GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 2, 1998, 139-156

ON SOME MULTIDIMENSIONAL VERSIONS OF A CHARACTERISTIC PROBLEM FOR SECOND-ORDER

DEGENERATING HYPERBOLIC EQUATIONS

S. KHARIBEGASHVILI

Abstract. Some multidimensional versions of a characteristic problem for second-order degenerating hyperbolic equations are considered. Using the technique of functional spaces with a negative norm, the correctness of these problems in the Sobolev weighted spaces are proved.

In the space of variables x_{1}, x_{2}, t let us consider a second-order degenerating hyperbolic equation of the kind

Lu ≡ u_{tt} − t^{m}(u_{x_{1}x_{1}} + u_{x_{2}x_{2}}) + a_{1}u_{x_{1}} + a_{2}u_{x_{2}} + a_{3}u_{t} + a_{4}u = F, (1)

where a_{j}, j = 1, …, 4, F are the given functions and u is the unknown real function, m = const > 0.

Denote by

D : 0 < t < [1 − \frac{2 + m}{2}r]^{\frac{2}{2+m}} , r = (x_{1}^{2} + x_{2}^{2})^{\frac{1}{2}} < \frac{2}{2 + m}

a bounded domain lying in a half-space t > 0, bounded above by the characteristic conoid

S : t = [1 − \frac{2 + m}{2}r]^{\frac{2}{2+m}} , r ≤ \frac{2}{2 + m}

of equation (1) with the vertex at the point (0, 0, 1), and below by the base

S_{0} : t = 0, r ≤ \frac{2}{2 + m}

139 of that conoid; equation (1) has on S_{0} a non-characteristic degeneration. In what follows, the coefficients a_{i}, i = 1, …, 4, of equation (1) in D are assumed to be the functions of the class C^{2}(\overline{D}).

1991 Mathematics Subject Classification. 35L80.

Key words and phrases. Degenerating hyperbolic equations, multidimensional versions of a characteristic problem, Sobolev weighted space, functional space with negative norm.

1072-947X/98/0300-0139$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

For equation (1), consider a multidimensional version of the characteristic problem which is formulated as follows: On the domain D, find a solution u(x_{1}, x_{2}, t) of equation (1) satisfying the boundary condition

u|_{S} = 0.  (2)

As will be shown below, the following Cauchy problem on finding in D a solution of equation

L^{∗}v ≡ v_{tt} − t^{m}(v_{x_{1}x_{1}} + v_{x_{2}x_{2}}) − (a_{1}v)_{x_{1}} − (a_{2}v)_{x_{2}} − (a_{3}v)_{t} + a_{4}v = F (3)

by the initial conditions

v|_{S_{0}} = 0, v_{t}|_{S_{0}} = 0  (4)

is the problem conjugate to problem (1), (2), where L^{∗} is the operator formally conjugate to the operator L.

Note that for m = 0, when equation (1) is non-degenerating and contains in its principal part a wave operator, some multidimensional Goursat and Darboux problems have been investigated in [1–6]. For a hyperbolic equation of second-order with non-characteristic degeneration of the kind

u_{tt} − |x_{2}|^{m}u_{x_{1}x_{1}} − u_{x_{2}x_{2}} + a_{1}u_{x_{1}} + a_{2}u_{x_{2}} + a_{3}u_{t} + a_{4}u = F,

as well as for a hyperbolic equation of second-order with characteristic degeneration

u_{tt} − u_{x_{1}x_{1}} − (|x_{2}|^{m}u_{x_{2}})_{x_{2}} + a_{1}u_{x_{1}} + a_{2}u_{x_{2}} + a_{3}u_{t} + a_{4}u = F

the multidimensional variants of the Darboux problem are respectively studied in [7] and [8]. Other variants of multidimensional Goursat and Darboux problems can be found in [9–11].

Denote by E and E^{∗} the classes of functions from the Sobolev space W_{2}^{2}(D), satisfying respectively the boundary condition (2) or (4) and vanishing in some (own for every function) three-dimensional neighborhood of the circle Γ = S ∩ S_{0} : r = \frac{2}{2+m}, t = 0 and of the segment I : x_{1} = x_{2} = 0, 0 ≤ t ≤ 1. Let W_{+}(W_{+}^{∗}) be a Hilbert space with weight, obtained by closing the space E(E^{∗}) in the norm

∥u∥_{1}^{2} = ∫_{D} [u_{t}^{2} + t^{m}(u_{x_{1}}^{2} + u_{x_{2}}^{2}) + u^{2}]dD.

Denote by W_{−}(W_{−}^{∗}) a space with negative norm which is constructed with respect to L_{2}(D) and W_{+}(W_{+}^{∗}) [12].

Let n = (ν_{1}, ν_{2}, ν_{0}) be the unit vector of the outer to ∂D normal, i.e., ν_{1} = cos({ \atop n, x_{1}}), ν_{2} = cos({ \atop n, x_{2}}), ν_{0} = cos(n̂, t). By definition, the derivative with respect to the conormal can be calculated on the boundary ∂D of the domain D for the operator L by the formula

\frac{∂}{∂N} = ν_{0} \frac{∂}{∂t} − t^{m}ν_{1} \frac{∂}{∂x_{1}} − t^{m}ν_{2} \frac{∂}{∂x_{2}} .

Remark 1. Since the derivative with respect to the conormal \frac{∂}{∂N} for the operator L is an interior differential operator on the characteristic surfaces of equation (1), by virtue of (2) and (4) we have for the functions u ∈ E and v ∈ E^{∗} that

\frac{∂u}{∂N} |_{S} = 0, \frac{∂v}{∂N} |_{S_{0}} = 0.  (5)

Impose on the lower coefficients a_{1} and a_{2} in equation (1) the following restrictions:

M_{i} = sup_{\overline{D}} |t^{− \frac{m}{2}} a_{i}(x_{1}, x_{2}, t)| < +∞, i = 1, 2.  (6)

Lemma 1. For all functions u ∈ E, v ∈ E^{∗} the following inequalities hold:

∥Lu∥_{W_{−}^{∗}} ≤ c_{1}∥u∥_{W_{+}},  (7)
∥L^{∗}v∥_{W_{−}} ≤ c_{2}∥v∥_{W_{+}^{∗}},  (8)

where the positive constants c_{1} and c_{2} do not depend respectively on u and v, ∥ · ∥_{W_{+}} = ∥ · ∥_{W_{+}^{∗}} = ∥ · ∥_{1}.

Proof. By the definition of a negative norm, for u ∈ E with regard for equalities (2), (4) and (5) we have

∥Lu∥_{W_{−}^{∗}} = sup_{v∈W_{+}^{∗}} ∥v∥_{W_{+}^{∗}}^{−1}(Lu, v)_{L_{2}(D)} = sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1}(Lu, v)_{L_{2}(D)} =

= sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1} ∫_{D} [u_{tt}v − t^{m}u_{x_{1}x_{1}}v − t^{m}u_{x_{2}x_{2}}v + a_{1}u_{x_{1}}v + a_{2}u_{x_{1}}v +

+a_{3}u_{t}v + a_{4}uv]dD = sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1} ∫_{∂D} [u_{t}vν_{0} − t^{m}u_{x_{1}}vν_{1} −

−t^{m}u_{x_{2}}vν_{2}]ds + sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1} ∫_{D} [ − u_{t}v_{t} + t^{m}(u_{x_{1}}v_{x_{1}} + u_{x_{2}}v_{x_{2}}) +

+a_{1}u_{x_{1}}v + a_{2}u_{x_{2}}v + a_{3}u_{t}v + a_{4}uv]dD = sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1} ∫_{∂D} \frac{∂u}{∂N} vds +

+ sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1} ∫_{D} [ − u_{t}v_{t} + t^{m}(u_{x_{1}}v_{x_{1}} + u_{x_{2}}v_{x_{2}}) + a_{1}u_{x_{1}}v + a_{2}u_{x_{2}}v +

+a_{3}u_{t}v + a_{4}uv]dD = sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1} ∫_{D} [ − u_{t}v_{t} + t^{m}(u_{x_{1}}v_{x_{1}} +

+u_{x_{2}}v_{x_{2}}) + a_{1}u_{x_{1}}v + a_{2}u_{x_{1}}v + a_{3}u_{t}v + a_{4}uv]dD.  (9)

Due to (6) as well as the Cauchy inequality, we have

| ∫_{D} [ − u_{t}v_{t} + t^{m}(u_{x_{1}}v_{x_{1}} + u_{x_{2}}v_{x_{2}})]dD| ≤ [ ∫_{D} (u_{t}^{2} + t^{m}u_{x_{1}}^{2} +

+t^{m}u_{x_{2}}^{2})dD]^{\frac{1}{2}} × [ ∫_{D} (v_{t}^{2} + t^{m}v_{x_{1}}^{2} + t^{m}v_{x_{2}}^{2})dD]^{\frac{1}{2}} ≤ ∥u∥_{W_{+}}∥v∥_{W_{+}^{∗}}, (10)

| ∫_{D} [a_{1}u_{x_{1}}v + a_{2}u_{x_{2}}v + a_{3}u_{t}v + a_{4}uv)]dD ≤

≤ M_{1}( ∫_{D} t^{m}u_{x_{1}}^{2}dD)^{\frac{1}{2}} ∥v∥_{L_{2}(D)} + M_{2}( ∫_{D} t^{m}u_{x_{2}}^{2}dD)^{\frac{1}{2}} ∥v∥_{L_{2}(D)} +

+ sup_{\overline{D}} |a_{3}| ∥u_{t}∥_{L_{2}(D)}∥v∥_{L_{2}(D)} + sup_{\overline{D}} |a_{4}| ∥u∥_{L_{2}(D)}∥v∥_{L_{2}(D)} ≤

≤ ( ∑_{i=1}^{2} (M_{i} + sup_{\overline{D}} |a_{2+i}|))∥u∥_{W_{+}}∥v∥_{W_{+}^{∗}} = c̃∥u∥_{W_{+}}∥v∥_{W_{+}^{∗}}. (11)

From (9)–(11) it follows that

∥Lu∥_{W_{−}^{∗}} ≤ (1 + c̃) sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1}∥u∥_{W_{+}}∥v∥_{W_{+}^{∗}} = c_{1}∥u∥_{W_{+}},

i.e., we get inequality (7). Since the proof of inequality (8) repeats that of inequality (7), therefore Lemma 1 is proved completely.

Remark 2. By virtue of inequality (7) ((8)), the operator L : W_{+} → W_{−}^{∗}(L^{∗} : W_{+}^{∗} → W_{−}) with a dense domain of definition E(E^{∗}) admits a closure, being a continuous operator from the space W_{+}(W_{+}^{∗}) to the space W_{−}(W_{−}^{∗}). Retaining for this operator the previous notation L(L^{∗}), we note that it is defined on the whole Hilbert space W_{+}(W_{+}^{∗}).

Lemma 2. Problem (1), (2) and problem (3), (4) are self-conjugate, i.e., for any u ∈ W_{+} and v ∈ W_{+}^{∗} the following equality holds:

(Lu, v) = (u, L^{∗}v).  (12)

Proof. According to Remark 2, it suffices to prove equality (12) in the case where u ∈ E and v ∈ E^{∗}. Obviously, in that case (Lu, v) = (Lu, v)_{L_{2}(D)}. Therefore we have

(Lu, v) = (Lu, v)_{L_{2}(D)} = ∫_{∂D} [u_{t}vν_{0} − t^{m}u_{x_{1}}vν_{1} − t^{m}u_{x_{2}}vν_{2}]ds +

+ ∫_{∂D} [a_{1}ν_{1} + a_{2}ν_{2} + a_{3}ν_{0}]uv ds + ∫_{D} [ − u_{t}v_{t} + t^{m}u_{x_{1}}v_{x_{1}} + t^{m}u_{x_{2}}v_{x_{2}} −

−u(a_{1}v)_{x_{1}} − u(a_{2}v)_{x_{2}} − u(a_{3}v)_{t} + a_{4}uv]dD = ∫_{∂D} [u_{t}vν_{0} − t^{m}u_{x_{1}}vν_{1} −

−t^{m}u_{x_{2}}vν_{2}]ds + ∫_{∂D} [a_{1}ν_{1} + a_{2}ν_{2} + a_{3}ν_{0}]uv ds − ∫_{∂D} [uv_{t}ν_{0} −

−t^{m}uv_{x_{1}}ν_{1} − t^{m}uv_{x_{2}}ν_{2}]ds + ∫_{D} [uv_{tt} − ut^{m}v_{x_{1}x_{1}} − ut^{m}v_{x_{2}x_{2}} −

−u(a_{1}v)_{x_{1}} − u(a_{2}v)_{x_{2}} − u(a_{3}v)_{t} + a_{4}uv]dD = ∫_{∂D} [(v \frac{∂u}{∂N} − u\frac{∂v}{∂N} ) +

+(u_{1}ν_{1} + a_{2}ν_{2} + a_{3}ν_{0})uv]ds + (u, L^{∗}v)_{L_{2}(D)}.  (13)

Equality (12) follows immediately from equalities (2), (4), (5), and (13).

Consider the conditions

Ω|_{S} ≤ 0, [tΩ_{t} − (λt + m)Ω]|_{D} ≥ 0,  (14)

where the second inequality holds for sufficiently large λ, and Ω = a_{1x_{1}} + a_{2x_{2}} + a_{3t} − a_{4}.

Remark 3. It can be easily seen that inequality (14) is the corollary of the condition

Ω|\overline{D} ≤ const < 0.

Lemma 3. Let conditions (6) and (14) be fulfilled. Then for any u ∈ W_{+} the inequality

c∥t^{\frac{1}{2} (m−1)}u∥_{L_{2}(D)} ≤ ∥Lu∥_{W_{−}^{∗}}  (15)

with the positive constant c independent of u is valid.

Proof. Due to Remark 2, it suffices to prove inequality (15) in the case where u ∈ E. If u ∈ E, then for α = const > 0 and λ = const > 0 the function

v(x_{1}, x_{2}, t) = ∫_{0}^{t} e^{λτ} τ^{α}u(x_{1}, x_{2}, τ)dτ  (16)

belongs to the space E^{∗}. The fact that for α ≥ 1 the function v ∈ E^{∗} can be easily verified, and for 0 < α < 1 this statement follows from the well-known Hardy’s inequality

∫_{0}^{1} t^{−2}g^{2}(t)dt ≤ 4 ∫_{0}^{1} f^{2}(t)dt,

where f(t) ∈ L_{2}(0, 1) and g(t) = ∫_{0}^{t} f(τ)dτ.

By (16), the inequalities

v_{t}(x_{1}, x_{2}, t) = e^{λt}t^{α}u(x_{1}, x_{2}, t), u(x_{1}, x_{2}, t) = e^{−λt}t^{−α}v_{t}(x_{1}, x_{2}, t) (17)

are valid.

With regard for (2), (4), (5), and (17) we have

(Lu, v)_{L_{2}(D)} = ∫_{∂D} [v \frac{∂u}{∂N} + (a_{1}ν_{1} + a_{2}ν_{2} + a_{3}ν_{0})uv]ds + ∫_{D} [−u_{t}v_{t} +

+t^{m}u_{x_{1}}v_{x_{1}} + t^{m}u_{x_{2}}v_{x_{2}} − u(a_{1}v)_{x_{1}} − u(a_{2}v)_{x_{2}} − u(a_{3}v)_{t} + a_{4}uv]dD =

= − ∫_{D} e^{λt}t^{α}uu_{t} dD + ∫_{D} e^{−λt}t^{−α}[t^{m}(v_{x_{1}t}v_{x_{1}} + v_{x_{2}t}v_{x_{2}}) −

−(a_{1}v_{x_{1}} + a_{2}v_{x_{2}})v_{t} − (a_{1x_{1}} + a_{2x_{2}} + a_{3t} − a_{4})v_{t}v − a_{3}v_{t}^{2}]dD. (18)

By virtue of (2) we find that

− ∫_{D} e^{−λt}t^{α}uu_{t} dD = −\frac{1}{2} ∫_{D} e^{λt}t^{α}(u^{2})_{t}dt = −\frac{1}{2} ∫_{∂D} e^{λt}t^{α}u^{2}ν_{0} ds +

+\frac{1}{2} ∫_{D} e^{λt}(αt^{α−1} + λt^{α})u^{2} dD = \frac{1}{2} ∫_{D} e^{λt}(αt^{α−1} + λt^{α})u^{2} dD =

= \frac{α}{2} ∫_{D} e^{λt}t^{α−1}u^{2} dD + \frac{1}{2} ∫_{D} λe^{−λt}t^{−α}v_{t}^{2} dD,  (19)

∫_{D} e^{−λt}t^{m−α}(−v_{x_{1}t}v_{x_{1}} + v_{x_{2}t}v_{x_{2}})dD = \frac{1}{2} ∫_{∂D} e^{−λt}t^{m−α}(v_{x_{1}}^{2} +

+v_{x_{2}}^{2})ν_{0} ds + \frac{1}{2} ∫_{D} e^{−λt}[λt^{m−α} + (α − m)t^{m−α−1}](v_{x_{1}}^{2} + v_{x_{2}}^{2})dD ≥

≥ \frac{1}{2} ∫_{D} e^{−λt}[λt^{m−α} + (α − m)t^{m−α−1}](v_{x_{1}}^{2} + v_{x_{2}}^{2})dD.  (20)

In deriving inequality (20) we have taken into account that

ν_{0}|_{S} ≥ 0, (v_{x_{1}}^{2} + v_{x_{2}}^{2})|_{S_{0}} = 0.

From (19) we have

− ∫_{D} e^{λt}t^{α}uu_{t} dD ≥ \frac{α}{2}∥t^{\frac{1}{2} (α−1)}u∥_{L_{2}(D)}^{2} + \frac{1}{2} ∫_{D} λe^{−λt}t^{−α}v_{t}^{2} dD. (21)

Below we assume that the parameter α = m. By (6) we obtain

| ∫_{D} e^{−λt}t^{−m}(a_{1}v_{x_{1}} + a_{2}v_{x_{2}})v_{t} dD| ≤ M ∫_{D} e^{−λt}t^{−m}[v_{t}^{2} + \frac{1}{2}t^{m}(v_{x_{1}}^{2} +

+v_{x_{2}}^{2})]dD ≤ M ∫_{D} e^{−λt}t^{−m}v_{t}^{2}dD + \frac{M}{2} ∫_{D} e^{−λt}(v_{x_{1}}^{2} + v_{x_{2}}^{2})dD, (22)

where M = max(M_{1}, M_{2}).

Since ν_{0}|_{S} ≥ 0, using conditions (4) and (14) and integrating them by parts, we obtain

− ∫_{D} e^{−λt}t^{−m}(a_{1x_{1}} + a_{2x_{2}} + a_{3t} − a_{4})v_{t}v dD =

−\frac{1}{2} ∫_{D} e^{−λt}t^{−m}Ω(v^{2})_{t}dD = −\frac{1}{2} ∫_{∂D} e^{−λt}t^{−m}Ωv^{2}ν_{0}ds +

+\frac{1}{2} ∫_{D} e^{−λt}t^{−m−1}[tΩ_{t} − (λt + m)Ω]v^{2}dD ≥ 0.  (23)

In deriving inequality (23) we have used the fact that the function t^{−m}v^{2} has on S_{0} a zero trace, i.e., t^{−m}v^{2}|_{S_{0}} = 0.

From (18) by virtue of (20)–(23) we have

(Lu, v)_{L_{2}(D)} ≥ \frac{m}{2}∥t^{\frac{1}{2} (m−1)}u∥_{L_{2}(D)}^{2} + \frac{1}{2} ∫_{D} λe^{−λt}t^{−m}v_{t}^{2}dD +

+\frac{1}{2} ∫_{D} λe^{−λt}(v_{x_{1}}^{2} + v_{x_{2}}^{2})dD − M ∫_{D} e^{−λt}t^{−m}v_{t}^{2}dD − \frac{M}{2} ∫_{D} e^{−λt}(v_{x_{1}}^{2} +

+v_{x_{2}}^{2})dD − sup_{\overline{D}} |a_{3}| ∫_{D} e^{−λt}t^{−m}v_{t}^{2}dD = \frac{m}{2}∥t^{\frac{1}{2} (m−1)}u∥_{L_{2}(D)}^{2} +

+(\frac{λ}{2} − M − sup_{\overline{D}} |a_{3}|) ∫_{D} e^{−λt}t^{−m}v_{t}^{2}dD + \frac{1}{2}(λ − M) ∫_{D} e^{−λt}(v_{x_{1}}^{2} +

+v_{x_{2}}^{2})dD ≥ \frac{m}{2}∥t^{\frac{1}{2} (m−1)}u∥_{L_{2}(D)}^{2} + σ ∫_{D} e^{−λt}(v_{t}^{2} + v_{x_{1}}^{2} + v_{x_{2}}^{2})dD ≥

≥ \sqrt{2mσ inf_{\overline{D}} e^{−λt}}∥t^{\frac{1}{2} (m−1)}u∥_{L_{2}(D)}^{2}( ∫_{D} [v_{t}^{2} + t^{m}(v_{x_{1}}^{2} + v_{x_{2}}^{2})]dD)^{\frac{1}{2}} , (24)

where σ = [\frac{λ}{2} −M −sup\overline{D} |a_{3}|] > 0 for sufficiently large λ, and inf\overline{D} e^{−λt} = e^{−λ} > 0. When deriving inequality (24), we have taken into account the fact that t^{−m}|_{D} ≥ 1.

If u ∈ W_{+}(W_{+}^{∗}) and because u|_{S} = 0 (u|_{S_{0}} = 0), we can easily prove the inequality

∫_{D} u^{2}dD ≤ c_{0} ∫_{D} u_{t}^{2}dD

for which c_{0} = const > 0 independent of u. Hence we find that in the space W_{+}(W_{+}^{∗}) the norm

∥u∥_{W_{+}(W_{+}^{∗})}^{2} = ∫_{D} [u_{t}^{2} + t^{m}(u_{x_{1}}^{2} + u_{x_{2}}^{2}) + u^{2}]dD

is equivalent to the norm

∥u∥^{2} = ∫_{D} [u_{t}^{2} + t^{m}(u_{x_{1}}^{2} + u_{x_{2}}^{2})]dD.  (25)

Therefore, retaining for norm (25) the previous designation ∥u∥_{W_{+}(W_{+}^{∗})}, from (24) we have

(Lu, v)_{L_{2}(D)} ≥ \sqrt{2mσe^{−λ}}∥t^{\frac{1}{2} (m−1)}u∥_{L_{2}(D)} ∥v∥_{W_{+}^{∗}}.  (26)

If now we apply the generalized Schwarz inequality

(Lu, v) ≤ ∥Lu∥_{W_{−}^{∗}} ∥v∥_{W_{+}^{∗}}

to the left-hand side of (26), then after reducing by ∥v∥_{W_{+}^{∗}} we get inequality (15) in which c = \sqrt{2mσe^{−λ}}.

Consider the conditions

a_{4}|_{S_{0}} ≥ 0, (λa_{4} + a_{4t})|_{D} ≥ 0,  (27)

of which the second one takes place for sufficiently large λ.

Lemma 4. Let conditions (6) and (27) be fulfilled. Then for any v ∈ W_{+}^{∗} the inequality

c∥v∥_{L_{2}(D)} ≤ ∥L^{∗}v∥_{W_{−}}  (28)

is valid for some c = const > 0 independent of v ∈ W_{+}^{∗}.

Proof. Just as in Lemma 3 and because of Remark 2, it suffices to prove the validity of inequality (28) for v ∈ E^{∗}. Let v ∈ E^{∗} and let us introduce into the consideration the function

u(x_{1}, x_{2}, t) = ∫_{t}^{φ(x_{1},x_{2})}e^{−λτ} v(x_{1}, x_{2}, τ)dτ, λ = const > 0,  (29)

where t = φ(x_{1}, x_{2}) is the equation of the characteristic conoid S. Although on the circle r = \frac{2}{2+m} the function

φ(x_{1}, x_{2}) = [1 − \frac{2 + m}{2}r]^{\frac{2}{2+m}}

has singularities and at the origin x_{1} = x_{2} = 0, but by the definition of the space E^{∗}, the function v vanishes in some neighborhood of the circle Γ = S ∩ S_{0} and of the segment I : x_{1} = x_{2} = 0, 0 ≤ t ≤ 1, the function u defined by equality (29) will belong to the space E. Moreover, it is obvious that the equalities

u_{t}(x_{1}, x_{2}, t) = −e^{−λt}v(x_{1}, x_{2}, t), v(x_{1}, x_{2}, t) = −e^{λt}u_{t}(x_{1}, x_{2}, t) (30)

hold.

Owing to (2), (4), (5), and (30), we have

(L^{∗}v, u)_{L_{2}(D)} = ∫_{∂D} [u\frac{∂v}{∂N} − (a_{1}ν_{1} + a_{2}ν_{2} + a_{3}ν_{0})vu]ds +

+ ∫_{D} [−v_{t}u_{t} + t^{m}v_{x_{1}}u_{x_{1}} + t^{m}v_{x_{2}}u_{x_{2}} + a_{1}vu_{x_{1}} + a_{2}vu_{x_{2}} + a_{3}vu_{t} +

+a_{4}uv]dD = ∫_{D} e^{−λt}v_{t}v dD − ∫_{D} e^{λt}[t^{m}(u_{x_{1}t}u_{x_{1}} + u_{x_{2}t}u_{x_{2}}) +

+(a_{1}u_{x_{1}} + a_{2}u_{x_{2}})u_{t} + a_{3}u_{t}^{2} + a_{4}uu_{t}] dD,  (31)

∫_{D} e^{−λt}v_{t}v dD = \frac{1}{2} ∫_{∂D} e^{−λt}v^{2}ν_{0} ds + \frac{1}{2} ∫_{D} e^{−λt}λv^{2} dD =

= \frac{1}{2} ∫_{S} e^{−λt}v^{2}ν_{0} ds + \frac{1}{2} ∫_{D} e^{−λt}λv^{2} dD =

= \frac{1}{2} ∫_{S} e^{λt}u_{t}^{2}ν_{0} ds + \frac{1}{2} ∫_{D} e^{λt}λu_{t}^{2} dD,  (32)

− ∫_{D} e^{λt}t^{m}(u_{x_{1}t}u_{x_{1}} + u_{x_{2}t}u_{x_{2}})dD = −\frac{1}{2} ∫_{∂D} e^{λt}t^{m}(u_{x_{1}}^{2} + u_{x_{2}}^{2})ν_{0} ds +

+\frac{1}{2} ∫_{D} e^{λt}[λt^{m} + mt^{m−1}](u_{x_{1}}^{2} + u_{x_{2}}^{2})dD ≥

≥ −\frac{1}{2} ∫_{∂D} e^{λt}t^{m}(u_{x_{1}}^{2} + u_{x_{2}}^{2})ν_{0} ds + \frac{1}{2} ∫_{D} λe^{λt}t^{m}(u_{x_{1}}^{2} + u_{x_{2}}^{2}) dD. (33)

Since u|_{S} = 0, for some α we have v_{t} = αν_{0}, v_{x_{1}} = αν_{1}, v_{x_{2}} = αν_{2} on S. Therefore the fact that the surface S is characteristic results in

[u_{t}^{2} − t^{m}(u_{x_{1}}^{2} + u_{x_{2}}^{2})]|_{S} = α^{2}[ν_{0}^{2} − t^{m}(ν_{1}^{2} + ν_{2}^{2})]|_{S} = 0.  (34)

Taking into account that m > 0 and hence t^{m}|_{S_{0}} = 0, equalities (2) and (34) imply

\frac{1}{2} ∫_{S} e^{λt}u_{t}^{2}ν_{0} ds − \frac{1}{2} ∫_{∂D} e^{λt}t^{m}(u_{x_{1}}^{2} + u_{x_{2}}^{2})ν_{0} ds =

= \frac{1}{2} ∫_{S} e^{λt}[u_{t}^{2} − t^{m}(u_{x_{1}}^{2} + u_{x_{2}}^{2})]ν_{0} ds = 0.  (35)

Due to (32), (33), and (35), equality (31) yields

(L^{∗}v, u)_{L_{2}(D)} ≥ \frac{1}{2} ∫_{D} e^{λt}[u_{t}^{2} + t^{m}(u_{x_{1}}^{2} + u_{x_{2}}^{2})]dD −

− ∫_{D} e^{λt}[(a_{1}u_{x_{1}} + a_{2}u_{x_{2}})u_{t} + a_{3}u_{t}^{2} + a_{4}uu_{t}]dD.  (36)

Since ν_{0}|_{S_{0}} < 0, by (27) we have

− ∫_{D} e^{λt}a_{4}uu_{t}dD = −\frac{1}{2} ∫_{D} e^{λt}a_{4}(u^{2})_{t}dD = −\frac{1}{2} ∫_{∂D} e^{λt}a_{4}u^{2}ν_{0}ds +

+\frac{1}{2} ∫_{D} e^{λt}(λa_{4} + a_{4t})u^{2}dD ≥ 0.  (37)

Using (6), we obtain

| ∫_{D} e^{λt}(a_{1}u_{x_{1}} + a_{2}u_{x_{2}})u_{t}dD| ≤ M ∫_{D} e^{λt}[u_{t}^{2} + \frac{1}{2}t^{m}(u_{x_{1}}^{2} + u_{x_{2}}^{2})]dD, (38)

where M = max(M_{1}, M_{2}).

With regard for (30), (37) and (38), from (36) we get

(L^{∗}v, u)_{L_{2}(D)} ≥ (\frac{λ}{2} − M − sup_{\overline{D}} |a_{3}|) ∫_{D} e^{λt}[u_{t}^{2} + t^{m}(u_{x_{1}}^{2} + u_{x_{2}}^{2})]dD ≥

≥ γ[ ∫_{D} e^{λt}u_{t}^{2}dD]^{\frac{1}{2}} [ ∫_{D} e^{λt}[u_{t}^{2} + t^{m}(u_{x_{1}}^{2} + u_{x_{2}}^{2})]dD]^{\frac{1}{2}} =

= γ[ ∫_{D} e^{−λt}v^{2}dD]^{\frac{1}{2}} [ ∫_{D} e^{λt}[u_{t}^{2} + t^{m}(u_{x_{1}}^{2} + u_{x_{2}}^{2})]dD]^{\frac{1}{2}} ≥

≥ γ inf_{\overline{D}} e^{−λt}∥v∥_{L_{2}(D)}[ ∫_{D} [u_{t}^{2} + t^{m}(u_{x_{1}}^{2} + u_{x_{2}}^{2})]dD]^{\frac{1}{2}} ,  (39)

where γ = (\frac{λ}{2} − M − sup\overline{D} |a_{3}|) > 0 for sufficiently large λ.

From (39), in just the same way as in obtaining inequality (26), we find that

(L^{∗}v, u)_{L_{2}(D)} ≥ c∥v∥_{L_{2}(D)} ∥u∥_{W_{+}},

which immediately implies (28).

Denote by L_{2,α}(D) a space of functions u such that t^{α}u ∈ L_{2}(D). Assume

∥u∥_{L_{2,α}(D)} = ∥t^{a}lu∥L_{2}(D), α_{m} = \frac{1}{2}(m − 1).

Definition 1. For F ∈ W_{−}^{∗} we call the function u a strong generalized solution of problem (1), (2) of the class L_{2,α_{m}}, if u ∈ L_{2,α_{m}}(D) and there exists a sequence of functions u_{n} ∈ E such that u_{n} → u in the space L_{2,α_{m}}(D) and Lu_{n} → F in the space W_{−}^{∗} as n → ∞, i.e.,

lim_{n→∞} ∥u_{n} − u∥_{L_{2,αm} (D)} = 0, lim_{n→∞} ∥Lu_{n} − F ∥_{W_{−}^{∗}} = 0.

Definition 2. For F ∈ L_{2}(D) we call the function u a strong generalized solution of problem (1), (2) of the class W_{+}, if u ∈ W_{+} and there exists a sequence of functions u_{n} ∈ E such that u_{n} → u and Lu_{n} → F in the spaces W_{+} and W_{−}^{∗}, respectively, i.e.,

lim_{n→∞} ∥u_{n} − u∥_{W_{+}} = 0, lim_{n→∞} ∥Lu_{n} − F ∥_{W_{−}^{∗}} = 0.

According to the results obtained in [13], the following theorems are the corollaries of Lemmas 1–4.

Theorem 1. Let conditions (6), (14), and (27) be fulfilled. Then for every F ∈ W_{−}^{∗} there exists a unique strong generalized solution u of problem (1), (2) of the class L_{2,α_{m}} for which the estimate

∥u∥_{L_{2,αm} (D)} ≤ c∥F ∥_{W_{−}^{∗}}  (40)

with the constant c independent of F is valid.

Theorem 2. Let conditions (6), (14), and (27) be fulfilled. Then for every F ∈ L_{2}(D) there exists a unique strong generalized solution u of problem (1), (2) of the class W_{+} for which estimate (40) is valid.

Consider now a second-order hyperbolic equation with a characteristic degeneration of the kind

L_{1}u ≡ (t^{m}u_{t})_{t} − u_{x_{1}x_{1}} − u_{x_{2}x_{2}} + a_{1}u_{x_{1}} + a_{2}u_{x_{2}} + a_{3}u_{t} + a_{4}u = F, (41)

where 1 ≤ m = const < 2.

Denote by

D_{1} : 0 < t < [1 − \frac{2 − m}{2}r]^{\frac{2}{2−m}} , r = (x_{1}^{2} + x_{2}^{2})^{\frac{1}{2}} < \frac{2}{2 − m}

a bounded domain lying in a half-space t > 0, bounded above by the characteristic conoid

S_{2} : t = [1 − \frac{2 − m}{2}r]^{\frac{2}{2−m}} , r ≤ \frac{2}{2 − m}

of equation (41) with the vertex at the point (0, 0, 1) and below by the base

S_{1} : t = 0, r ≤ \frac{2}{2 − m}

of the same conoid where equation (41) has a characteristic degeneration. Just as in the case of equation (1), in what follows, the coefficients a_{i}, i = 1, …, 4, of equation (41) in the domain D_{1} are assumed to be the functions of the class C^{2}(\overline{D}).

For equation (41) let us consider a multidimensional version of the characteristic problem which is formulated as follows: Find in the domain D_{1} a solution u(x_{1}, x_{2}, t) of equation (41) satisfying the boundary condition

u|_{S_{1}} = 0  (42)

on the plane characteristic surface S_{1}.

The characteristic problem for the equation

L_{1}^{∗}v ≡(t^{m}v_{t})_{t} − v_{x_{1}x_{1}} − v_{x_{2}x_{2}} − (a_{1}v)_{x_{1}} − (a_{2}v)_{x_{2}} − (a_{3}v)_{t} + a_{4}v =F (43)

in the domain D_{1} is formulated analogously for the boundary condition

v|_{S_{2}} = 0,  (44)

where L_{1}^{∗} is the operator conjugate formally to the operator L_{1}.

Denote by E_{1} and E_{1}^{∗} the classes of functions from the Sobolev space W_{2}^{2}(D_{1}), satisfying the corresponding boundary condition (42) or (44) and vanishing in some (own for every function) three-dimensional neighborhood of the segment I : x_{1} = 0, x_{2} = 0, 0 ≤ t ≤ 1. Let W_{1+}(W_{1+}^{∗}) be the Hilbert space obtained by closing the space E_{1}(E_{1}^{∗}) in the norm

∥u∥^{2} = ∫_{D_{1}} [u_{t}^{2} + u_{x_{1}}^{2} + u_{x_{2}}^{2} + u^{2}]dD_{1}.

Denote by W_{1−}(W_{1−}^{∗}) a space with a negative norm, constructed with respect to L_{2}(D_{1}) and W_{1+}(W_{1+}^{∗}).

The following lemma can be proved analogously to Lemmas 1 and 2.

Lemma 5. For all functions u ∈ E_{1} and v ∈ E_{1}^{∗} the inequalities

∥L_{1}u∥_{W_{1−}^{∗}} ≤ c_{1}∥u∥_{W_{1+}}, ∥L_{1}^{∗}v∥_{W_{1−}} ≤ c_{1}∥v∥_{W_{1+}^{∗}},

are fulfilled and problems (41), (42) and (43), (44) are self-conjugate, i.e., for every u ∈ W_{1+} and v ∈ W_{1+}^{∗} the equality

(L_{1}u, v) = (u, L_{1}^{∗}v)

holds.

Let us consider the conditions

inf_{\overline{D}_{1}}(a_{4} − a_{1x_{1}} − a_{2x_{2}} − a_{3t}) > 0,  (45)
inf_{S_{1}} a_{3} > \frac{1}{2} for m = 1, inf_{S_{1}} a_{3} > 0 for m > 1.  (46)

Lemma 6. Let conditions (45) and (46) be fulfilled. Then for any u ∈ W_{1+} the inequality

c∥u∥_{L_{2}(D_{1})} ≤ ∥L_{1}u∥_{W_{1−}^{∗}},  (47)

with the positive constant c independent of u, is valid.

Consider now the conditions

inf_{\overline{D}_{1}} a_{4} > 0,  (48)
inf_{S_{1}} a_{3} > −\frac{1}{2} for m = 1, inf_{S_{1}} a_{3} > 0 for m > 1.  (49)

Note that condition (46) results in condition (49).

Lemma 7. Let conditions (48) and (49) be fulfilled. Then for any v ∈ W_{1+}^{∗} the inequality

c∥v∥_{L_{2}(D_{1})} ≤ ∥L_{1}^{∗}v∥_{W_{1−}}  (50)

with the positive constant c independent of v holds.

Below we will restrict ourselves to proving only Lemma 6. Let u ∈ E_{1}. We introduce into consideration the function

v(x_{1}, x_{2}, t) = ∫_{t}^{ψ(x_{1},x_{2})}e^{−λτ} u(x_{1}, x_{2}, τ)dτ, λ = const > 0,  (51)

where t = ψ(x_{1}, x_{2}) is the equation of the characteristic conoid S_{2} of equation (41). Since 1 ≤ m < 2, the first and second-order derivatives of the function

ψ(x_{1}, x_{2}) = [1 − \frac{2 − m}{2}r]^{\frac{2}{2−m}}

with respect to the variables x_{1} and x_{2} will have singularities at the origin only. But by the definition of the space E_{1}, the function u vanishes in some neighborhood of the segment I : x_{1} = x_{2} = 0, 0 ≤ t ≤ 1. Therefore the function v defined by equality (51) belongs to the space E_{1}^{∗}, and the equalities

v_{t}(x_{1}, x_{2}, t) = −e^{−λt}u(x_{1}, x_{2}, t), u_{t}(x_{1}, x_{2}, t) = −e^{λt}v_{t}(x_{1}, x_{2}, t) (52)

hold.

Since the derivative with respect to the conormal

\frac{∂}{∂N} = t^{m}ν_{0} \frac{∂}{∂t} − ν_{1} \frac{∂}{∂x_{1}} − ν_{2} \frac{∂}{∂x_{2}}

for the operator L_{1} is an interior differential operator on the characteristic surfaces of equation (41), because of (42) and (44) for the functions u ∈ E_{1} and v ∈ E_{1}^{∗} we have

\frac{∂u}{∂N} |_{S_{1}} = 0, \frac{∂u}{∂N} |_{S_{2}} = 0.  (53)

ON SOME MULTIDIMENSIONAL VERSIONS 153

By (42), (44), (52) and (53) we arrive at

(Lu, v)_{L_{2}(D_{1})} = ∫_{∂D_{1}} [v \frac{∂u}{∂N} + (a_{1}ν_{1} + a_{2}ν_{2} + a_{3}ν_{0})uv]ds +

+ ∫_{D_{1}} [ − t^{m}u_{t}v_{t} + u_{x_{1}}v_{x_{1}} + u_{x_{2}}v_{x_{2}} − u(a_{1}v)_{x_{1}} − u(a_{2}v)_{x_{2}} − u(a_{3}v)_{t} +

+a_{4}uv]dD_{1} = ∫_{D_{1}} e^{−λt}t^{m}uu_{t}dD_{1} + ∫_{D_{1}} e^{λt}[ − v_{x_{1}t}v_{x_{1}} − v_{x_{2}t}v_{x_{2}} +

+(a_{1}v_{x_{1}} + a_{2}v_{x_{2}})v_{t} + (a_{1x_{1}} + a_{2x_{2}} + a_{3t} − a_{4})v_{t}v + +a_{3}v_{t}^{2}]dD, (54)

∫_{D_{1}} e^{−λt}t^{m}uu_{t}dD_{1} = \frac{1}{2} ∫_{D_{1}} e^{−λt}t^{m}(u^{2})_{t}dD_{1} = \frac{1}{2} ∫_{∂D_{1}} e^{−λt}t^{m}u^{2}ν_{0}ds +

+\frac{1}{2} ∫_{D_{1}} e^{−λt}(λt^{m} − mt^{m−1})u^{2}dD_{1} = \frac{1}{2} ∫_{S_{2}} e^{−λt}t^{m}u^{2}ν_{0}ds +

+\frac{1}{2} ∫_{D_{1}} e^{λt}(λt^{m} − mt^{m−1})v_{t}^{2}dD_{1} = \frac{1}{2} ∫_{S_{2}} e^{λt}t^{m}v_{t}^{2}ν_{0}ds +

+\frac{1}{2} ∫_{D_{1}} e^{λt}(λt^{m} − mt^{m−1})v_{t}^{2}dD_{1},  (55)

∫_{D_{1}} e^{λt}[−v_{x_{1}t}v_{x_{1}} − v_{x_{2}t}v_{x_{2}}]dD_{1} = −\frac{1}{2} ∫_{∂D_{1}} e^{λt}[v_{x_{1}}^{2} + v_{x_{2}}^{2}]ν_{0}ds +

+\frac{1}{2} ∫_{D_{1}} e^{λt}λ[v_{x_{1}}^{2} + v_{x_{2}}^{2}]dD_{1}.  (56)

Since v|_{S_{2}} = 0 and the surface S_{2} is characteristic, similarly to equality (34) we have

(t^{m}v_{t}^{2} − v_{x_{1}}^{2} − v_{x_{2}}^{2})|_{S_{2}} = 0.  (57)

Taking into account that ν_{0}|_{S_{1}} < 0, with regard for equalities (54)–(57) we find that

(Lu, v)_{L_{2}(D)} = −\frac{1}{2} ∫_{S_{1}} e^{λt}[v_{x_{1}}^{2} + v_{x_{2}}^{2}]ν_{0}ds + \frac{1}{2} ∫_{S_{2}} e^{λt}[t^{m}v_{t}^{2} − v_{x_{1}}^{2} −

−v_{x_{2}}^{2}]ν_{0}ds + \frac{1}{2} ∫_{D_{1}} e^{λt}[2a_{3} − mt^{m−1} + λt^{m}]v_{t}^{2}dD_{1} + \frac{1}{2} ∫_{D_{1}} e^{λt}λ[v_{x_{1}}^{2} +

154 S. KHARIBEGASHVILI

+v_{x_{2}}^{2}]dD_{1} + ∫_{D_{1}} e^{λt}[a_{1}v_{x_{1}} + a_{2}v_{x_{2}}]v_{t}dD_{1} + ∫_{D_{1}} (a_{1x_{1}} + a_{2x_{2}} + a_{3t} −

−a_{4})v_{t}vdD_{1} ≥ \frac{1}{2} ∫_{D_{1}} e^{λt}[2a_{3} − mt^{m−1} + λt^{m}]v_{t}^{2}dD_{1} +

+\frac{1}{2} ∫_{D_{1}} e^{λt}λ[v_{x_{1}}^{2} + v_{x_{2}}^{2}]dD_{1} − | ∫_{D_{1}} e^{λt}[a_{1}v_{x_{1}} + a_{2}v_{x_{2}}]v_{t}dD_{1} +

+ ∫_{D_{1}} e^{λt}(a_{1x_{1}} + a_{2x_{2}} + a_{3t} − a_{4})v_{t}vdD_{1}.  (58)

Since a_{3} ∈ C(\overline{D}), it follows from condition (46) that for sufficiently large λ

(2a_{3} − mt^{m−1} + λt^{m}|_{D_{1}} ≥ 4δ = const > 0

and thus

\frac{1}{2} ∫_{D_{1}} e^{λt}[2a_{3} − mt^{m−1} + λt^{m}]v_{t}^{2}dD_{1} ≥ 2δ ∫_{D_{1}} e^{λt}v_{t}^{2}dD_{1}.  (59)

Integration by parts gives

∫_{D_{1}} e^{λt}(a_{1x_{1}} + a_{2x_{2}} + a_{3t} − a_{4})v_{t}vdD_{1} = \frac{1}{2} ∫_{∂D_{1}} e^{λt}(a_{1x_{1}} + a_{2x_{2}} +

+a_{3t} − a_{4})v^{2}ν_{0}ds − \frac{1}{2} ∫_{D_{1}} e^{λt}[λ(a_{1x_{1}} + a_{2x_{2}} + a_{3t} − a_{4}) +

+(a_{1x_{1}} + a_{2x_{2}} + a_{3t} − a_{4})_{t}]v^{2}dD_{1},

whence by condition (44) and inequalities ν_{0}|_{S_{1}} < 0 and (45) we find that for sufficiently large λ the inequality

∫_{D_{1}} e^{λt}(a_{1x_{1}} + a_{2x_{2}} + a_{3t} − a_{4})v_{t}vdD_{1} ≥ 0  (60)

is valid.

Using the inequality

(a + b)^{2} ≤ 2a^{2} + 2b^{2}, |ab| ≤ δ|a|^{2} + \frac{1}{4δ} |b|^{2},

we obtain

| ∫_{D_{1}} e^{λt}[a_{1}v_{x_{1}} + a_{2}v_{x_{2}}]v_{t}dD_{1} ≤ δ ∫_{D_{1}} e^{λt}v_{t}^{2}dD_{1} +

+\frac{γ}{2δ} ∫_{D_{1}} e^{λt}(v_{x_{1}}^{2} + v_{x_{2}}^{2})dD_{1},  (61)

where γ = max ( sup_{\overline{D}_{1}} |a_{1}|^{2}, sup_{\overline{D}_{1}} |a_{2}|^{2}).

With regard for (59), (60), and (61), for sufficiently large λ we get from (58) that

(Lu, v)_{L_{2}(D)} ≥ δ ∫_{D_{1}} e^{λt}v_{t}^{2}dD_{1} + (\frac{λ}{2} − \frac{λ}{2δ} ) ∫_{D_{1}} e^{λt}(v_{x_{1}}^{2} + v_{x_{2}}^{2})dD_{1},

which for λ ≥ 2δ + \frac{γ}{δ} yields

(Lu, v)_{L_{2}(D)} ≥ δ ∫_{D_{1}} e^{λt}[v_{t}^{2} + v_{x_{1}}^{2} + v_{x_{2}}^{2}]dD_{1}.  (62)

In the same way as in proving inequality (15) in Lemma 3, from (62) follows inequality (47) which proves Lemma 6.

Definition 3. For F ∈ W_{1−}^{∗}(W_{1−}) we call the function u(v) a strong generalized solution of problem (41), (42) (of problem (43), (44)) of the class L_{2}, if u(v) ∈ L_{2}(D_{1}) and there exists a sequence of functions u_{n}(v_{n}) ∈ E_{1}(E_{1}^{∗}) such that u_{n} → u (v_{n} → v) in the space L_{2}(D_{1}) and L_{1}u_{n} → F (L_{1}^{∗}v_{n} → F) in the space W_{1−}^{∗}(W_{1−}) as n → ∞.

Definition 4. For F ∈ L_{2}(D) we call the function u(v) a strong generalized solution of problem (41), (42) (of problem (43), (44)) of the class W_{1+}(W_{1+}^{∗}), if u(v) ∈ W_{1+}(W_{1+}^{∗}) and there exists a sequence of functions u_{n}(v_{n}) ∈ E_{1}(E_{1}^{∗}) such that u_{n} → u (v_{n} → v) and L_{1}u_{n} → F (L_{1}^{∗}v_{n} → F) in the spaces W_{1+}(W_{1+}^{∗}) and W_{1−}^{∗}(W_{1−}), respectively.

The following theorems are the corollaries of Lemmas 5–7.

Theorem 3. Let conditions (45), (46), and (48) be fulfilled. Then for any F ∈ W_{1−}^{∗}(W_{1−}) there exists a unique strong generalized solution u(v) of problem (41), (42) (of problem (43), (44)) of the class L_{2} for which the estimate

∥u∥_{L_{2}(D_{1})} ≤ c∥F ∥_{W_{1−}^{∗}} (∥v∥_{L_{2}(D_{1})} ≤ c∥F ∥_{W_{1−}})  (63)

with the positive constant c independent of F holds.

Theorem 4. Let conditions (45), (46), and (48) be fulfilled. Then for any F ∈ L_{2}(D_{1}) there exists a unique strong generalized solution u(v) of problem (41), (42) (of problem (43), (44)) of the class W_{1+}(W_{1+}^{∗}) for which estimate (63) is valid.

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(Received 2.02.1995)

Author’s address:

Department of Theoretical Mechanics (4)

Georgian Technical University

77, M. Kostava St., Tbilisi 380075

Georgia