneider [23] and Grammatikopoulos et al. [11] proved that for the right-hand side f = W n,i 0 the corresponding unique generalized solution of problem (P α ) behaves like (x 2 1 + x 2 2 + t 2 ) −n/2 around the origin O (for more comments about this subject, we refer to Remarks 1.4 and 1.6).Now we know some solutions, W n,i k , of the homogeneous adjoined problem (P2 * ), and if we take one of these solutions in the right-hand side of (1.1), then we have to expect that the generalized solution of problem (P α ) will also be singular, possibly with a different power type of singularity.An analogous result, in the case of problem (P1) and functions V n,i k , has been proved by Popivanov and Popov in [21].Having this in mind, here we are looking for some new singular solutions of problem (P α ), which are different from those found in [11].
In the case of problem (P α ) with α(x) = 0, there are only few publications, while for problem (P α ), concerning the wave equation (1.1), see the results of [11].Moreover, some results of this type can also be found in Section 3.
For the homogeneous problem (P * α ) even for the wave equation (except the case α ≡ 0, i.e., except problem (P2 * )), we do not know nontrivial solutions analogous to (1.7) and (1.8).In Section 2, we give an approach for finding nontrivial solutions.Relatively, we refer to Khe [18], who found nontrivial solutions for the homogeneous problems (P1 * ) and (P2 * ), but in the case of the Euler-Poisson-Darboux equation.These results are closely connected to such ones of Lemma 1.1.
In order to obtain our results, we formulate the following definition of a generalized solution of problem (P α ) with a possible singularity at O.
holds for all v in Existence and uniqueness results for a generalized solution of problems (P1) and (P2) can be found in [23], while for problem (P α ), see [11].
In order to deal successfully with the encountered difficulties, as are singularities of generalized solutions on the cone Σ 2,0 , we introduce the region 1), (1.11) which in polar coordinates becomes Note that a generalized solution u, which belongs to C 1 ( Ωε )∩C 2 (Ω ε ) and satisfies the wave equation (1.1) in Ω ε , is called a classical solution of problem (P α ) in Ω ε , ε ∈ (0,1).It should be pointed out that the case ε = 0 is totally different from the case ε = 0.This paper is an extension of some results obtained in [11,12] and, besides the introduction, involves two more sections.In Section 2, we formulate the 2D BVPs corresponding to the 3D Protter's problems.Using Riemann functions, we show the way for finding nontrivial solutions.For the same goal, we consider functions orthogonal to the Legendre one and formulate some open questions for finding more functions of this type in the frame of nontrivial solutions of problems (P1 * ), (P2 * ), and (P * α ).Also, using the results of Sections 1 and 2, in Section 3, we study the existence of a singular generalized solution of 3D problem (P α ).To investigate the behavior of such singular solutions, we need some information about them.In Theorem 3.1, we state a maximum principle for the singular generalized solution of 2D problem (P α,2 ), corresponding to problem (P α ) in Ω 0 .This solution is a classical one in each domain Ω ε , ε ∈ (0,1).Note that this maximum principle can be applied even in the cases where the right-hand side changes its sign in the domain.(Theorem 1.3 deals exactly with this special situation.)Other maximum principles can be found in [6,25].Using information of this kind, we present singular generalized solutions which are smooth enough away from the point O, while at the point O, they have power-type singularity.More precisely, in Section 3, we prove the following theorem.1] and let α(ρ) ≥ 0 be an arbitrary function.Then, for each n ∈ N, n ≥ 4, there exists a function f n ∈ C n−3 ( Ω0 ) ∩ C ∞ (Ω 0 ), for which the corresponding unique generalized solution u n of problem (P α ) belongs to C n−1 ( Ω0 \O) and satisfies the estimates where the constant n 1 ∈ (0,1) depends only on n.
Remark 1.4.For the right-hand side of the wave equation equals W n,2 0 , the exact behavior of the corresponding singular solution [11,12]), while for the right-hand side equals W n,2 1 = ∂ 2 /∂t 2 {W n,2 0 }, the singularities are at least of type (x 2 1 + x 2 2 + t 2 ) −(n−2)/2 cos n(arctanx 2 /x 1 ) (see Theorem 1.3 ).The following open question arises: is this the exact type of singularity or not?If the last case is true, it would be possible, using an appropriate linear combination of both right-hand sides, to find a solution of the last lower-type singularity.Then the result of this kind could give an answer to Open Question (1).
Remark 1.5.It is interesting that for any parameter α(x) ≥ 0, involved in the boundary condition (P α ) on Σ 0 , there are infinitely many singular solutions of the wave equation.Note that all these solutions have strong singularities at the vertex O of the cone Σ 2,0 .These singularities of generalized solutions do not propagate in the direction of the bicharacteristics on the characteristic cone.It is traditionally assumed that the wave equation with right-hand side sufficiently smooth in Ω0 cannot have a solution with an isolated singular point.For results concerning the propagation of singularities for secondorder operators, see Hörmander [13,Chapter 24.5].For some related results in the case of the plane Darboux problem, see [20].
Remark 1.6.Considering problems (P1) and (P2), Popivanov and Schneider [22] announced the existence of singular solutions for both wave and degenerate hyperbolic equations.First a priori estimates for singular solutions of Protter's problems (P1) and (P2), concerning the wave equation in R 3 , were obtained in [23].In [1], Aldashev mentioned the results of [22] and, for the case of the wave equation in R m+1 , showed that there exist solutions of problem (P1) (resp., (P2)) in the domain Ω ε , which grow up on the cone Σ 2,ε like ε −(n+m−2) (resp., ε −(n+m−1) ), and the cone Σ 2,ε := {ρ = t + ε} approximates Σ 2,0 when ε → 0. It is obvious that, for m = 2, these results can be compared to the estimates of [11].Finally, we point out that in the case of an equation which involves the wave operator and nonzero lower-order terms, Karatoprakliev [15] obtained a priori estimates, but only for the enough smooth solutions of problem (P1) in Ω 0 .
We fix the right-hand side as a trigonometric polynomial of the order l: We already know that the corresponding solution u(x 1 ,x 2 ,t) may have behavior of type (x 2  1 + x 2 2 + t 2 ) −l/2 at the point O.We conclude this section with the following questions.Open Questions.(1) Find the exact behavior of all singular solutions at the point O, which differ from those of Theorem 1.3.In other words, (i) are there generalized solutions for the right-hand side (1.14) with a higher order of singularity, for example, of the form (x 2 1 + x 2 2 + t 2 ) −k−l/2 , k > 0? (ii) are there generalized solutions for the right-hand side (1.14) with a lower order of singularity, for example, of the form (x 2 1 + x 2 2 + t 2 ) k−l/2 , k > 0? (2) Find appropriate conditions for the function f under which problem (P α ) has only classical solutions.We do not know any kind of such results even for problem (P2).
(3) From the a priori estimates, obtained in [11], for all solutions of problem (P α ), including singular ones, it follows that, as ρ → 0, none of these solutions can grow up faster than the exponential one.The arising question is: are there singular solutions of problem (P α ) with exponential growth as ρ → 0 or any such solution is of polynomial growth less than or equal to (x 2 1 + x 2 2 + t 2 ) −l/2 ?(4) Why there appear singularities for smooth right-hand side, even for the wave equation?Can we explain this phenomenon numerically?
2. Nontrivial solutions for the homogeneous problems (P1 * ), (P2 * ), and (P * α ) Suppose that the right-hand side f of the wave equation is of the form Then we are seeking solutions of the wave equation of the same form and due to this fact, the wave equation reduces to M. K. Grammatikopoulos et al. 321 Now introduce the new coordinates x = (ρ + t)/2, y = (ρ − t)/2 and set Then, denoting ν = n − (1/2), problems (P1 * ), (P2 * ), and (P * α ) transform into the following problems.
Problems (P31), (P32), and (P3 α ).Find a solution v(x, y) of the equation in the domain D = {0 < x < 1/2; 0 < y < x} with the following corresponding boundary conditions: A basic tool for our treatment of problems (P3) is the Legendre functions P ν (for more information, see [9]).Note that the function is a Riemann one for (2.5) (see Copson [7]), that is, with respect to the variables (x 1 , y 1 ), it is a solution of (2.5) with g = 0, and R x, y 1 ;x, y = 1, R x 1 , y;x, y = 1. (2.7) Therefore, we can construct the function u(x, y) in the following way.Integrating (2.5) over the characteristic triangle with vertices M(x, y) ∈ D, P(y, y), and Q(x,x), and using the properties (2.7) of the Riemann function, we see that (2.9) 322 New singular solutions of Protter's problem In the case of g = 0, we obtain v(x, y) = v(y, y) (2.10) Using the condition v(x,0) = 0, finally we find that if we suppose, in addition, that limt −1 v(t,t) = 0, t → +0.Thus, Suppose that there exist two functions ψ and ψ 1 such that Then we are looking for a solution ψ(t) of the equation Now we are ready to formulate the following useful lemma.
Lemma 2.1.The following identity holds: (2.15) Proof.As known, the Legendre functions P ν (t) are solutions of the Legendre differential equation and so the proof of the lemma follows by induction.
(2) Using the way described above, find nontrivial solutions of problem (P * α ), when α(x) is a nonzero function.
The representation (2.10), together with (2.21) and (2.23), gives us exact formulae for the solution of the homogeneous problems (P1 * ) and (P2 * ).Using Lemma 1.1, we obtain a different representation of the same solutions.The solutions V n,i 0 and W n,i 0 were found by Popivanov and Schneider, while the functions H n k and E n k can be found in [18] with a different presentation, where they are defined by using the Gauss hypergeometric function.
The following result implies Lemma 1.1.

M. K. Grammatikopoulos et al. 325
Proof of Lemma 1.1.We already know (see [23]) that V n,i 0 and W n,i 0 (i = 1,2) are solutions of the wave equation (1.1).Using formulae (2.26) and (2.27), we conclude that V n,i k and W n,i k are also solutions of the wave equation.Thus, the functions ρ 1/2 H n k (t,ρ) and ρ 1/2 E n k (t,ρ) are solutions of the 2D equation (2.5).It is easy to see directly that (2.29) These Cauchy conditions on {x = y} (i.e., on {t = 0}) coincide with the conditions of (2.23) for p = n − 2k − 1/2 with the accuracy of a multiplicative constant.Moreover, because of the uniqueness of the solution of Cauchy problem for (2.5), the function v(x, y) defined by (2.10), together with the conditions of (2.23) for p = n − 2k − 1/2, coincides with the function (

New singular solutions of problem (P α )
We are seeking a generalized solution of BVP (P α ) for the wave equation which has some power type of singularity at the origin O.While in [11,23] the function W n,i 0 (ρ,t,ϕ) has been used systematically as the right-hand side function, we will try to use here, for the same reason, the function W n,i 1 (ρ,t,ϕ).Due to the fact that the function E n 1 (ρ,t) changes its sign inside the domain, the appearing situation causes some complications.Note first that, by Lemma 1.1, the functions with W n,2 1 ∈ C n−3 ( Ω0 ), are classical solutions of problem (P * α ) when α ≡ 0. To prove Theorem 1.3, consider now the special case of problem (P α ): ) Theorem 5.1 of [11] declares that problem (3.3), (3.4) has at most one generalized solution.On the other hand, by [11,Theorem 5.2], we know that for this right-hand side there exists a generalized solution in Ω 0 of the form which is a classical solution in Ω ε , ε ∈ (0,1).By introducing a new function u (2) (ρ,t) = ρ 1/2 u (1) (ρ,t), (3.6) 326 New singular solutions of Protter's problem we transform (3.3) into the equation u (2)  ρρ − u (2)   tt − with the string operator in the main part.The domain, corresponding to Ω ε in this case, is In order to use directly the results of [11], we introduce the new coordinates and transform the singular point O into the point (1,1).From (3.7), we derive that In order to investigate the smoothness or the singularity of a solution for the original 3D problem (P α ) on Σ 2,0 , we are seeking a classical solution of the corresponding 2D problem (P α,2 ), not only in the domain D ε but also in the domain Clearly, D ε ⊂ D (1)  ε .Thus, we arrive at the Goursat-Darboux problem.
Note that, according to [11], solving problem (P α,2 ) is equivalent to solving the following integral equation: ε . ( For this reason, we define (see [11]) the following sequence of successive approximations U (m) : ( In [11], the uniform convergence of U (m) in each domain D (1)  ε , ε > 0, has been proved.To use this fact here, we now formulate the following maximum principle, which is very important for the investigation of the singularity of a generalized solution of problem (P α ).
Remark 3.4.In our opinion, the analogous result follows for all functions E n k (ρ,t), k = 2,3,...,[n/2] −1.As before, for k > 0, the function E n k (ρ,t) changes its sign in the domain, but due to the monotonicity of the solution U(ξ,η), the desired result would follow.Also, by using the more general formula this result could be obtained for k > 1 too.Now we are ready to prove Theorem 1.3 formulated in the introduction.
.22) By Lemma 2.1, possible solutions of the above equation are the functions v(x,x) = x p , v x (x,x) =