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 x1 , x2 , t let us consider a second-order degenerating hyperbolic equation of the kind
Lu ≡ utt − tm (ux1 x1 + ux2 x2 ) + a1 ux1 + a2 ux2 + a3 ut + a4 u = F, (1)
where aj , j = 1, . . . , 4, F are the given functions and u is the unknown real
function, m = const > 0.
Denote by
2
h
1
2 + m i 2+m
2
D :0<t< 1−
r
, r = (x21 + x22 ) 2 <
2
2+m
a bounded domain lying in a half-space t > 0, bounded above by the characteristic conoid
2
h
2 + m i 2+m
2
S :t= 1−
r
, r≤
2
2+m
of equation (1) with the vertex at the point (0, 0, 1), and below by the base
S0 : t = 0, r ≤
2
2+m
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.
139
c 1998 Plenum Publishing Corporation
1072-947X/98/0300-0139$15.00/0 
140
S. KHARIBEGASHVILI
of that conoid; equation (1) has on S0 a non-characteristic degeneration.
In what follows, the coefficients ai , i = 1, . . . , 4, of equation (1) in D are
assumed to be the functions of the class C 2 (D).
For equation (1), consider a multidimensional version of the characteristic
problem which is formulated as follows: On the domain D, find a solution
u(x1 , x2 , t) of equation (1) satisfying the boundary condition
Œ
(2)
uŒS = 0.
As will be shown below, the following Cauchy problem on finding in D a
solution of equation
L∗ v ≡ vtt − tm (vx1 x1 + vx2 x2 ) − (a1 v)x1 − (a2 v)x2 − (a3 v)t + a4 v = F (3)
by the initial conditions
Œ
Œ
v ŒS = 0, vt ŒS0 = 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
utt − |x2 |m ux1 x1 − ux2 x2 + a1 ux1 + a2 ux2 + a3 ut + a4 u = F,
as well as for a hyperbolic equation of second-order with characteristic degeneration

€
utt − ux1 x1 − |x2 |m ux2 x2 + a1 ux1 + a2 ux2 + a3 ut + a4 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
2
W2 (D), satisfying respectively the boundary condition (2) or (4) and vanishing in some (own for every function) three-dimensional neighborhood of
2
, t = 0 and of the segment I : x1 = x2 = 0,
the circle Γ = S ∩ S0 : r = 2+m
∗
0 ≤ t ≤ 1. Let W+ (W+ ) be a Hilbert space with weight, obtained by closing
the space E(E ∗ ) in the norm
Z
ƒ
‚ 2
2
ut + tm (u2x1 + u2x2 ) + u2 dD.
kuk1 =
D
Denote by W− (W−∗ ) a space with negative norm which is constructed
with respect to L2 (D) and W+ (W+∗ ) [12].
ON SOME MULTIDIMENSIONAL VERSIONS
141
Let n = (ν1 , ν2 , ν0 ) be the unit vector of the outer to ∂D normal, i.e.,
ct). By definition, the derivative
ν1 = cos([
n, x1 ), ν2 = cos([
n, x2 ), ν0 = cos(n,
with respect to the conormal can be calculated on the boundary ∂D of the
domain D for the operator L by the formula
∂
∂
∂
∂
= ν0 − tm ν1
− tm ν 2
.
∂N
∂t
∂x1
∂x2
∂
for the
Remark 1. Since the derivative with respect to the conormal ∂N
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
∂v ŒŒ
∂u ŒŒ
(5)
Π= 0,
Π= 0.
∂N S
∂N S0
Impose on the lower coefficients a1 and a2 in equation (1) the following
restrictions:
Πm
Œ
Mi = sup Œt− 2 ai (x1 , x2 , t)Œ < +∞, i = 1, 2.
(6)
D
Lemma 1. For all functions u ∈ E, v ∈ E ∗ the following inequalities
hold:
(7)
kLukW−∗ ≤ c1 kukW+ ,
∗
kL vkW− ≤ c2 kvkW+∗ ,
(8)
where the positive constants c1 and c2 do not depend respectively on u and
v, k · kW+ = k · kW+∗ = k · k1 .
Proof. By the definition of a negative norm, for u ∈ E with regard for
equalities (2), (4) and (5) we have
−1
kLukW−∗ = sup kvk−1
W ∗ (Lu, v)L2 (D) = sup kvkW ∗ (Lu, v)L2 (D) =
+
∗
v∈W+
−1
= sup kvkW
∗
v∈E ∗
+
Z
D
v∈E ∗
‚
utt v − tm ux1 x1 v − tm ux2 x2 v + a1 ux1 v + a2 ux1 v +
ƒ
−1
+a3 ut v + a4 uv dD = sup kvkW
∗
+
v∈E ∗
ƒ
+
−1
−tm ux2 vν2 ds + sup kvkW
∗
v∈E ∗
+
Z
D
‚
Z
∂D
‚
ut vν0 − tm ux1 vν1 −
− ut vt + tm (ux1 vx1 + ux2 vx2 ) +
ƒ
+a1 ux1 v + a2 ux2 v + a3 ut v + a4 uv dD = sup
v∈E ∗
kvk−1
∗
W+
Z
∂D
∂u
vds +
∂N
142
S. KHARIBEGASHVILI
Z
+ sup kvk−1
W∗
+
v∈E ∗
D
‚
− ut vt + tm (ux1 vx1 + ux2 vx2 ) + a1 ux1 v + a2 ux2 v +
ƒ
+a3 ut v + a4 uv dD = sup
v∈E ∗
kvk−1
∗
W+
Z
D
‚
− ut vt + tm (ux1 vx1 +
ƒ
+ux2 vx2 ) + a1 ux1 v + a2 ux1 v + a3 ut v + a4 uv dD.
(9)
Due to (6) as well as the Cauchy inequality, we have
Z
ŒZ ‚
ƒ ŒŒ h
Œ
(u2t + tm ux2 1 +
− ut vt + tm (ux1 vx1 + ux2 vx2 ) dDŒ ≤
Œ
D
+tm u2x2 )dD
≤ M1
Z
i 12
×
hZ
D
(vt2 + tm vx21 + tm vx22 )dD
D
i 12
≤ kukW+ kvkW+∗ , (10)
ŒZ
ƒ
Œ
Œ [a1 ux1 v + a2 ux2 v + a3 ut v + a4 uv) dD ≤
D
tm u2x1 dD
D
‘ 21
kvkL2 (D) + M2
Z
tm u2x2 dD
D
‘ 12
kvkL2 (D) +
+ sup |a3 | kut kL2 (D) kvkL2 (D) + sup |a4 | kukL2 (D) kvkL2 (D) ≤
D
≤
D
2
X
€
i=1
‘
Mi + sup |a2+i | kukW+ kvkW+∗ = e
ckukW+ kvkW+∗ .
(11)
D
From (9)–(11) it follows that
−1
c) sup kvkW
kLukW−∗ ≤ (1 + e
∗ kukW+ kvkW ∗ = c1 kukW+ ,
+
v∈E ∗
+
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)
ON SOME MULTIDIMENSIONAL VERSIONS
143
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)L2 (D) .
Therefore we have
Z
(Lu, v) = (Lu, v)L2 (D) = [ut vν0 − tm ux1 vν1 − tm ux2 vν2 ]ds +
+
Z
∂D
[a1 ν1 + a2 ν2 + a3 ν0 ]uv ds +
Z
D
∂D
‚
− ut vt + tm ux1 vx1 + tm ux2 vx2 −
ƒ
−u(a1 v)x1 − u(a2 v)x2 − u(a3 v)t + a4 uv dD =
m
−t ux2 vν2 ]ds +
Z
Z
[ut vν0 − tm ux1 vν1 −
∂D
[a1 ν1 + a2 ν2 + a3 ν0 ]uv ds −
∂D
−tm uvx1 ν1 − tm uvx2 ν2 ]ds +
Z
[uvt ν0 −
∂D
Z
‚
uvtt − utm vx1 x1 − utm vx2 x2 −
D
ƒ
−u(a1 v)x1 − u(a2 v)x2 − u(a3 v)t + a4 uv dD =
i
Z h
∂u
∂v ‘
v
−u
+
∂N
∂N
∂D
+(u1 ν1 + a2 ν2 + a3 ν0 )uv ds + (u, L∗ v)L2 (D) .
(13)
Equality (12) follows immediately from equalities (2), (4), (5), and (13).
Consider the conditions
Ω|S ≤ 0,
‚
ƒŒ
tΩt − (λt + m)Ω ŒD ≥ 0,
(14)
where the second inequality holds for sufficiently large λ, and Ω = a1x1 +
a2x2 + a3t − a4 .
Remark 3. It can be easily seen that inequality (14) is the corollary of
the condition
Œ
ΩŒ ≤ const < 0.
D
Lemma 3. Let conditions (6) and (14) be fulfilled. Then for any u ∈ W+
the inequality
 1

ct 2 (m−1) uL
2 (D)
≤ kLukW−∗
with the positive constant c independent of u is valid.
(15)
144
S. KHARIBEGASHVILI
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(x1 , x2 , t) =
Zt
eλτ τ α u(x1 , x2 , τ )dτ
(16)
0
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
Z1
0
t
−2 2
g (t)dt ≤ 4
where f (t) ∈ L2 (0, 1) and g(t) =
By (16), the inequalities
Rt
0
Z1
f 2 (t)dt,
0
f (τ )dτ .
vt (x1 , x2 , t) = eλt tα u(x1 , x2 , t), u(x1 , x2 , t) = e−λt t−α vt (x1 , x2 , t) (17)
are valid.
With regard for (2), (4), (5), and (17) we have
Z
Z h
i
∂u
+ (a1 ν1 + a2 ν2 + a3 ν0 )uv ds + [−ut vt +
(Lu, v)L2 (D) =
v
∂N
D
∂D
+tm ux1 vx1 + tm ux2 vx2 − u(a1 v)x1 − u(a2 v)x2 − u(a3 v)t + a4 uv]dD =
Z
Z
= − eλt tα uut dD + e−λt t−α [tm (vx1 t vx1 + vx2 t vx2 ) −
D
D
−(a1 vx1 + a2 vx2 )vt − (a1x1 + a2x2 + a3t − a4 )vt v − a3 vt2 ]dD.
(18)
By virtue of (2) we find that
Z
Z
Z
1
1
eλt tα (u2 )t dt = −
eλt tα u2 ν0 ds +
− e−λt tα uut dD = −
2
2
D
D
∂D
Z
Z
1
1
+
eλt (αtα−1 + λtα )u2 dD =
eλt (αtα−1 + λtα )u2 dD =
2
2
D
D
Z
Z
α
1
=
(19)
eλt tα−1 u2 dD +
λe−λt t−α vt2 dD,
2
2
D
D
Z
Z
1
−λt m−α
e−λt tm−α (vx21 +
e t
(−vx1 t vx1 + vx2 t vx2 )dD =
2
D
∂D
ON SOME MULTIDIMENSIONAL VERSIONS
+vx22 )ν0 ds +
1
≥
2
1
2
Z
‚
ƒ
e−λt λtm−α + (α − m)tm−α−1 (vx21 + vx22 )dD ≥
D
Z
‚
ƒ
e−λt λtm−α + (α − m)tm−α−1 (vx21 + vx22 )dD.
D
145
(20)
In deriving inequality (20) we have taken into account that
ν0 |S ≥ 0,
(vx21 + vx22 )|S0 = 0.
From (19) we have
Z
Z
 1 (α−1) 2
α
1
λt α

2
− e t uut dD ≥
t
λe−λt t−α vt2 dD.
u L (D) +
2
2
2
D
(21)
D
Below we assume that the parameter α = m.
By (6) we obtain
Z
Œ
ŒZ
h
1
Œ
Œ
Œ e−λt t−m (a1 vx1 + a2 vx2 )vt dDŒ ≤ M e−λt t−m vt2 + tm (vx21 +
2
D
D
Z
Z
i
M
2
−λt −m 2
+vx2 ) dD ≤ M e t vt dD +
e−λt (vx21 + vx22 )dD, (22)
2
D
D
where M =Πmax(M1 , M2 ).
Since ν0 ŒS ≥ 0, using conditions (4) and (14) and integrating them by
parts, we obtain
Z
− e−λt t−m (a1x1 + a2x2 + a3t − a4 )vt v dD =
1
−
2
Z
D
−λt −m
e
D
1
+
2
t
Z
D
1
Ω(v )t dD = −
2
2
Z
e−λt t−m Ωv 2 ν0 ds +
∂D
‚
ƒ
e−λt t−m−1 tΩt − (λt + m)Ω v 2 dD ≥ 0.
(23)
−m 2
In deriving inequality (23) we have
Πused the fact that the function t v
−m 2 Œ
has on S0 a zero trace, i.e., t v S0 = 0.
From (18) by virtue of (20)–(23) we have
Z
2
m 1
1
λe−λt t−m vt2 dD +
(Lu, v)L2 (D) ≥ t 2 (m−1) uL2 (D) +
2
2
D
Z
Z
Z
1
M
−λt 2
−λt −m 2
2
λe (vx1 + vx2 )dD − M e t vt dD −
e−λt (vx21 +
+
2
2
D
D
D
146
S. KHARIBEGASHVILI
Z
 1 (m−1) 2
m
t2
uL2 (D) +
2
D
D
Z
λ
‘Z
1
−λt −m 2
+
− M − sup |a3 |
e t vt dD + (λ − M ) e−λt (vx21 +
2
2
D
D
D
Z
2
1
m
2
(m−1)
−λt
2
2
uL (D) + σ e (vt + vx1 + vx22 )dD ≥
+vx2 )dD ≥ t 2
2
2
D
r
Z ‚
 1 (m−1) 2
ƒ ‘ 12
vt2 + tm (vx21 + vx22 ) dD , (24)
≥ 2mσ inf e−λt t 2
uL2 (D)
+vx22 )dD − sup |a3 |
e−λt t−m vt2 dD =
D
D
ƒ
‚λ
where σ = 2 − M − supD |a3 | > 0 for sufficiently large λ, and inf D e−λt =
e−λ > 0. When
Πderiving inequality (24), we have taken into account the
fact that t−m ŒD ≥ 1.
If u ∈ W+ (W+∗ ) and because u|S = 0 (u|S0 = 0), we can easily prove the
inequality
Z
Z
u2 dD ≤ c0 u2t dD
D
D
for which c0 = const > 0 independent of u. Hence we find that in the space
W+ (W+∗ ) the norm
Z
ƒ
‚ 2
ut + tm (u2x1 + u2x2 ) + u2 dD
kuk2W+ (W ∗ ) =
+
D
is equivalent to the norm
2
kuk =
Z
D
ƒ
‚ 2
ut + tm (u2x1 + u2x2 ) dD.
(25)
Therefore, retaining for norm (25) the previous designation kukW+ (W+∗ ) ,
from (24) we have
(Lu, v)L2 (D) ≥
√
1
2mσe−λ kt 2 (m−1) ukL2 (D) kvkW+∗ .
(26)
If now we apply the generalized Schwarz inequality
(Lu, v) ≤ kLukW−∗ kvkW+∗
to the left-hand side of (26), then after reducing by kvkW+∗ we get inequality
√
(15) in which c = 2mσe−λ .
ON SOME MULTIDIMENSIONAL VERSIONS
Consider the conditions
Œ
a4 ŒS ≥ 0,
0
Œ
(λa4 + a4t )ŒD ≥ 0,
147
(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
ckvkL2 (D) ≤ kL∗ vkW−
(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(x1 , x2 , t) =
ϕ(x
Z1 ,x2 )
e−λτ v(x1 , x2 , τ )dτ,
λ = const > 0,
(29)
t
where t = ϕ(x1 , x2 ) is the equation of the characteristic conoid S. Although
2
the function
on the circle r = 2+m
2
h
2 + m i 2+m
ϕ(x1 , x2 ) = 1 −
r
2
has singularities and at the origin x1 = x2 = 0, but by the definition of
the space E ∗ , the function v vanishes in some neighborhood of the circle
Γ = S ∩ S0 and of the segment I : x1 = x2 = 0, 0 ≤ t ≤ 1, the function u
defined by equality (29) will belong to the space E. Moreover, it is obvious
that the equalities
ut (x1 , x2 , t) = −e−λt v(x1 , x2 , t), v(x1 , x2 , t) = −eλt ut (x1 , x2 , t) (30)
hold.
Owing to (2), (4), (5), and (30), we have
Z h
i
∂v
∗
− (a1 ν1 + a2 ν2 + a3 ν0 )vu ds +
(L v, u)L2 (D) =
u
∂N
∂D
Z
+ [−vt ut + tm vx1 ux1 + tm vx2 ux2 + a1 vux1 + a2 vux2 + a3 vut +
D
+a4 uv]dD =
Z
D
e−λt vt v dD −
Z
D
‚
eλt tm (ux1 t ux1 + ux2 t ux2 ) +
ƒ
+(a1 ux1 + a2 ux2 )ut + a3 u2t + a4 uut dD,
(31)
148
S. KHARIBEGASHVILI
Z
e−λt vt v dD =
D
Z
e−λt v 2 ν0 ds +
1
2
Z
e−λt λv 2 dD =
D
∂D
Z
1
−λt 2
e v ν0 ds +
e−λt λv 2 dD =
2
S
D
Z
Z
1
1
λt 2
(32)
=
e ut ν0 ds +
eλt λu2t dD,
2
2
S
D
Z
Z
1
λt m
− e t (ux1 t ux1 + ux2 t ux2 )dD = −
eλt tm (ux2 1 + u2x2 )ν0 ds +
2
D
∂D
Z
1
eλt [λtm + mtm−1 ](u2x1 + u2x2 )dD ≥
+
2
D
Z
Z
1
1
λt m 2
e t (ux1 + ux2 2 )ν0 ds +
λeλt tm (u2x1 + u2x2 ) dD. (33)
≥−
2
2
1
=
2
Z
1
2
D
∂D
Since u|S = 0, for some α we have vt = αν0 , vx1 = αν1 , vx2 = αν2 on S.
Therefore the fact that the surface S is characteristic results in
‚ 2
ƒŒ
‚
ƒŒ
ut − tm (u2x1 + u2x2 ) ŒS = α2 ν02 − tm (ν12 + ν22 ) ŒS = 0.
(34)
Taking into account that m > 0 and hence tm |S0 = 0, equalities (2) and
(34) imply
Z
Z
1
1
λt 2
e ut ν0 ds −
eλt tm (u2x1 + u2x2 )ν0 ds =
2
2
S
∂D
Z
1
(35)
eλt [u2t − tm (ux2 1 + ux2 2 )]ν0 ds = 0.
=
2
S
Due to (32), (33), and (35), equality (31) yields
Z
‚
ƒ
1
∗
(L v, u)L2 (D) ≥
eλt u2t + tm (ux2 1 + u2x2 ) dD −
2
D
Z
‚
ƒ
− eλt (a1 ux1 + a2 ux2 )ut + a3 u2t + a4 uut dD.
D
Œ
Since ν0 ŒS0 < 0, by (27) we have
−
Z
D
1
e a4 uut dD = −
2
λt
Z
D
1
e a4 (u )t dD = −
2
λt
2
Z
∂D
eλt a4 u2 ν0 ds +
(36)
ON SOME MULTIDIMENSIONAL VERSIONS
+
1
2
Z
eλt (λa4 + a4t )u2 dD ≥ 0.
149
(37)
D
Using (6), we obtain
Z
Œ
ŒZ
h
i
1
Œ
Œ
λt
Œ e (a1 ux1 + a2 ux2 )ut dDŒ ≤ M eλt u2t + tm (u2x1 + ux2 2 ) dD, (38)
2
D
D
where M = max(M1 , M2 ).
With regard for (30), (37) and (38), from (36) we get
λ
‘Z
ƒ
‚
eλt u2t + tm (u2x1 + u2x2 ) dD ≥
(L∗ v, u)L2 (D) ≥
− M − sup |a3 |
2
D
D
hZ
i 21 h Z
‚
ƒ i 21
≥γ
eλt ut2 dD
eλt u2t + tm (u2x1 + u2x2 ) dD =
D
=γ
hZ
D
e−λt v 2 dD
D
i 12 h Z
D
≥ γ inf e−λt kvkL2 (D)
D
€λ
‚
ƒ i 12
eλt u2t + tm (u2x1 + ux2 2 ) dD ≥
hZ ‚
ƒ i 21
ut2 + tm (u2x1 + ux2 2 ) dD ,
(39)
D

where γ = 2 − M − supD |a3 | > 0 for sufficiently large λ.
From (39), in just the same way as in obtaining inequality (26), we find
that
(L∗ v, u)L2 (D) ≥ ckvkL2 (D) kukW+ ,
which immediately implies (28).
Denote by L2,α (D) a space of functions u such that tα u ∈ L2 (D). Assume
kukL2,α (D) = kta lukL2 (D), αm =
1
(m − 1).
2
Definition 1. For F ∈ W−∗ we call the function u a strong generalized
solution of problem (1), (2) of the class L2,αm , if u ∈ L2,αm (D) and there
exists a sequence of functions un ∈ E such that un → u in the space
L2,αm (D) and Lun → F in the space W−∗ as n → ∞, i.e.,
lim kun − ukL2,αm (D) = 0,
n→∞
lim kLun − F kW−∗ = 0.
n→∞
Definition 2. For F ∈ L2 (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
150
S. KHARIBEGASHVILI
sequence of functions un ∈ E such that un → u and Lun → F in the spaces
W+ and W−∗ , respectively, i.e.,
lim kun − ukW+ = 0,
n→∞
lim kLun − F kW−∗ = 0.
n→∞
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 L2,αm for which the estimate
kukL2,αm (D) ≤ ckF kW−∗
(40)
with the constant c independent of F is valid.
Theorem 2. Let conditions (6), (14), and (27) be fulfilled. Then for
every F ∈ L2 (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
L1 u ≡ (tm ut )t − ux1 x1 − ux2 x2 + a1 ux1 + a2 ux2 + a3 ut + a4 u = F, (41)
where 1 ≤ m = const < 2.
Denote by
2
h
1
2
2 − m i 2−m
, r = (x21 + x22 ) 2 <
r
D1 : 0 < t < 1 −
2
2−m
a bounded domain lying in a half-space t > 0, bounded above by the characteristic conoid
2
h
2 − m i 2−m
2
r
S2 : t = 1 −
, r≤
2
2−m
of equation (41) with the vertex at the point (0, 0, 1) and below by the base
S1 : t = 0, r ≤
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 ai ,
i = 1, . . . , 4, of equation (41) in the domain D1 are assumed to be the
functions of the class C 2 (D).
For equation (41) let us consider a multidimensional version of the characteristic problem which is formulated as follows: Find in the domain D1 a
solution u(x1 , x2 , t) of equation (41) satisfying the boundary condition
Œ
uŒS = 0
(42)
1
ON SOME MULTIDIMENSIONAL VERSIONS
151
on the plane characteristic surface S1 .
The characteristic problem for the equation
L1∗ v ≡ (tm vt )t − vx1 x1 − vx2 x2 − (a1 v)x1 − (a2 v)x2 − (a3 v)t + a4 v = F (43)
in the domain D1 is formulated analogously for the boundary condition
Œ
v ŒS = 0,
(44)
2
L∗1
where
is the operator conjugate formally to the operator L1 .
Denote by E1 and E1∗ the classes of functions from the Sobolev space
W22 (D1 ), satisfying the corresponding boundary condition (42) or (44) and
vanishing in some (own for every function) three-dimensional neighborhood
∗
of the segment I : x1 = 0, x2 = 0, 0 ≤ t ≤ 1. Let W1+ (W1+
) be the Hilbert
∗
space obtained by closing the space E1 (E1 ) in the norm
Z
2
kuk = [u2t + u2x1 + u2x2 + u2 ]dD1 .
D1
∗
Denote by W1− (W1−
) a space with a negative norm, constructed with re∗
).
spect to L2 (D1 ) and W1+ (W1+
The following lemma can be proved analogously to Lemmas 1 and 2.
Lemma 5. For all functions u ∈ E1 and v ∈ E1∗ the inequalities
∗
≤ c1 kukW1+ ,
kL1 ukW1−
∗ ,
kL∗1 vkW1− ≤ c1 kvkW1+
are fulfilled and problems (41), (42) and (43), (44) are self-conjugate, i.e.,
∗
for every u ∈ W1+ and v ∈ W1+
the equality
(L1 u, v) = (u, L∗1 v)
holds.
Let us consider the conditions
inf (a4 − a1x1 − a2x2 − a3t ) > 0,
(45)
D1
inf a3 >
S1
1
for m = 1, inf a3 > 0 for m > 1.
S1
2
(46)
Lemma 6. Let conditions (45) and (46) be fulfilled. Then for any u ∈
W1+ the inequality
∗ ,
ckukL2 (D1 ) ≤ kL1 ukW1−
with the positive constant c independent of u, is valid.
(47)
152
S. KHARIBEGASHVILI
Consider now the conditions
inf a4 > 0,
(48)
D1
inf a3 > −
S1
1
for m = 1, inf a3 > 0 for m > 1.
S1
2
(49)
Note that condition (46) results in condition (49).
Lemma 7. Let conditions (48) and (49) be fulfilled. Then for any v ∈
∗
W1+
the inequality
ckvkL2 (D1 ) ≤ kL∗1 vkW1−
(50)
with the positive constant c independent of v holds.
Below we will restrict ourselves to proving only Lemma 6. Let u ∈ E1 .
We introduce into consideration the function
v(x1 , x2 , t) =
ψ(x
Z1 ,x2 )
e−λτ u(x1 , x2 , τ )dτ,
λ = const > 0,
(51)
t
where t = ψ(x1 , x2 ) is the equation of the characteristic conoid S2 of equation (41). Since 1 ≤ m < 2, the first and second-order derivatives of the
function
2
h
2 − m i 2−m
ψ(x1 , x2 ) = 1 −
r
2
with respect to the variables x1 and x2 will have singularities at the origin
only. But by the definition of the space E1 , the function u vanishes in
some neighborhood of the segment I : x1 = x2 = 0, 0 ≤ t ≤ 1. Therefore
the function v defined by equality (51) belongs to the space E1∗ , and the
equalities
vt (x1 , x2 , t) = −e−λt u(x1 , x2 , t),
ut (x1 , x2 , t) = −eλt vt (x1 , x2 , t)
(52)
hold.
Since the derivative with respect to the conormal
∂
∂
∂
∂
− ν2
= tm ν0 − ν1
∂N
∂t
∂x1
∂x2
for the operator L1 is an interior differential operator on the characteristic
surfaces of equation (41), because of (42) and (44) for the functions u ∈ E1
and v ∈ E1∗ we have
∂u ŒŒ
∂u ŒŒ
(53)
Π= 0,
Π= 0.
∂N S1
∂N S2
ON SOME MULTIDIMENSIONAL VERSIONS
153
By (42), (44), (52) and (53) we arrive at
Z h
i
∂u
+ (a1 ν1 + a2 ν2 + a3 ν0 )uv ds +
(Lu, v)L2 (D1 ) =
v
∂N
∂D1
Z
‚
+
− tm ut vt + ux1 vx1 + ux2 vx2 − u(a1 v)x1 − u(a2 v)x2 − u(a3 v)t +
D1
ƒ
+a4 uv dD1 =
Z
−λt m
e
t uut dD1 +
D1
Z
D1
‚
eλt − vx1 t vx1 − vx2 t vx2 +
ƒ
+(a1 vx1 + a2 vx2 )vt + (a1x1 + a2x2 + a3t − a4 )vt v + +a3 vt2 dD, (54)
Z
Z
Z
1
1
e−λt tm (u2 )t dD1 =
e−λt tm u2 ν0 ds +
e−λt tm uut dD1 =
2
2
D1
∂D1
D1
Z
Z
1
1
e−λt (λtm − mtm−1 )u2 dD1 =
e−λt tm u2 ν0 ds +
+
2
2
D1
S2
Z
Z
1
1
λt
m
m−1 2
+
)vt dD1 =
e (λt − mt
eλt tm vt2 ν0 ds +
2
2
D1
S2
Z
1
+
eλt (λtm − mtm−1 )vt2 dD1 ,
(55)
2
D1
Z
Z
1
eλt [vx21 + vx22 ]ν0 ds +
eλt [−vx1 t vx1 − vx2 t vx2 ]dD1 = −
2
D1
∂D1
Z
1
+
eλt λ[vx21 + vx22 ]dD1 .
(56)
2
D1
Œ
Since v ŒS2 = 0 and the surface S2 is characteristic, similarly to equality
(34) we have
Œ
(tm vt2 − vx21 − vx22 )ŒS2 = 0.
(57)
Œ
Taking into account that ν0 ŒS1 < 0, with regard for equalities (54)–(57)
we find that
Z
Z
1
1
2
λt 2
(Lu, v)L2 (D) = −
e [vx1 + vx2 ]ν0 ds +
eλt [tm vt2 − vx21 −
2
2
S1
S2
Z
Z
1
1
−vx22 ]ν0 ds +
eλt [2a3 − mtm−1 + λtm ]vt2 dD1 +
eλt λ[vx21 +
2
2
D1
D1
154
S. KHARIBEGASHVILI
+vx22 ]dD1 +
Z
eλt [a1 vx1 + a2 vx2 ]vt dD1 +
Z
(a1x1 + a2x2 + a3t −
D1
D1
Z
1
−a4 )vt vdD1 ≥
eλt [2a3 − mtm−1 + λtm ]vt2 dD1 +
2
D1
Z
ŒZ
1
Œ
λt
2
2
+
e λ[vx1 + vx2 ]dD1 − Œ eλt [a1 vx1 + a2 vx2 ]vt dD1 +
2
D1
D1
Z
+ eλt (a1x1 + a2x2 + a3t − a4 )vt vdD1 .
(58)
D1
Since a3 ∈ C(D), it follows from condition (46) that for sufficiently large λ
Œ
(2a3 − mtm−1 + λtm ŒD ≥ 4δ = const > 0
1
and thus
1
2
Z
eλt [2a3 − mtm−1 + λtm ]vt2 dD1 ≥ 2δ
D1
Z
eλt vt2 dD1 .
(59)
D1
Integration by parts gives
Z
Z
1
eλt (a1x1 + a2x2 +
eλt (a1x1 + a2x2 + a3t − a4 )vt vdD1 =
2
D1
∂D1
Z
‚
1
eλt λ(a1x1 + a2x2 + a3t − a4 ) +
+a3t − a4 )v 2 ν0 ds −
2
D1
ƒ
+(a1x1 + a2x2 + a3t − a4 )t v 2 dD1 ,
Œ
whence by condition (44) and inequalities ν0 ŒS < 0 and (45) we find that
1
for sufficiently large λ the inequality
Z
eλt (a1x1 + a2x2 + a3t − a4 )vt vdD1 ≥ 0
(60)
D1
is valid.
Using the inequality
(a + b)2 ≤ 2a2 + 2b2 ,
we obtain
|ab| ≤ δ|a|2 +
1 2
|b| ,
4δ
Z
ŒZ
Œ
Œ eλt [a1 vx1 + a2 vx2 ]vt dD1 ≤ δ eλt vt2 dD1 +
D1
D1
ON SOME MULTIDIMENSIONAL VERSIONS
+
γ
2δ
Z
eλt (vx21 + vx22 )dD1 ,
155
(61)
D1
€

where γ = max supD1 |a1 |2 , supD1 |a2 |2 .
With regard for (59), (60), and (61), for sufficiently large λ we get from
(58) that
Z
Z
λ
λ‘
(Lu, v)L2 (D) ≥ δ eλt vt2 dD1 +
−
eλt (vx21 + vx22 )dD1 ,
2
2δ
D1
which for λ ≥ 2δ +
γ
δ
D1
yields
(Lu, v)L2 (D) ≥ δ
Z
eλt [vt2 + vx21 + vx22 ]dD1 .
(62)
D1
In the same way as in proving inequality (15) in Lemma 3, from (62) follows
inequality (47) which proves Lemma 6.
∗
(W1− ) we call the function u(v) a strong
Definition 3. For F ∈ W1−
generalized solution of problem (41), (42) (of problem (43), (44)) of the
class L2 , if u(v) ∈ L2 (D1 ) and there exists a sequence of functions un (vn ) ∈
E1 (E1∗ ) such that un → u (vn → v) in the space L2 (D1 ) and L1 un → F
∗
(W1− ) as n → ∞.
(L1∗ vn → F ) in the space W1−
Definition 4. For F ∈ L2 (D) we call the function u(v) a strong generalized solution of problem (41), (42) (of problem (43), (44)) of the class
∗
∗
) and there exists a sequence of functions
W1+ (W1+
), if u(v) ∈ W1+ (W1+
∗
un (vn ) ∈ E1 (E1 ) such that un → u (vn → v) and L1 un → F (L∗1 vn → F ) in
∗
∗
(W1− ), respectively.
) and W1−
the spaces W1+ (W1+
The following theorems are the corollaries of Lemmas 5–7.
Theorem 3. Let conditions (45), (46), and (48) be fulfilled. Then for
∗
any F ∈ W1−
(W1− ) there exists a unique strong generalized solution u(v)
of problem (41), (42) (of problem (43), (44)) of the class L2 for which the
estimate
€

∗
kvkL2 (D1 ) ≤ ckF kW1−
kukL2 (D1 ) ≤ ckF kW1−
(63)
with the positive constant c independent of F holds.
Theorem 4. Let conditions (45), (46), and (48) be fulfilled. Then for
any F ∈ L2 (D1 ) there exists a unique strong generalized solution u(v) of
∗
problem (41), (42) (of problem (43), (44)) of the class W1+ (W1+
) for which
estimate (63) is valid.
156
S. KHARIBEGASHVILI
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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