Generalized Euler–Poisson–Darboux equation and singular Klein–Gordon equation

Solutions of the Cauchy problem for generalized Euler–Poisson–Darboux equation were obtained by Hankel transform method.


Introduction
The Euler-Poisson-Darboux equation is the one of the most extensively studied singular linear hyperbolic equation. This equation is closely related to the spherical mean transform, which integrates functions over spheres of arbitrary radius and centers located on a sphere and to the Riesz potential theory (see [1], [2]). In our article we study generalized Euler-Poisson-Darboux equation with both timelike and spacelike singularities and with a function multiplied by a constant on the right-hand side. In the particular case, such an equation is the singular Klein-Gordon equation and must be interesting for relativists ([3], [4]). We treat the problem with sufficient generality to develop generalized and regular solutions.
We will call (1) the generalized Euler-Poisson-Darboux equation. We obtain the distributional solution of (1)- (2) in convenient space. Besides, we give formulas for regular solution of (1)- (2) in particular case of k and of Cauchy the the singular Klein-Gordon equation. Equation appears in Leonard Euler's work (see [5] p. 227) and later was studied by Siméon Denis Poisson in [6], by Gaston Darboux in [7] and by Bernhard Riemann in [8]. Alexander Weinstein (see [9]) and then his students from "University of Maryland College Park" made significant progress in studies of (1) when γ i = 0, i = 1, ..., n and c = 0 in the classical sense. David Fox (one of the students of Weinstein) obtained the solution of (1)- (2) when c = 0 and k = −1, −3, −5, ... in terms of the Lauricella function (see [10]). Carrol

Definitions and propositions
We deal with the subset of the Euclidean space x 2 i and Ω be finite or infinite open set in R n symmetric with respect to The function f is said to be of class C ∞ (Ω + ) if it has derivatives of all orders on Ω + . Function f ∈ C ∞ (Ω + ) we will call even with respect to = 0 for all nonnegative integer k (see [19], p. 21). Class C ∞ ev (Ω + ) consists of functions from C ∞ (Ω + ) even with respect to each variable x i , i = 1, ..., n. Let • C ∞ ev (Ω + ) be the space of all functions f ∈C ∞ ev (Ω + ) with a compact support. Elements of • C ∞ ev (Ω + ) we will call test functions and use the notation • C ∞ ev (Ω + )=D + (Ω + ). We will use notation D + = D + (R n + ). As the space of basic functions we will use the subspace of the space of rapidly decreasing functions: where α = (α 1 , ..., α n ), β = (β 1 , ..., β n ), α 1 , ..., α n , β 1 , ..., β n are integer nonnegative numbers, We deal with multi-index γ=(γ 1 , . . ., γ n ) consists of positive fixed reals γ i > 0, i=1, ..., n, |γ|=γ 1 +. . .+γ n . Let L γ p (Ω + ), 1≤p<∞, be the space of all measurable in Ω + functions even with respect to each variable x i , i = 1, ..., n such that where and further For a real number p ≥ 1, the L γ p (Ω + )-norm of f is defined by Weighted measure of Ω + is denoted by mes γ (Ω) and is defined by formula For every measurable function f (x) defined on R n + we consider We will call the function µ γ = µ γ (f, t) a weighted distribution function |f (x)|. A space L γ ∞ (Ω + ) is defined as a set of measurable on Ω + and even with respect to each variable functions f (x) such as For 1 ≤ p ≤ ∞ the L γ p,loc (Ω + ) is the set of functions u(x) defined almost everywhere in Ω + such that uf ∈ L γ p (Ω + ) for any f ∈ • C ∞ ev (Ω + ). Each function u(x) ∈ L γ 1,loc (Ω + ) will be identified with the functional u ∈ D + (Ω + ) acting according to the formula Functionals u ∈ D + (Ω + ) acting by the formula 3 will be called regular weighted functionals. All other linear functionals u ∈ D + (Ω + ) will be called singular weighted functionals.
3. Singular Bessel differential operator, Laplace-Bessel operator, Bessel functions and Hankel transform We will deal with the singular Bessel differential operator B ν (see, for example, [19], p. 5): and the elliptical singular operator or the Laplace-Bessel operator γ : The operator 4 belongs to the class of B-elliptic operators by I. A. Kipriyanovs' classification (see [19]). The symbol j ν is used for the normalized Bessel function: where J ν (t) is the Bessel function of the first kind of order ν (see [20]): Using formulas 9.1.27 from [21] we obtain that the function j ν (t) is an eigenfunction of a linear operator B ν : We also will need some other Bessel functions (see [20]). Bessel functions of the second kind Y α for non-integer α is related to J α by: In the case of integer order n, the function Y n is defined by taking the limit as a non-integer α tends to n, Hankel functions of the first and second kind H (1) α (x) and H (2) α (x), defined by: and Modified Bessel functions of the first and second kind I α (x) and K α (x) are defined by: when α is not an integer and when α is an integer, then the limit is used. The multidimensional Hankel (Fourier-Bessel) transform of a function f (x) is given by (see [22]): For f ∈ S ev inverse multidimensional Hankel transform is defined by We will use the generalized convolution operator defined by the formula It is easy to see that Proof. We have Integrating by parts by variable x i and using formula (6), we obtain This completes the proof. Lemma 2. The integral Proof. It is well known that the function j γ (x, ξ) is related to e −i x,ξ by the formula (see [24], p. 32): j γ (rθ, ξ) = P γ ξ e −ir θ,ξ therefore, one can use formula which is proved in [25]. We obtain Replacing p by −p, we get It is well known that the above-obtained integral satisfies the next formula (e.g., see [26], formula That gives (14).
To the function P λ we compare the generalized function P λ γ = (P 1 + iP 2 ) λ γ by formula Let us consider the non-degenerate quadratic form with real coefficients Suppose that the quadratic form P has p positive summands and q negative, p + q = n. Let where P is positive definite quadratic form with real coefficients. Without loss of generality, we assume P = ε(x 2 1 + ... + x 2 n ), ε > 0.
If P λ γ = (P + iP ) λ γ then (P + i0) λ γ and (P − i0) λ γ when Re λ > 0 are defined by formulas in which we pass to the limit under the integral sign and then we pass to the limit as ε → 0 (see [27]). In addition, we will consider similarly defined functions (c 2 + P + i0) λ γ and (c 2 + P − i0) λ γ , where P is an arbitrary quadratic form and c ∈ R does not depends on x.
The residue of at λ = −m, m ∈ N is (see [27], [31]) This completes the proof. Lemma 4. We have the following formulas (22) and α and H (2) α are the Hankel functions of the first and second kind and K α is modified Bessel function.
Using formula 2.12.4.28 from [28] ∞ 0 where λ < 1−n−|γ| 4 . For other values of λ the Hankel transform F γ (w 2 + P ) λ γ remains valid by analytic continuation in λ. Now let P be any real quadratic form. We wish to consider the generalized functions (w 2 + P + i0) λ γ and (w 2 + P − i0) λ γ defined by where ε > 0 and P 1 is a positive definite quadratic form. According to the uniqueness of analytic continuation, (24) implies that a i x 2 i . Considering the definitions of modified Bessel functions of the first and second kind (9) and (10) we get Weighted generalized functions (Q + i0) λ γ and (Q − i0) λ γ can be expressed through weighted generalized functions Q λ γ,+ and Q λ γ,− defined by (see [27]): Then we get γ,+ ) λ+ n+|γ| Noticing that we have which, taking into account formula (25), gives (22) and (23) Corollary. If P is positive definite then we have and while if P is negative definite and α and H (2) α are the Hankel functions of the first and second kind and K α is modified Bessel function.

Solution of the Cauchy problem using the Hankel transform
The main tool used here consist of using the Hankel transform for solving Cauchy problem for generalized Euler-Poisson-Darboux equation. The greatest advantage of this method is that it gives solutions for all values of k. The case when γ i = 0, i = 1, ..., n was studied in [13]. We will be concerned with the solutions of the following initial value hyperbolic problem We will call (31) the generalized Euler-Poisson-Darboux equation. We are looking for the solution u ∈ S ev (R n + ) × C 2 (0, ∞) of (31)- (32). Notation u ∈ S ev (R n + ) × C 2 (0, ∞) means that u(x, t; k) belongs to S ev (R n + ) by variable x and belongs to C 2 (0, ∞) by variable t.