GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 4, 1997, 341-354

ON THE SOLVABILITY OF THE MULTIDIMENSIONAL VERSION OF THE FIRST DARBOUX PROBLEM FOR A MODEL SECOND-ORDER DEGENERATING HYPERBOLIC

EQUATION

S. KHARIBEGASHVILI

Abstract. A multidimensional version of the first Darboux problem is considered for a model second order degenerating hyperbolic equation. Using the technique of functional spaces with a negative norm, the correct formulation of this problem in the Sobolev weighted space is proved.

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

Lu ≡ 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, (1)

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

Denote by D : x_{2} < t < 1 − x_{2}, 0 < x_{2} < \frac{1}{2}, an unbounded domain lying in a space x_{2} > 0 and bounded by characteristic surfaces S_{1} : t − x_{2} = 0, 0 < x_{2} < \frac{1}{2}, S_{2} : t + x_{2} − 1 = 0, 0 < x_{2} < \frac{1}{2}, of equation (1) and by a plane surface S_{0} : x_{2} = 0, 0 < t < 1, of time type with an equation degenerating on it. The coefficients a_{i}, i = 1, …, 4, of equation (1) in the domain D are assumed to be bounded functions of the class C^{1}(\overline{D}).

For equation (1) let us consider a multidimensional version of the first Darboux problem formulated as follows: find in the domain D a solution u(x_{1}, x_{2}, t) of equation (1) satisfying the boundary condition

u|_{S_{0}∪S_{1}} = 0.  (2)

341 The problem in the domain D for the equation

1991 Mathematics Subject Classification. 35L80.

Key words and phrases. Degenerating hyperbolic equation, multidimensional version of the first Darboux problem, Sobolev’s weighted space, functional space with negative norm.

1072-947X/97/0700-0341$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

L^{∗}v ≡ v_{tt} − x_{2}^{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)

is posed analogously by the boundary condition

v|_{S_{0}∪S_{2}} = 0,  (4)

where L^{∗} is a formally conjugate operator of L.

Note that similar problems for m = 0, when equation (1) is not degenerating and contains, in its principal part, a wave operator, were studied in [1–6]. Other versions of multidimensional Darboux problems can be found in [7–9].

Denote by E, E^{∗} the classes of functions from the Sobolev space W_{2}^{2}(D) satisfying respectively the boundary condition (2) and the boundary condition (4). Let W_{+}(W_{+}^{∗}) be the weighted Hilbert space obtained by closing the space E(E^{∗}) in the norm

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

Denote by W_{−}(W_{−}^{∗}) the space with a negative norm constructed with respect to L_{2}(D) and W_{+}(W_{+}^{∗}) [10].

Consider the condition

M = sup_{\overline{D}} |x_{2}^{− \frac{m}{2}} a_{1}(x_{1}, x_{2}, t)| < ∞  (5)

imposed an the lowest coefficient a_{1} in equation (1).

The uniqueness theorem required for solving problem (1), (2) of the class W_{2}^{2}(D) follows from

Lemma 1. Let condition (5) be fulfilled. Then for every u ∈ W_{2}^{2}(D) satisfying the homogeneous boundary condition

u|_{S_{0}} = 0  (6)

the a priori estimate

∥u∥_{1,+} ≤ c(∥f∥_{1,∗} + ∥F ∥_{L_{2}(D)})  (7)

is valid, where the positive constant c does not depend on u; f = u|_{S_{1}}, F = Lu,

∥f∥_{1,∗}^{2} = ∫_{S_{1}} [x_{2}^{m}f_{x_{1}}^{2} + (\frac{∂f}{∂N} )^{2}]ds,

and \frac{∂}{∂N} = −\frac{1}{2}(\frac{∂}{∂x_{2}} + \frac{∂}{∂t}) is the derivative with respect to the conormal

which is the inner differential operator on the characteristic surface S_{1}.

MULTIDIMENSIONAL VERSION OF THE FIRST DARBOUX PROBLEM 343

Proof. Let n = (ν_{1}, ν_{2}, ν_{0}) be the outer unit vector to ∂D, i.e., ν_{1} = cos({ \atop n, x_{1}}), ν_{2} = cos({ \atop n, x_{2}}), ν_{0} = cos(n̂, t). For the function u ∈ W_{2}^{2}(D) satisfying the boundary condition (6) and λ = const > 0 a simple integration by parts gives

2 ∫_{D} e^{−λt}u_{tt}u_{t} dD = ∫_{∂D} e^{−λt}u_{t}^{2}ν_{0} ds + ∫_{D} λe^{−λt}u_{t}^{2} dD,  (8)

−2 ∫_{D} e^{−λt}(x_{2}^{m}u_{x_{1}x_{1}}u_{t} + u_{x_{2}x_{2}}u_{t})dD =

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

+ ∫ De^{−λt}(x_{2}^{m}u_{x_{1}}^{2} + u_{x_{2}}^{2})ν_{0} ds + ∫_{D} e^{−λt}(λx_{2}^{m}u_{x_{1}}^{2} + λu_{x_{2}}^{2})dD. (9)

It can be easily seen that

\begin{matrix}u|_{S_{0}} = u_{t}|_{S_{0}} = ν_{0}|_{S_{0}} = 0, n|_{S_{1}} = (0, \frac{1}{\sqrt{2}}, −\frac{1}{\sqrt{2}}), \\ ν_{0}|_{S_{2}} > 0, (ν_{0}^{2} − x_{2}^{m}ν_{1}^{2} − ν_{2}^{2})|_{S_{1}∪S_{2}} = 0. \end{matrix}  (10)

Taking into account (8)–(10), multiplying both parts of equation (1) by 2e^{−λt}u_{t}, where F = Lu, and integrating the obtained expression with respect to D, we obtain

2(Lu, e^{−λt}u_{t})_{L_{2}(D)} = ∫_{D} e^{−λt}[λ(u_{t}^{2} + x_{2}^{m}u_{x_{1}}^{2} + u_{x_{2}}^{2}) +

+2(a_{1}u_{x_{1}} + a_{2}u_{x_{2}} + a_{3}u_{t} + a_{4}u)u_{t}]dD + ∫_{S_{1}∪S_{2}} e^{−λt}ν_{0}^{−1}[x_{2}^{m}(ν_{0}u_{x_{1}} −

−ν_{1}u_{t})^{2} + (ν_{0}u_{x_{2}} − ν_{2}u_{t})^{2} + (ν_{0}^{2} − x_{2}^{m}ν_{1}^{2} − ν_{2}^{2})u_{t}^{2}]ds ≥

≥ ∫_{D} e^{−λt}[λ(u_{t}^{2} + x_{2}^{m}u_{x_{1}}^{2} + u_{x_{2}}^{2}) + 2(a_{1}u_{x_{1}} + a_{2}u_{x_{2}} + a_{3}u_{t} +

+a_{4}u)u_{t}]dD − \sqrt{2} ∫_{S_{1}} e^{−λt}[x_{2}^{m}u_{x_{1}}^{2} + (\frac{∂u}{∂N} )^{2}]ds.  (11)

Owing to condition (6) and the structure of D, one can easily verify that the inequality

∫_{D} u^{2}dD ≤ c_{0} ∫_{D} u_{x_{2}}^{2}dD  (12)

is valid for some c_{0} = const > 0 not depending on u ∈ W_{2}^{2}(D).

Using inequality (5), we can show that

|2a_{1}u_{x_{1}}u_{t}| ≤ 2M(x2^{\frac{m}{2}} u_{x_{1}})u_{t} ≤ M(x_{2}^{m}u_{x_{1}}^{2} + u_{t}^{2}).  (13)

By virtue of (12) and (13), from (11) for sufficiently large λ we get

2(Lu, e^{−λt}u_{t})_{L_{2}(D)} ≥ c_{1} ∫_{D} (u_{t}^{2} + x_{2}^{m}u_{x_{1}}^{2} + u_{x_{2}}^{2} + u^{2})dD −

− c_{2} ∫_{S_{1}} [x_{2}^{m}u_{x_{1}}^{2} + (\frac{∂u}{∂N} )^{2}]ds,  (14)

where the positive constants c_{1} and c_{2} do not depend on u; note that depending on λ, the constant c_{1} can be chosen arbitrarily large. Therefore estimate (7) follows obviously from (14).

Remark 1. Since for the principal part of the operator L the derivative

with respect to the conormal \frac{∂}{∂N} = ν_{0} \frac{∂}{∂t} − x_{2}^{m}ν_{1} \frac{∂}{∂x_{1}} − ν_{2} \frac{∂}{∂x_{2}} is the inner

differential operator on the characteristic surfaces of equation (1), by (2) and (4) we have

\frac{∂u}{∂N} |_{S_{1}} = 0, \frac{∂v}{∂N} |_{S_{2}} = 0  (15)

for the functions u ∈ E and v ∈ E^{∗}.

Lemma 2. Let condition (5) be fulfilled. Then for all functions u ∈ E, v ∈ E^{∗} the inequalities

∥Lu∥_{W_{−}^{∗}} ≤ c_{1}∥u∥_{W_{+}},  (16)
∥L^{∗}v∥_{W_{−}} ≤ c_{2}∥v∥_{W_{+}^{∗}}  (17)

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

Proof. According to the definition of a negative norm and because of (2), (4), and (15), 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_{t}vν_{0} − x_{2}^{m}u_{x_{1}}vν_{1} − u_{x_{2}}vν_{2}]ds + sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1} ∫_{D} [−u_{t}v_{t} +

+x_{2}^{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} ∫_{S_{1}∪S_{2}} \frac{∂}{∂N} v ds + sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1} ∫_{D} [−u_{t}v_{t} + x_{2}^{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} +

+x_{2}^{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. (18)

Due to condition (5) and the Schwartz inequality we obtain

| ∫_{D} [−u_{t}v_{t} + x_{2}^{m}u_{x_{1}}v_{x_{1}} + u_{x_{2}}v_{x_{2}}]dD| ≤ 3[ ∫_{D} (u_{t}^{2} + x_{2}^{m}u_{x_{1}}^{2} + u_{x_{2}}^{2})dD]^{\frac{1}{2}} ×

×[ ∫_{D} (v_{t}^{2} + x_{2}^{m}v_{x_{1}}^{2} + v_{x_{2}}^{2})dD]^{\frac{1}{2}} ≤ 3∥u∥_{W_{+}}∥v∥_{W_{+}^{∗}},  (19)

| ∫_{D} (a_{1}u_{x_{1}}v + a_{2}u_{x_{2}}v + a_{3}u_{t}v + a_{4}uv]dD| ≤ M( ∫_{D} x_{2}^{m}u_{x_{1}}^{2}dD)^{\frac{1}{2}} ∥v∥_{L_{2}(D)} +

+ sup_{D} |a_{2}| ∥u_{x_{2}}∥_{L_{2}(D)}∥v∥_{L_{2}(D)} + sup_{D} |a_{3}| ∥u_{t}∥_{L_{2}(D)}∥v∥_{L_{2}(D)} +

+ sup_{D} |a_{4}| ∥u∥_{L_{2}(D)}∥v∥_{L_{2}(D)} ≤ (M + ∑_{i=1}^{4} sup_{D} |a_{i}|)∥u∥_{W_{+}}∥v∥_{W_{+}^{∗}}. (20)

From (18)–(20) it follows that

∥Lu∥_{W_{−}^{∗}} ≤ (3 + M + ∑_{i=2}^{4} sup_{D} |a_{i}|) sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1}∥u∥_{W_{+}}∥v∥_{W_{+}^{∗}} =

= (3 + M + ∑_{i=2}^{4} sup_{D} |a_{i}|)∥u∥_{W_{+}},

which proves inequality (16). Thus Lemma 2 is completely proved, since the proof of inequality (17) repeats that of the inequality (16).

Remark 2. By inequalities (16) and (17), the operator L : W_{+} → W_{−}^{∗}(L : W_{+}^{∗} → W_{−}) with a dense domain of definition E(E^{∗}) admits a closure which is a continuous operator from W_{+}(W_{+}^{∗}) to W_{−}^{∗}(W_{−}). Denoting this closure as previously by L(L^{∗}), we note that it is defined on the whole Hilbert space W_{+}(W_{+}^{∗}).

Lemma 3. Problems (1), (2) and (3), (4) are self-conjugate, i.e., the equality

(Lu, v) = (u, L^{∗}v)  (21)

holds for every u ∈ W_{+} and v ∈ W_{+}^{∗}.

Proof. By Remark 2 it suffices to prove equality (21) when u ∈ E and v ∈ E^{∗}. We have

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

−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} − x_{2}^{m}uv_{x_{1}}ν_{1} −

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

+(a_{1}ν_{1} + a_{2}ν_{2} + a_{3}ν_{3}]uv]ds + (u, L^{∗}v)_{L_{2}(D)}.  (22)

By (2), (4), and (15), equality (21) follows directly from (22), which proves Lemma 3.

Lemma 4. Let condition (5) be fulfilled. Then for every u ∈ W_{+} we have the inequality

c∥u∥_{L_{2}(D)} ≤ ∥Lu∥_{W_{−}^{∗}}  (23)

with the positive constant c not depending on u.

Proof. By Remark 2 it suffices to prove the inequality (23) when u ∈ E. If u ∈ E, then it can be easily verified that the function

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

where t = φ_{2}(x_{1}, x_{2}) is the equation of the characteristic surface S_{2}, belongs to the space E^{∗}, and the equalities

v_{t}(x_{1}, x_{2}, t) = −e^{−λt}u(x_{1}, x_{2}, t), u(x_{1}, x_{2}, t) = −e^{λt}v_{t}(x_{1}, x_{2}, t) (24)

are valid. By (10), (15) and (24) 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} + x_{2}^{m}u_{x_{1}}v_{x_{1}} + u_{x_{2}}v_{x_{2}} − ua_{1x_{1}}v − ua_{1}v_{x_{1}} − ua_{2x_{2}}v −

−ua_{2}v_{x_{2}} − ua_{3t}v − ua_{3}v_{t} + a_{4}uv]dD = ∫_{D} e^{−λt}u_{t}u dD +

+ ∫_{D} e^{λt}[−x_{2}^{m}v_{x_{1}t}v_{x_{1}} − v_{x_{2}t}v_{x_{2}} + a_{1x_{1}}v_{t}v + a_{1}v_{t}v_{x_{1}} + a_{2x_{2}}v_{t}v +

+a_{2}v_{t}v_{x_{2}} + a_{3t}v_{t}v + a_{3}v_{t}^{2} − a_{4}v_{t}v]dD.  (25)

Analogously to (8) and (9), because of (2) we have

∫_{D} e^{−λt}u_{t}u dD = \frac{1}{2} ∫_{∂D} e^{−λt}u^{2}ν_{0} ds + \frac{1}{2} ∫_{D} e^{−λt}λu^{2} dD =

= \frac{1}{2} ∫_{S_{2}} e^{−λt}u^{2}ν_{0} ds + \frac{1}{2} ∫_{D} e^{λt}λv_{t}^{2} dD =

= \frac{1}{2} ∫_{S_{2}} e^{λt}v_{t}^{2}ν_{0} ds + \frac{1}{2} ∫_{D} e^{λt}λv_{t}^{2} dD,  (26)

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

+\frac{1}{2} ∫_{D} e^{λt}λ[x_{2}^{m}v_{x_{1}}^{2} + v_{x_{2}}^{2}]dD.  (27)

348 S. KHARIBEGASHVILI

Since v|_{S_{2}} = 0, for some α on S_{2} we have

v_{t} = αν_{0}, v_{x_{1}} = αν_{1}, v_{x_{2}} = αν_{2}.

Therefore, since the surface S_{2} is characteristic, we have

(v_{t}^{2} − x_{2}^{m}v_{x_{1}}^{2} − v_{x_{2}}^{2})|_{S_{2}} = α^{2}(ν_{0}^{2} − x_{2}^{m}ν_{1}^{2} − ν_{2}^{2})|_{S_{2}} = 0.  (28)

Due to the fact that ν_{0}|_{S_{0}} = 0, ν_{0}|_{S_{1}} < 0 and owing to equalities (4) and (28), we find that

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

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

−\frac{1}{2} ∫_{S_{2}} e^{λt}[x_{2}^{m}v_{x_{1}}^{2} + v_{x_{2}}^{2}]ν_{0} ds ≥ \frac{1}{2} ∫_{S_{2}} e^{λt}v_{t}^{2}ν_{0} ds − \frac{1}{2} ∫_{S_{2}} e^{λt}[x_{2}^{m}v_{x_{1}}^{2} +

+v_{x_{2}}^{2}]ν_{0} ds = \frac{1}{2} ∫_{S_{2}} e^{λt}[v_{t}^{2} − x_{2}^{m}v_{x_{1}}^{2} − v_{x_{2}}^{2}]ν_{0} ds = 0.  (29)

Taking into consideration (26), (27), and (29), from (25) we get

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

−\frac{1}{2} ∫_{∂D} e^{λt}[x_{2}^{m}v_{x_{1}}^{2} + v_{x_{2}}^{2}]ν_{0} ds + \frac{1}{2} ∫_{D} e^{λt}λ[x_{2}^{m}v_{x_{1}}^{2} + v_{x_{2}}^{2}]dD +

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

≥ \frac{λ}{2} ∫_{D} e^{λt}[v_{t}^{2} + x_{2}^{m}v_{x_{1}}^{2} + v_{x_{2}}^{2}]dD − | ∫_{D} e^{λt}[a_{1}v_{t}v_{x_{1}} +

+a_{2}v_{t}v_{x_{2}} + a_{3}v_{t}^{2} + (a_{1x_{1}} + a_{2x_{2}} + a_{3t} − a_{4})v_{t}v]dD|.  (30)

Putting

μ = max ( sup_{D} |a_{2}|, sup_{D} |a_{3}|, sup_{D} |a_{1x_{1}} + a_{2x_{2}} + a_{3t} − a_{4}|)

MULTIDIMENSIONAL VERSION OF THE FIRST DARBOUX PROBLEM 349

and taking into account (5), we find that

| ∫_{D} e^{λt}[a_{1}v_{t}v_{x_{1}} + a_{2}v_{t}v_{x_{2}} + a_{3}v_{t}^{2} + (a_{1x_{1}} + a_{2x_{2}} + a_{3t} − a_{4})v_{t}v]dD| ≤

≤ ∫_{D} e^{λt}[M \frac{1}{2}(x_{2}^{m}v_{x_{1}}^{2} + v_{t}^{2}) + \frac{μ}{2}(v_{x_{2}}^{2} + v_{t}^{2}) + μv_{t}^{2} + \frac{μ}{2}(v^{2} + v_{t}^{2})]dD =

= ∫_{D} e^{λt}[(\frac{1}{2}M + 2μ)v_{t}^{2} + \frac{1}{2}Mx_{2}^{m}v_{x_{1}}^{2} + \frac{μ}{2}v_{x_{2}}^{2} + \frac{μ}{2}v^{2}]dD ≤

≤ (\frac{1}{2}M + 2μ) ∫_{D} e^{λt}[v_{t}^{2} + x_{2}^{m}v_{x_{1}}^{2} + v_{x_{2}}^{2} + v^{2}]dD.  (31)

Since the function e^{\frac{λ}{2} t}v|_{S_{0}} = 0, by virtue of inequality (12) we have

∫_{D} e^{λt}v^{2}dD ≤ c_{0} ∫_{D} e^{λt}v_{x_{2}}^{2}dD ≤ c_{0} ∫_{D} e^{λt}[v_{t}^{2} + x_{2}^{m}v_{x_{1}}^{2} + v_{x_{2}}^{2}]dD

and, consequently,

∫_{D} e^{λt}[v_{t}^{2} + x_{2}^{m}v_{x_{1}}^{2} + v_{x_{2}}^{2}]dD ≥ \frac{1}{1 + c_{0}} ∫_{D} e^{λt}[v_{t}^{2} + x_{2}^{m}v_{x_{1}}^{2} + v_{x_{2}}^{2} + v^{2}]dD. (32)

By (31), (32), and (24) from (30) we obtain

(Lu, v)_{L_{2}(D)} ≥ \frac{λ}{2(1 + c_{0})} ∫_{D} e^{λt}[v_{t}^{2} + x_{2}^{m}v_{x_{1}}^{2} + v_{x_{2}}^{2} + v^{2}]dD −

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

= (\frac{λ}{2(1 + c_{0})} − \frac{1}{2}M − 2μ) ∫_{D} e^{λt}[v_{t}^{2} + x_{2}^{m}v_{x_{1}}^{2} + v_{x_{2}}^{2} + v^{2}]dD ≥

≥ σ[ ∫_{D} e^{λt}v_{t}^{2}dD]^{\frac{1}{2}} [ ∫_{D} [v_{t}^{2} + x_{2}^{m}v_{x_{1}}^{2} + v_{x_{2}}^{2} + v^{2}]dD]^{\frac{1}{2}} =

= σ[ ∫_{D} e^{−λt}u^{2}dD]^{\frac{1}{2}} ∥v∥_{W_{+}^{∗}} ≥ σ · inf_{D} e^{−λt}∥u∥_{L_{2}(D)}∥v∥_{W_{+}^{∗}},  (33)

where σ = (\frac{λ}{2(1 + c_{0})} − \frac{1}{2}M − 2μ) > 0 for sufficiently large λ, while inf_{D} e^{−λt} = const > 0 owing to the structure of the domain D.

If now we apply the generalized Schwartz inequality

(Lu, v)_{L_{2}(D)} ≤ ∥Lu∥_{W_{−}^{∗}} ∥v∥_{W_{−}^{∗}}

to the left-hand side of (33), then after reduction by ∥v∥_{W_{+}^{∗}} we obtain inequality (23), where c = σ inf_{D} e^{−λt} = const > 0.

Lemma 5. Let condition (5) be fulfilled. Then for every v ∈ W_{+}^{∗} the inequality

c∥v∥_{L_{2}(D)} ≤ ∥L^{∗}v∥_{W_{−}}  (34)

is valid for some c = const > 0 not depending on v ∈ W_{+}^{∗}.

Proof. As in Lemma 4, by Remark 2 it suffices to prove the validity of inequality (34) for v ∈ E^{∗}. Let v ∈ E^{∗} and introduce into consideration the function

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

where t = φ_{1}(x_{1}, x_{2}) is the equation of the characteristic surface S_{1}. It is easily seen that the function u(x_{1}, x_{2}, t) belongs to the class E, and we have 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). (35)

Because of (10), (15), and (35) 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})uv]ds +

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

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

Similarly to (26)–(29) we can prove the equalities

− ∫_{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_{1}} e^{−λt}u_{t}^{2}ν_{0} ds + \frac{1}{2} ∫_{D} e^{−λt}λu_{t}^{2} dD,  (37)

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

+\frac{1}{2} ∫_{D} e^{−λt}λ[x_{2}^{m}u_{x_{1}}^{2} + u_{x_{2}}^{2}]dD,  (38)
(u_{t}^{2} − x_{2}^{m}u_{x_{1}}^{2} − u_{x_{2}}^{2})|_{S_{1}} = 0,  (39)

−\frac{1}{2} ∫_{S_{1}} e^{−λt}u_{t}^{2}ν_{0} ds + \frac{1}{2} ∫_{∂D} e^{−λt}[x_{2}^{m}u_{x_{1}}^{2} + u_{x_{2}}^{2}]ν_{0} ds =

= −\frac{1}{2} ∫_{S_{1}} e^{−λt}u_{t}^{2}ν_{0} ds + \frac{1}{2} ∫_{S_{1}} e^{−λt}[x_{2}^{m}u_{x_{1}}^{2} + u_{x_{2}}^{2}]ν_{0} ds +

+\frac{1}{2} ∫_{S_{2}} e^{−λt}[x_{2}^{m}u_{x_{1}}^{2} +u_{x_{2}}^{2}]ν_{0} ds ≥

≥ −\frac{1}{2} ∫_{S_{1}} e^{−λt}[u_{t}^{2} − x_{2}^{m}u_{x_{1}}^{2} − u_{x_{2}}^{2}]ν_{0} ds = 0.  (40)

To obtain inequality (40) we have used the fact that ν_{0}|_{S_{2}} > 0. Owing to (37)–(40), from (36) we get

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

+ ∫_{D} e^{−λt}[a_{1}u_{x_{1}} + a_{2}u_{x_{2}} + a_{3}u_{t} + a_{4}u]u_{t} dD ≥ \frac{λ}{2} ∫_{D} e^{−λt}[u_{t}^{2} + x_{2}^{m}u_{x_{1}}^{2} +

+u_{x_{2}}^{2}]dD − | ∫_{D} e^{−λt}[a_{1}u_{x_{1}} + a_{2}u_{x_{2}} + a_{3}u_{t} + a_{4}u]u_{t} dD|.

whence, as in obtaining inequality (33) from (30), we have

(L^{∗}v, u)_{L_{2}(D)} ≥ [\frac{λ}{2(1 + c_{0})} − (\frac{1}{2}M + max_{i=2,3,4} sup_{D} |a_{i}|)] ×

× inf_{D} e^{−λt}∥v∥_{L_{2}(D)}∥u∥_{W_{+}}.

Inequality (34) follows directly from the above inequality for sufficiently large λ.

Definition 1. If F ∈ L_{2}(D), then the function u will be called 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 respectively in the spaces W_{+} and W_{−}^{∗} as n → ∞, i.e.,

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

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

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

According to the results of [11], the theorems below are consequences of Lemmas 2–5.

Theorem 1. Let condition (5) be fulfilled. Then for every F ∈ W_{−}^{∗} there exists a unique strong generalized solution u of problem (1), (2) of the class L_{2} for which the estimate

∥u∥_{L_{2}(D)} ≤ c∥F ∥_{W_{−}^{∗}},  (41)

with a positive constant c not depending on F, is valid.

Theorem 2. Let condition (5) 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 (41) is valid.

Proof. The existence of a solution of problem (1), (2) in Theorem 2 follows, for example, from the arguments as follows. By virtue of inequality (34), the functional (F, v)_{L_{2}(D)} can be regarded as a linear continuous functional of L^{∗}v, where v ∈ E^{∗}, F ∈ L_{2}(D). Indeed, using this inequality, we have

|(F, v)_{L_{2}(D)}| ≤ ∥F ∥_{L_{2}(D)}∥v∥_{L_{2}(D)} ≤ c^{∗}∥L^{∗}v∥_{W_{−}}, c^{∗} = const > 0.

By the Khan–Banach theorem, this functional can be linearly and continuously extended into the whole space W_{−}. Following the theorem on a general type of a linear continuous functional over W_{−}, there exists a function u ∈ W_{+} such that

(u, L^{∗})_{L_{2}(D)} = (F, v)_{L_{2}(D)}, v ∈ E^{∗}.  (42)

Equality (42) means that u is a weak generalized solution of the problem (1), (2). Let us now show that this solution is also a strong generalized solution of problem (1), (2) of the class W_{+}.

Since the space E is dense in W_{+}, there exists a sequence u_{n} ∈ E of functions such that

lim_{n→∞} ∥u_{n} − u∥_{W_{+}} = 0.  (43)

Using equalities (21) and (42), we have

(u_{n} − u, L^{∗}v)_{L_{2}(D)} = (Lu_{n} − F, v)_{L_{2}(D)}.  (44)

Now, according to the generalized Schwartz inequality,

|(u_{n} − u, L^{∗}v)_{L_{2}(D)}| ≤ ∥u_{n} − u∥_{W_{+}}∥L^{∗}v∥_{W_{−}}.  (45)

It follows from (43)–(45) that in the space W_{−}^{∗} the sequence Lu_{n} of functions converges weakly to the function F. But since this sequence, because of (16) and (43), converges in the norm of the space W_{−}^{∗}, we obtain

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

Consequently, the function u is a strong generalized solution of problem (1), (2) of the class W_{+}.

This fact can be proved in a different way. Indeed, using equalities (21) and (42) and inequality (17), we have

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

= sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1}[(Lu_{n}, v)_{L_{2}(D)} − (F, v)_{L_{2}(D)}] =

= sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1}[(u_{n}, L^{∗}v)_{L_{2}(D)} − (u, L^{∗}v)_{L_{2}(D)}] =

= sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1}(u_{n} − u, L^{∗}v)_{L_{2}(D)} ≤

≤ sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1}∥u_{n} − u∥_{W_{+}}∥L^{∗}v∥_{W_{−}} ≤

≤ sup_{v∈E^{∗}} ∥v∥_{W_{+}^{∗}}^{−1}∥u_{n} − u∥_{W_{+}}c_{2}∥v∥_{W_{+}^{∗}} = c_{2}∥u_{n} − u∥_{W_{+}},

whence lim_{n→∞} ∥Lu_{n} − F ∥_{W_{−}^{∗}} = 0.

The uniqueness of a strong generalized solution of problem (1), (2) of the class W_{+} in Theorem 2 as well as estimate (41) follow from inequality (23).

As for Theorem 1, it can be proved as follows. Since the space L_{2}(D) is dense in the space W_{−}^{∗}, for every element F ∈ W_{−}^{∗} there exists a sequence F_{n} ∈ L_{2}(D) of functions such that lim_{n→∞} ∥F_{n} − F ∥_{W_{−}^{∗}} = 0. According to Theorem 2, for every function F_{n} ∈ L_{2}(D) there exists a unique strong generalized solution u_{n} of problem (1), (2) of the class W_{+}. Furthermore, using inequality (23) and passing to the limit, we obtain the existence and the uniqueness of a strong generalized solution of problem (1), (2) of the class L_{2} as well as estimate (41).

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(Received 23.11.1994)

Author’s address: Department of Theoretical Mechanics (4) Georgian Technical University 77, M. Kostava Str., Tbilisi 380075 Georgia