GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 4, 1995, 385-394

ON THE SOLVABILITY OF A SPATIAL PROBLEM OF DARBOUX TYPE FOR THE WAVE EQUATION

S. KHARIBEGASHVILI

Abstract. The question of the correct formulation of one spatial problem of Darboux type for the wave equation has been investigated. The correct formulation of that problem in the Sobolev space has been proved for surfaces having a quite definite orientation on which are given the boundary value conditions of the problem of Darboux type.

In the space of variables x_{1}, x_{2}, t we consider the wave equation

□u ≡ \frac{∂^{2}u}{∂t^{2}} − \frac{∂^{2}u}{∂x_{1}^{2}} − \frac{∂^{2}u}{∂x_{2}^{2}} = F,  (1)

where F is the known and u is the unknown function.

Denote by D_{+} : 0 < x_{2} < t, 0 < t < t_{0}, the domain lying in a half-space t > 0 bounded by a time-type plane surface S_{0} : x_{2} = 0, 0 ≤ t ≤ t_{0}, a characteristic surface S_{1} : t − x_{2} = 0, 0 ≤ t ≤ t_{0}, of equation (1), and a plane t = t_{0}.

Consider the problem of Darboux type formulated as follows: find in the domain D_{+} the solution u(x_{1}, x_{2}, t) of equation (1) by the following boundary conditions:

u|_{S_{1}} = f_{1}  (2)

and

\frac{∂u}{∂n}|_{S_{0}} = 0,  (3)

where f_{1} is a given real function and \frac{∂}{∂n} is the derivative with respect to the outer normal to S_{0}.

Note that in the case where S_{0} is either a characteristic surface S_{2} : t + x_{2} = 0, 0 ≤ t ≤ t_{0} or a plane surface S_{2} : kt + x_{2} = 0, 0 ≤ t ≤ t_{0},

385 |k| < 1 of timetype, the problem (1)-(3) in which the boundary condition (3) is replaced by the condition

1991 Mathematics Subject Classification. 35L20.

Key words and phrases. Characteristic, spatial problem of Darboux type, wave equation, a priori estimate, Riemann function, Bessel function.

1072-947X/95/0700-0385$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

u|_{S_{2}} = f_{2}  (4)

is studied in [1-4]. Other multidimensional analogues of the Darboux problem are considered in [5-7].

Denote by C_{∗}^{∞}(\overline{D}_{+}) the space of functions in the class C^{∞}(\overline{D}_{+}) having bounded supports, i.e.,

C_{∗}^{∞}(\overline{D}_{+}) = \{u ∈ C^{∞}(\overline{D}_{+}) : diam supp u < +∞\}.

The spaces C_{∗}^{∞}(S_{i}), i = 0, 1, 2, are defined in a similar manner.

It is known that the spaces C_{∗}^{∞}(\overline{D}_{+}), C_{∗}^{∞}(S_{i}), i = 0, 1, 2, are dense everywhere in the Sobolev spaces W_{2}^{k}(D_{+}), W_{2}^{k}(S_{i}), i = 0, 1, 2, where k ≥ 0 is integer [8].

Lemma 1. For any u ∈ W_{2}^{2}(D_{+}), satisfying the homogeneous boundary condition (3), the a priori estimate

∥u∥_{W_{2}^{1}(D_{+})} ≤ C(∥f_{1}∥_{W_{2}^{1}(S_{1})} + ∥F ∥_{L_{2}(D_{+})})  (5)

is valid, where f_{1} = u|_{S_{1}}, F = □u, and C is a positive constant not depending on u.

Proof. Denote by D_{−} : −t < x_{2} < 0, 0 < t < t_{0}, the domain symmetric to D_{+} with respect to the plane x_{2} = 0 and by D : −t < x_{2} < t, 0 < t < t_{0}, the domain which is the union of the domains D_{+} and D_{−} with a part of a plane surface x_{2} = 0, 0 < t < t_{0}.

It is easy to verify that if one continue evenly the function satisfying the boundary condition (3), then the function u_{0} obtained in D

u_{0}(x_{1}, x_{2}, t) = \begin{cases}u(x_{1}, x_{2}, t), & x_{2} ≥ 0, \\ u(x_{1}, −x_{2}, t), & x_{2} < 0 \end{cases}  (6)

will belong to the class W_{2}^{2}(D). According to the results in [3], the function u_{0} ∈ W_{2}^{2}(D) satisfies the following a priori estimate:

∥u_{0}∥_{W_{2}^{1}(D)} ≤ C(∥f_{1}∥_{W_{2}^{1}(S_{1})} + ∥f_{2}∥_{W_{2}^{1}(S_{2})} + ∥F_{0}∥_{L_{2}(D)}),  (7)

where f_{i} = u_{0}|_{S_{i}}, i = 1, 2, F_{0} = □u_{0}, and S_{2} : t + x_{2} = 0, 0 ≤ t ≤ t_{0}, is a part of a boundary of D, appearing in the boundary condition (4).

It remains only to note that in virtue of (6) in the estimate (7)

∥u_{0}∥_{W_{2}^{1}(D)} = 2∥u∥_{W_{2}^{1}(D_{+})}, ∥f_{2}∥_{W_{2}^{1}(S_{2})} = ∥f_{1}∥_{W_{2}^{1}(S_{1})},
∥F_{0}∥_{L_{2}(D)} = 2∥F ∥_{L_{2}(D_{+})}.

Below we shall prove the following

Lemma 2. For any f_{1} ∈ C_{∗}^{∞}(S_{1}) and F ∈ C_{∗}^{∞}(\overline{D}_{+}) satisfying the conditions

\frac{∂^{k}F}{∂n^{k}} |_{S_{0}} = 0, k = 1, 3, 5, …,  (8)

the problem (1)–(3) can be solved uniquely in the class C_{∗}^{∞}(\overline{D}_{+}).

If one continues evenly the function F ∈ C_{∗}^{∞}(\overline{D}_{+}) in D_{−}, then in virtue of (8) the function F_{0} obtained in D

F_{0}(x_{1}, x_{2}, t) = \begin{cases}F (x_{1}, x_{2}, t), & x_{2} ≥ 0, \\ F (x_{1}, −x_{2}, t), & x_{2} < 0 \end{cases}

will belong to the class C_{∗}^{∞}(\overline{D}). Denote by f_{2} the function defined on S_{2} : t + x_{2} = 0, 0 ≤ t ≤ t_{0}, by the equality

f_{2}|_{S_{2}} = f_{2}(x_{1}, x_{2}, −x_{2}) = f_{1}(x_{1}, −x_{2}, −x_{2}) = f_{1}|_{S_{1}}.  (9)

Consider now in D the problem of determining the solution u_{0}(x_{1}, x_{2}, t) of the equation

□u_{0} ≡ \frac{∂^{2}u_{0}}{∂t^{2}} − \frac{∂^{2}u_{0}}{∂x_{1}^{2}} − \frac{∂^{2}u_{0}}{∂x_{2}^{2}} = F_{0}  (10)

belonging to the class C_{∗}^{∞}(\overline{D}) by the boundary conditions

u_{0}|_{S_{i}} = f_{i}, i = 1, 2.  (11)

Note that the integral representation for regular solutions of the problem (10),(11) is obtained in [3]. On the basis of this representation the conclusion on the solvability of the problem in the class C_{∗}^{∞}(\overline{D}) is made without proof. To prove the conclusion completely, below we shall reduce the spatial problem (10),(11) to the plane Goursat problem with a parameter. For the solution of the problem, necessary estimates depending on the parameter will be obtained.

If u_{0} is a solution of the problem (10),(11) of the class C_{∗}^{∞}(\overline{D}), then after the Fourier transform with respect to the variable x_{1}, equation (10) and the boundary conditions (11) take the form

\frac{∂^{2}v}{∂t^{2}} − \frac{∂^{2}v}{∂x_{2}^{2}} + λ^{2}v = Φ,  (12)
v|_{ℓ_{i}} = g_{i}, i = 1, 2,  (13)

where

v(λ, x_{2}, t) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} u_{0}(x_{1}, x_{2}, t)e^{−ix_{1}λ} dx_{1}

is the Fourier transform of the function u_{0}(x_{1}, x_{2}, t) and Φ, g_{1}, g_{2} is the Fourier transform of the functions F_{0}, f_{1}, f_{2} with respect to x_{1}. Here ℓ_{1} : t − x_{2} = 0, 0 ≤ t ≤ t_{0}, ℓ_{2} : t + x_{2} = 0, 0 ≤ t ≤ t_{0}, are segments of rays lying in the plane of variables x_{2}, t and coming from the origin O(0, 0).

Thus after the Fourier transform with respect to x_{1} the spatial problem (10),(11) is reduced to the plane Goursat problem (12),(13) with a parameter λ in the domain D_{0} : −t < x_{2} < t, 0 < t < t_{0}, of the plane of variables x_{2}, t.

Remark 1. If u_{0}(x_{1}, x_{2}, t) is the solution of problem (10),(11) of the class C_{∗}^{∞}(\overline{D}), then v(λ, x_{2}, t) will be the solution of the problem (12),(13) of the class C^{∞}(\overline{D}_{0}) which at the same time according to Paley–Wiener theorem is an entire analytic function with respect to λ satisfying the following growth condition: for an integer N ≥ 0 there is a constant K_{N} such that [8,9]

|v(λ, x_{2}^{0}, t^{0})| ≤ K_{N}(1 + |λ|^{2})^{−N}e^{d| Im λ|},  (14)

where

d = d(x_{2}^{0}, t^{0}) = max_{(x_{1},x_{2}^{0},t^{0})∈supp u_{0}} |x_{1}|;

moreover, as the constant K_{N} one can take the value [9]

K_{N} = K_{N}(x_{2}^{0}, t^{0}) = \frac{1}{\sqrt{2π}} ∫_{|x_{1}|<d} |(1 − \frac{∂^{2}}{∂x_{1}^{2}} )^{N}u_{0}(x_{1}, x_{2}^{0}, t^{0})| dx_{1}.

According to the same theorem, if v(λ, x_{2}, t) belongs to the class C^{∞}(\overline{D}_{0}) with respect to the variables x_{2}, t for fixed λ, while with respect to λ it is an entire analytic function satisfying the estimate (14) for some d = const > 0, then the function u_{0}(x_{1}, x_{2}, t), being the inverse Fourier transform of the function v(λ, x_{2}, t), belongs to the class C_{∗}^{∞}(\overline{D}).

According to our assumptions, estimates analogous to (14) are valid for the functions Φ, g_{1}, g_{2} which belong respectively to the classes C^{∞}(\overline{D}_{0}), C^{∞}(ℓ_{1}), C^{∞}(ℓ_{2}) and are entire analytic functions with respect to λ.

In the new variables

ξ = \frac{1}{2}(t + x_{2}), η = \frac{1}{2}(t − x_{2}),  (15)

retaining the same notations for the functions v, Φ, g_{i}, the problem (12),(13) takes the form

\frac{∂^{2}v}{∂ξ∂η} + λ^{2}v = Φ,  (16)
v|_{γ_{i}} = g_{i}, i = 1, 2.  (17)

Here the solution v = v(λ, ξ, η) of equation (16) is considered in the domain Ω_{0} of the plane of variables ξ, η being the image of the domain D_{0} for linear transform (15), and γ_{i} being the image of ℓ_{i} for the same transform. Obviously, the domain Ω_{0} is a triangle OP_{1}P_{2} witha vertices O(0, 0), P_{1}(t_{0}, 0), P_{2}(0, t_{0}) and

γ_{1} : η = 0, 0 ≤ ξ ≤ t_{0}, γ_{2} : ξ = 0, 0 ≤ η ≤ t_{0}

are the sides of OP_{1}, OP_{2}.

As is well known, under the assumptions with respect to the functions Φ, g_{i} the problem (16),(17) has a unique solution of the class C^{∞}(\overline{Ω_{0}}) which can be represented in the form [10]

v(λ, ξ, η) = R(ξ, 0; ξ, η)g_{1}(λ, ξ) + R(0, η; ξ, η)g_{2}(λ, η) −

− R(0, 0; ξ, η)g_{1}(λ, 0) − ∫_{0}^{ξ} \frac{∂R(σ, 0; ξ, η)}{∂σ} g_{1}(λ, σ) dσ −

− ∫_{0}^{η} \frac{∂R(0, τ; ξ, η)}{∂τ} g_{2}(λ, τ) dτ +

+ ∫_{0}^{ξ} dσ ∫_{0}^{η} R(σ, τ; ξ, η)Φ(λ, σ, τ) dτ,  (18)

where g_{1}(λ, ξ) = v(λ, ξ, 0), 0 ≤ ξ ≤ t_{0}, g_{2}(λ, η) = v(λ, 0, η), 0 ≤ η ≤ t_{0} are the Goursat data for v and R(ξ_{1}, η_{1}; ξ, η) are the Riemann functions for equation (16).

The Riemann function R(ξ_{1}, η_{1}; ξ, η) for equation (16), as is known, can be expressed in terms of the Bessel function I_{0} of zero order as [11]

R(ξ_{1}, η_{1}; ξ, η) = I_{0}(2λ\sqrt{(ξ − ξ_{1})(η − η_{1})} ).  (19)

Remark 2. Since the Bessel function I_{0}(z) of the complex argument z is an entire analytic function, the formula (18) in virtue of the equality (19) gives the solution of equation (16) satisfying the Goursat data

\begin{matrix}v(λ, ξ, 0) = g_{1}(ξ), 0 ≤ ξ ≤ t_{0}, \\ v(λ, 0, η) = g_{2}(η), 0 ≤ η ≤ t_{0}. \end{matrix}  (20)

The solution is an entire analytic function with respect to the complex parameter λ.

From the known representation of the Bessel function [12]

I_{0}(z) = \frac{1}{2π} ∫_{−π}^{π} exp(iz sin θ) dθ  (21)

we can easily get that

I′_{0}(z) = −\frac{z}{2π} ∫_{−π}^{π} cos^{2} θ exp(iz sin θ) dθ,

whence

\frac{dI_{0}(2λ\sqrt{νx})}{dx} = −\frac{λ^{2}ν}{π} ∫_{−π}^{π} cos^{2} θ exp(i2λ\sqrt{νx} sin θ) dθ.  (22)

Now, from (19),(21) and (22) we immediately get the following equalities and estimates

R(ξ, 0; ξ, η) = R(0, η; ξ, η) = 1,
R(0, 0; ξ, η) ≤ exp(2\sqrt{ξη} | Im λ|) ≤ exp(2t_{0}| Im λ|),

|\frac{∂R(σ, 0; ξ, η)}{∂σ}| ≤ 2|λ|^{2}η exp(2\sqrt{ξη} | Im λ|) ≤ 2|λ|^{2}t_{0} exp(2t_{0}| Im λ|),

|\frac{∂R(0, τ; ξ, η)}{∂τ}| ≤ 2|λ|^{2}ξ exp(2\sqrt{ξη} | Im λ|) ≤ 2|λ|^{2}t_{0} exp(2t_{0}| Im λ|),

|R(σ, τ; ξ, η)| ≤ exp(2\sqrt{ξη} | Im λ|) ≤ exp(2t_{0}| Im λ|).

From this, without loss of generality and assuming the the estimate (14) with respect to λ and the same constants K_{N} and d are valid in virtue of our assumptions for the functions Φ, g_{1}, g_{2}, for the solution v(λ, ξ, η) of the problem (16),(17) representable in the form of (18)m we obtain the following estimates:

|v(λ, ξ, η)| ≤ |g_{1}(λ, ξ)| + |g_{2}(λ, η)| + |g_{1}(λ, 0)| exp(2t_{0}| Im λ|) +

+ 2|λ|^{2}t_{0} exp(2t_{0}| Im λ|) ∫_{0}^{ξ} |g_{1}(λ, σ)|dσ +

+ 2|λ|^{2}t_{0} exp(2t_{0}| Im λ|) ∫_{0}^{η} |g_{2}(λ, τ)|dτ +

+ exp(2t_{0}| Im λ|) ∫_{0}^{ξ} dσ ∫_{0}^{η} |Φ(λ, σ, τ)|dτ ≤

≤ 2K_{N}(1 + |λ|^{2})^{−N} exp(d| Im λ|) +

+ exp(2t_{0}| Im λ|)K_{N}(1 + |λ|^{2})^{−N} exp(d| Im λ|) +

+ 2|λ|^{2}t_{0} exp(2t_{0}| Im λ|)ξK_{N}(1 + |λ|^{2})^{−N} exp(d| Im λ|) +

+ 2|λ|^{2}t_{0} exp(2t_{0}| Im λ|)ηK_{N}(1 + |λ|^{2})^{−N} exp(d| Im λ|) +

+ exp(2t_{0}| Im λ|)ξηK_{N}(1 + |λ|^{2})^{−N} exp(d| Im λ|) ≤

≤ K̃_{N−1}(1 + |λ|^{2})^{N−1} exp(d̃| Im λ|).  (23)

Here

K̃_{N−1} = (3 + 5t_{0}^{2})K_{N}, d̃= 2t_{0} + d,

d = max_{(x_{1},x_{2},t)∈I} |x_{1}|, I = supp F_{0} ∪ supp f_{1} ∪ supp f_{2},

K_{N} = \frac{1}{2π} ∫_{|x_{1}|<d} max_{0≤i≤2} max_{(x_{2}^{0},t^{0})∈D_{0}} |φ_{i}(x_{1}, x_{2}^{0}, t^{0})| dx_{1},

φ_{0} = (1 − \frac{∂^{2}}{∂x_{1}^{2}} )^{N}F_{0}, φ_{i} = (1 − \frac{∂^{2}}{∂x_{1}^{2}} )^{N}f_{i}, i = 1, 2.

In virtue of the estimate (23), the function v(λ, ξ, η) according to the Paley–Wiener theorem, turning to the original variables x_{2}, t, by the formula (15) will be the Fourier transform of a function u_{0}(x_{1}, x_{2}, t) of the class C_{∗}^{∞}(\overline{D}); moreover, in virtue of (12),(13) the function u_{0}(x_{1}, x_{2}, t) ∈ C_{∗}^{∞}(\overline{D}) will be the solution of the problem (10),(11). Let us show now that the restriction of that function to the domain D_{+}, i.e., u = u_{0}|_{D_{+}}, is the solution of the problem (1)–(3) of the class C_{∗}^{∞}(\overline{D}_{+}). To this end let us prove that the function u_{0}(x_{1}, x_{2}, t) is even with respect to x_{2}. Because the function F_{0} is even with respect to x_{2} and the functions f_{1} and f_{2} are connected by the equality (9), we can easily verify that the function ũ(x_{1}, x_{2}, t) = u_{0}(x_{1}, −x_{2}, t) is also the solution of the problem (10),(11) of the class C_{∗}^{∞}(\overline{D}). But in virtue of a priori estimate (7) the problem (10),(11) cannot have more than one solution of the above-mentioned class. Therefore ũ(x_{1}, x_{2}, t) ≡ u_{0}(x_{1}, x_{2}, t), i.e., the solution u_{0}(x_{1}, x_{2}, t) of equation (10) is an even function with respect to x_{2}. From this it immediately follows that \frac{∂u_{0}}{∂n} |_{x_{2}=0} = 0, i.e., the boundary condition (3) is fulfilled for u = u_{0}|_{D_{+}}. Thus the function u = u_{0}|_{D_{+}} ∈ C_{∗}^{∞}(\overline{D}_{+}) is the solution of the problem (1)-(3). The uniqueness of the solution follows from a priori estimate (5).

Definition. Let f_{1} ∈ W_{2}^{1}(S_{1}), F ∈ L_{2}(D_{+}). The function u ∈ W_{2}^{1}(D) will be called a strong solution of the problem (1)–(3) of the class W_{2}^{1}, if there exists a sequence u_{n} ∈ C_{∗}^{∞}(\overline{D}_{+}) such that \frac{∂u_{n}}{∂n} |_{S_{0}} = 0, u_{n} → u in the space W_{2}^{1}(D_{+}) and □u → F in the space L_{2}(D_{+}), i.e., for n → ∞

∥u_{n} − u∥_{W_{2}^{1}(D_{+})} → 0, ∥□u_{n} − F ∥_{L_{2}(D_{+})} → 0,
∥u_{n}|_{S_{1}} − f_{1}∥_{W_{2}^{1}(S_{1})} → 0.

We have the following

Theorem 1. For any f_{1} ∈ W_{2}^{1}(S_{1}), F ∈ L_{2}(D_{+}) there exists a unique strong solution u of the problem (1)–(3) of the class W_{2}^{1} for which the estimate (5) is valid.

Proof. It is known that the space C_{0}^{∞}(D_{+}) ⊂ C_{∗}^{∞}(\overline{D}_{+}) of infinitely differentiable finite functions in D_{+} is everywhere dense in L_{2}(D_{+}), while the space C_{∗}^{∞}(S_{1}) is dense in W_{2}^{1}(S_{1}). Therefore there exist sequences F_{n} ∈ C_{0}^{∞}(D_{+}) and f_{1n} ∈ C_{∗}^{∞}(S_{1}) such that

lim_{n→∞} ∥F − F_{n}∥_{L_{2}(D_{+})} = lim_{n→∞} ∥f_{1} − f_{1n}∥_{W_{2}^{1}(S_{1})} = 0.  (24)

Since the functions F_{n} ∈ C_{0}^{∞}(D_{+}) satisfy the conditions (8), according to Lemma 2 there exists a sequence u_{n} ∈ C_{∗}^{∞}(\overline{D}_{+}) of solutions of the problem (1)–(3) for F = F_{n}, f_{1} = f_{1n}.

In virtue of the inequality (5) we have

∥u_{n} − u_{m}∥_{W_{2}^{1}(D_{+})} ≤ C(∥f_{1n} − f_{1m}∥_{W_{2}^{1}(S_{1})} +
+ ∥F_{n} − F_{m}∥_{L_{2}(D_{+})}).  (25)

It follows from (24) and (25) that the sequence of functions u_{n} is fundamental in the space W_{2}^{1}(D_{+}). Therefore because the space W_{2}^{1}(D_{+}) is complete, there exists a function u ∈ W_{2}^{1}(D_{+}) such that \frac{∂u_{n}}{∂n} |_{S_{0}} = 0, u_{n} → u in W_{2}^{1}(D_{+}), □u_{n} → F in L_{2}(D_{+}) and u_{n}|_{S_{1}} → f_{1} in W_{2}^{1}(S_{1}) for n → ∞. Hence the function u is a strong solution of the problem (1)–(3) of the class W_{2}^{1}. The uniqueness of the strong solution of the problem (1)–(3) of the class W_{2}^{1} follows from the inequality (5).

Consider now the case where in equation (1) we have the lowest terms

Lu ≡ □u + au_{x_{1}} + bu_{x_{2}} + cu_{t} + du = F,  (26)

where the coefficients a, b, c, and d are the given bounded measurable functions in the domain D_{+}.

In the space W_{2}^{1}(D_{+}) let us introduce an equivalent norm

∥u∥_{D_{+},1,γ}^{2} = ∫_{D_{+}} e^{−γt}(u + u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2}) dx dt, γ > 0,

depending on the parameter γ. The norms ∥F ∥_{D_{+},0,γ} and ∥f_{1}∥_{S_{1},1,γ} in the spaces L_{2}(D_{+}) and W_{2}^{1}(S_{1}) are introduced analogously.

Arguments similar to those given in [4] allow us to prove the validity of the following

Lemma 3. For any u ∈ W_{2}^{1}(D_{+}) satisfying the boundary condition (3), the a priori estimate

∥u∥_{D_{+},1,γ} ≤ \frac{C}{\sqrt{γ}} (∥f_{1}∥_{S_{1},1,γ} + ∥F ∥_{D_{+},0,γ})  (27)

holds, where f_{1} = u|_{S_{1}}, F = □u, and the positive constant C does not depend on u and the parameter γ.

In virtue of the estimate (27), the lowest term of the above-introduced equivalent norms of spaces L_{2}(D_{+}), W_{2}^{1}(D_{+}), W_{2}^{1}(S_{1}) in equation (26) for sufficiently large parameter γ give arbitrarily small perturbations which on the basis of Theorem 1 and the results of [4] allow us to prove that the problems (26),(2),(3) are uniquely solvable in the class W_{2}^{1}.

The following theorem holds.

Theorem 2. For any f_{1} ∈ W_{2}^{1}(S_{1}), F ∈ L_{2}(S_{+}) there exists a unique strong solution u of the problem (26), (2), (3) of the class W_{2}^{1} for which the estimate (5) is valid.

References

1. J. Hadamard, Lectures on Cauchy’s problem in linear partial differential equations. Yale Univ. Press, New Haven; Oxford Univ. Press, London, 1923.

2. J. Tolen, Probleme de Cauchy sur la deux hipersurfaces caracteristiques secantes. C. R. Acad. Sci. Paris, Ser. A-B 291(1980), No. 1, 49–52.

3. S. S. Kharibegashvili, On a characteristic problem for the wave equation. Proc. I. Vekua Inst. Appl. Math. Tbilisi St. Univ. 47(1992), 76–82.

4. S. Kharibegashvili, On a spatial problem of Darboux type for second order hyperbolic equation. Georgian Math. J. 2(1995), No. 3, 299–311.

5. A. V. Bitsadze, On mixed type equations on three-dimensional domains. (Russian) Dokl. Akad. Nauk SSSR 143(1962), No. 5, 1017–1019.

6. A. M. Nakhushev, A multidimensional analogy of the Darboux problem for hyperbolic equations. (Russian) Dokl. Akad. Nauk SSSR 194(1970), No. 1, 31–34.

7. T. Sh. Kalmenov, On multidimensional regular boundary value problems for the wave equation. (Russian) Izv. Akad. Nauk Kazakh. SSR, Ser. Fiz.-Mat. (1982), No. 3, 18–25.

8. L. Hörmander, Linear partial differential operators. Grundl. Math. Wiss. Band 116, Springer-Verlag, Berlin–Heidelberg–New York, 1963.

9. V. S. Vladimirov, Generalized functions in mathematical physics. (Russian) Nauka, Moscow, 1976.

10. A. V. Bitsadze, Some classes of partial differential equations. (Russian) Nauka, Moscow, 1981.

11. R. Courant, Partial differential equations (Methods of mathematical physics, ed. R. Courabt and D. Hilbert, Vol. 2), Interscience, New York-London, 1962.

12. F. Olver, Introduction to asymptotics and special functions. Academic Press, New York-London, 1974.

(Received 06.12.1993)

Author’s address: I. Vekua Institute of Applied Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Republic of Georgia