Electronic Journal of Differential Equations, Vol. 2006(2006), No. 103, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
ON A MIXED NONLOCAL PROBLEM FOR A WAVE
EQUATION
SERGEI A. BEILIN
Abstract. A nonlocal problem for a wave equation with integral condition is
studied in a cylinder. The uniqueness and existence of a generalized solution
is proved with the help of an a priori estimate and the Galerkin procedure,
respectively.
1. Introduction
Certain problems of modern physics and technology can be effectively described
in terms of nonlocal problems for partial differential equations. The history of
nonlocal problems with integral conditions for partial differential equations goes
back to [3]. Cannon studied a problem for a heat equation, and in most papers,
devoted to nonlocal problems, parabolic and elliptic equations were studied. Mixed
problems with nonlocal integral conditions for one-dimensional hyperbolic equations
were considered in [7, 2, 4, 1, 8, 9].
Nonlocal problem for a hyperbolic equation with n space variables with a different integral condition was considered in [5]
In this paper we consider an n-dimensional wave equation with weighted integral
nonlocal condition. Consider a wave equation
utt − ∆u + c(x, t)u = f (x, t)
(1.1)
n
in a cylinder Q = Ω × (0, T ), where Ω ∈ R is a bounded domain with a smooth
boundary, with Cauchy conditions
u(x, 0) = φ(x), ut (x, 0) = ψ(x),
and an integral nonlocal condition
Z tZ
∂u +
K(x, ξ, τ )u(ξ, τ )dξdτ = 0,
∂n S
0
Ω
(1.2)
x ∈ ∂Ω,
(1.3)
where φ(x), ψ(x), K(x, ξ, τ ) are given functions.
c 1 (Q) = {v(x, t) : v ∈ W 1 (Q), v(x, T ) = 0}. Let u(x, t) be a solution
Denote W
2
2
c 1 (Q) and
of the problem. Multiply the equation (1.1) by the function v(x, t) ∈ W
2
2000 Mathematics Subject Classification. 35L05, 35L20, 35L99.
Key words and phrases. Nonlocal problem; integral condition; wave equation.
c
2006
Texas State University - San Marcos.
Submitted April 13, 2006. Published September 8, 2006.
1
2
S. A. BEILIN
EJDE-2006/103
integrate it over the cylinder Q. After integrating by parts, we obtain
Z TZ
(∇u∇v − ut vt + cuv) dx dt
0
Ω
T Z
Z
T
Z
Z
f vdx dt +
=
0
0
Ω
∂Ω
∂u
vds dt +
∂n
Z
ψ(x)v(x, 0)dx.
Ω
After substituting the condition (1.3), we have:
Z TZ
(∇u∇v − ut vt + cuv) dx dt
0
Ω
T Z
Z
Z tZ
+
v(x, t)
0
Z
T
0
Ω
Z
Z
=
f vdx dt +
0
(1.4)
K(x, ξ, τ )u(ξ, τ )dξdτ ds dt
∂Ω
Ω
ψ(x)v(x, 0)dx.
Ω
Definition 1.1. A function u(x, t) ∈ W21 (Q) is called a generalized solution of
c 1 (Q) and u(x, 0) = φ(x).
(1.1)-(1.2)-(1.3), if it satisfies (1.4) for every v ∈ W
2
The main result of this paper can be formulated as the following theorem.
Theorem 1.2. If φ(x) ∈ W21 (Ω), ψ(x) ∈ L2 (Ω), f (x, t) ∈ L2 (Q), K(x, ξ, t) ∈
C(Ω × Ω × (0, T )), there exist ∂K
∂ξi , i = 1 . . . n and
max |K| ≤ K0 ,
Q̄
∂K ≤ K1
max ∂ξi
Q̄
then there exists a unique generalized solution to the problem (1.1)-(1.2)-(1.3).
The following two sections provide the proof of the above theorem.
2. Uniqueness
Let u1 6= u2 be solutions of (1.1)-(1.2)-(1.3); then u = u1 − u2 is a solution of
the same problem with f = φ = ψ = 0; so, u(x, t) satisfies
Z TZ
(∇u∇v − ut vt + cuv) dx dt
0
Ω
(2.1)
Z TZ
Z tZ
+
v(x, t)
0
∂Ω
K(x, ξ, τ )u(ξ, τ )dξdτ ds dt = 0.
0
Ω
Consider the function
(R τ
v(x, t) =
u(x, η)dη, 0 ≤ t ≤ τ,
0,
τ ≤ t ≤ T.
t
(2.2)
c 1 (Q), and vt (x, t) = −u(x, t) ∀t ∈ [0, τ ]. Integration by parts
Note, that v(x, t) ∈ W
2
in (2.1) will give us:
Z TZ
Z τZ
Z τZ
Z
∇vt ∇vdx dt =
∇v∇vt dx dt −
∇u∇vdx dt = −
|∇v|2 |τ0 dx,
0
Ω
0
Ω
0
Ω
Ω
EJDE-2006/103
ON A MIXED NONLOCAL PROBLEM
3
hence,
T
Z
Z
∇u∇vdx dt = −
0
T
Z
Ω
Z
Z
τ
|∇v(x, 0)|2 dx;
Ω
Z
τ
Z
ut vt dx dt = −
0
Z
1
2
Z
Ω
0
Z
uut dx dt −
ut udx dt =
Ω
0
Ω
u2 |τ0 dx,
Ω
hence,
T
Z
Z
ut vt dx dt =
0
Ω
1
2
Z
u2 (x, τ )dx.
Ω
Thus, (2.1) will take the form:
Z
1
|∇v(x, 0)|2 + u2 (x, τ ) dx dt
2 Ω
Z TZ
Z tZ
=
v(x, t)
K(x, ξ, τ )u(ξ, τ )dξ dτ ds dt.
0
∂Ω
0
Ω
With the help of elementary inequalities we obtain the following estimates:
Z
1
|∇v(x, 0)|2 + u2 (x, τ ) dx dt
2 Ω
Z TZ
Z tZ
≤
|v(x, t)|
|K(x, ξ, τ )||u(ξ, τ )|dξdτ ds dt
0
Z
∂Ω
T Z
0
Ω
Z tZ
≤ K0
|v(x, t)|
|u(ξ, τ )|dξdτ ds dt
0
Ω
Z TZ
Z tZ
2
2
≤ K0
v (x, t) + T |Ω|
u (ξ, τ )dξdτ ds dt
0
Ω
Z0 τ Z ∂Ω
Z τZ
2
2
= K0
v (x, t)ds dt + K0 T |Ω||∂Ω|
u2 (x, t)dx dt.
0
∂Ω
0
∂Ω
0
(2.3)
Ω
Using the inequality
Z
v 2 (x, t)ds ≤
∂Ω
Z
|∇v|2 + c()v 2 dx
Ω
and denoting L = T 2 |Ω||∂Ω|, we obtain
Z
Z
|∇v(x, 0)|2 + u2 (x, τ ) dx dt ≤ K0
Ω
τ
Z
0
It’s trivial that
v2 =
Z
τ
u(x, η)dη
2
|∇v|2 + c()v 2 + Lu2 dx dt.
Ω
Z
≤τ
t
τ
u2 (x, η)dη,
0
and
Z
τ
Z
2
Z
v dx dt ≤
0
Ω
0
τ
Z Z
τ
Ω
τ
2
u (x, η)dη dx dt ≤ τ 2
0
Z
0
τ
Z
u2 (x, t)dx dt.
Ω
Using this result and denoting C0 = K0 max{, c(), L}, we obtain the following
inequality:
Z
Z τZ
2
2
|∇v(x, 0)| + u (x, τ ) dx dt ≤ C0
|∇v(x, t)|2 + u2 (x, t) dx dt. (2.4)
Ω
0
Ω
4
S. A. BEILIN
EJDE-2006/103
Consider the function
t
Z
w(x, t) =
u(x, η)dη.
0
Then v(x, t) = w(x, τ ) − w(x, t), ∇v(x, 0) = ∇w(x, τ ), and also
|∇v|2 = |∇w( x, τ ) − ∇w(x, t)|2 ≤ 2|∇w(x, τ )|2 + 2|∇w(x, t)|2 ,
so we have
Z
0
τ
Z
|∇v|2 dx dt ≤ 2τ
Ω
Z
|∇w(x, τ )|2 dx + 2
Z
0
Ω
Thus, the inequality (2.4) takes the form
Z
|∇w(x, τ )|2 + u2 (x, τ ) dx
Ω
Z
Z
2
≤ 2C0 τ
|∇w(x, τ )| dx + 2C0
τ
Z
|∇w|2 dx dt.
Ω
(2.5)
Z
0
Ω
τ
|∇w|2 + u2 dx dt.
Ω
Because of arbitrariness of τ , let τ satisfy 2C0 τ < 1. Then it follows from (2.5):
Z
Z τZ
(1 − 2C0 τ )
|∇w(x, τ )|2 + u2 (x, τ ) dx ≤ 2C0
|∇w|2 + u2 dx dt. (2.6)
Ω
0
Ω
Applying the Gronwall inequality, we conclude that
Z
1
],
|∇w(x, τ )|2 + u2 (x, τ ) dx ≤ 0 ∀τ ∈ [0,
2C0
Ω
whence it follows that
1
].
2C0
k
Obtaining the same inequality for the “slices” τ ∈ (k−1)
2C0 , 2C0 to cover the [0, T ],
we ensure that
u(x, τ ) = 0 ∀τ ∈ [0, T ].
Thus, the uniqueness is proved.
u(x, τ ) = 0
∀τ ∈ [0,
3. Existence
To prove the existence, we will apply the Galerkin method. Let wk (x) be a
fundamental system in W21 (Ω) such that (wk , wl )L2 (Ω) = δk,l .
We will try to find approximate solutions in the form
m
X
um (x, t) =
dk (t)wk (x),
(3.1)
k=1
where coefficients dk (t) are to be determined, from
Z
m
m
(um
tt wl + ∇u ∇wl + cu wl ) dx
Ω
Z
Z tZ
Z
+
wl (x)
K(x, ξ, τ )um (ξ, τ )dξdτ ds =
f wl dx.
∂Ω
0
Ω
dk (0) = αk ,
where αk are coefficients of sums φm (x) =
φ(x) as m → ∞ with respect to the norm
(3.2)
Ω
d0 (0) = βk ,
(3.3)
Pkm
k=1 αk wk , approximating the function
of W21 (Ω), and βk = (ψ, wk )L2 (Ω) .
EJDE-2006/103
ON A MIXED NONLOCAL PROBLEM
5
Substituting (3.1) into (3.2), we obtain
Z X
m
(d00k wk wl + dk ∇wk ∇wl + cdk wk wl ) dx
Ω k=1
Z tZ
Z
+
wl
∂Ω
K(x, ξ, τ )
0
Ω
m
X
dk (τ )wk (ξ)dξdτ ds = fl (t),
k=1
where fl (t) = (f, wl )L2 (Ω) . Changing the order of summation and integration, we
have
m
X
(d00k (t)(wk , wl )L2 + dk (t)(∇wk , ∇wl )L2 + dk (t)(cwk , wl )L2 ))
k=1
+
m Z t
X
k=1
Z
dk (τ )
0
K(x, ξ, τ )wk (ξ)dξ ds dτ = fl (t).
Z
wl (x)
∂Ω
Ω
Denote, for short,
γkl (t) = (∇wk , ∇wl ) + (cwk , wl );
Z
Z
κkl (τ ) =
wl (x)
K(x, ξ, τ )wk (ξ)dξ ds.
∂Ω
(3.4)
(3.5)
Ω
and obtain
m
X
d00k (t)δkl + dk (t)γkl (t) +
Z
t
dk (τ )κkl (τ )dτ = fl (t).
(3.6)
0
k=1
The resulting system of integro–differential equations can be reduced to a system
of differential equations of the third order by differentiating (3.6) with respect to t,
m
X
0
0
0
d000
k (t)δkl + dk (t)γkl (t) + dk (t)(κkl (t) + γkl (t)) = fl (t).
(3.7)
k=1
Together with the initial conditions
dk (0) = αk ,
d0k (0) = βk ,
d00k (0) = fl (0) − αk γk (0),
(3.8)
we have a Cauchy problem for the system of linear differential equations with
smooth coefficients, that is uniquely solvable.
Thus, for every m there exists a unique um (x, t), that satisfies (3.2), in other
words, the sequence {um } is defined. We shall investigate it’s convergence. Multiply
(3.2) by d0l (t), sum by l from 1 to m and, finally, integrate by t from 0 to τ ,
Z τZ
m
m
m
m m
(um
tt ut + ∇u ∇ut + cu ut ) dx dt
0
Ω
Z τZ
Z tZ
m
+
ut
K(x, ξ, η)um (ξ, η)dξ dηds dt
0
∂Ω
0
Ω
Z τZ
=
f um
t dx dt.
0
Ω
Integration by parts on the left will give us
Z τZ
Z τZ
Z
2 τ
m m
m m
utt ut dx dt = −
ut utt dx dt +
(um
t ) |0 dx,
0
Ω
0
Ω
Ω
(3.9)
6
S. A. BEILIN
EJDE-2006/103
hence,
Z
τ
Z
Z
1
=
2
m
um
tt ut dx dt
2
(um
t (x, τ ))
1
dx −
2
Z
2
(um
t (x, 0)) dx;
Ω
Ω
Z τ 0Z Ω
Z τZ
Z
2
∇um ∇um
∇um ∇um
(∇um ) |τ0 dx,
t dx dt = −
t dx dt +
0
hence,
Z
τ
Ω
Z
0
0
∇um ∇um
t dx dt =
Ω
τ
Z
Z
0
cum um
t dx dt
Z
1
2
Ω
Ω
2
(∇um (x, τ )) dx −
Ω
Z
τ
Z
m
cum
t u dx dt
=−
0
Ω
Z
+
1
2
Z
Ω
τ
Z
2
(∇um (x, 0)) dx;
Z
−
0
Ω
ct (um )2 dx dt
Ω
2
c (um (x, t)) |τ0 dx;
Ω
hence,
Z
0
τ
Z
cum um
t dx dt
Ω
1
=
2
Z
1
c (u (x, τ )) dx −
2
Ω
Z Z
1 τ
−
ct (um )2 dx dt;
2 0 Ω
m
2
Z
2
c (um (x, 0)) dx
Ω
τ
Z tZ
um
(x,
t)
K(x, ξ, η)um (ξ, η)dξ dη dt ds
t
∂Ω 0
0
Ω
Z Z τ
Z
=−
um (x, t)
K(x, ξ, t)um (ξ, η)dξ dt ds
∂Ω 0
Ω
Z Z tZ
τ
+
um (x, t)
K(x, ξ, η)um (ξ, η)dξ dη ds
0
0
Z∂Ω Z τ
Z Ω
=−
um (x, t)
K(x, ξ, t)um (ξ, η)dξ dt ds
∂Ω 0
Ω
Z
Z τZ
m
+
u (x, τ )
K(x, ξ, η)um (ξ, η)dξ dη ds.
Z
Z
∂Ω
0
Ω
Finally, we obtain
Z 1
2
2
2
m
m
(um
dx
t (x, τ )) + |∇u (x, τ )| + c(x, τ ) (u (x, τ ))
2 Ω
Z 1
2
2
2
m
m
=
(um
dx
t (x, 0)) + |∇u (x, 0)| + c(x, 0) (u (x, 0))
2 Ω
Z Z
Z τZ
1 τ
2
+
ct (um ) dx dt +
f um
t dx dt
2 0 Ω
0
Ω
Z τZ
Z
+
um (x, t)
K(x, ξ, t)um (ξ, t)dξds dt
0
∂Ω
Ω
Z
Z τZ
m
−
u (x, τ )
K(x, ξ, η)um (ξ, η)dξ dηds.
∂Ω
0
Ω
(3.10)
EJDE-2006/103
ON A MIXED NONLOCAL PROBLEM
7
Consider the right-most integral in (3.10). Applying the Cauchy inequality, we have
Z τZ
Z
um (x, t)
K(x, ξ, t)um (ξ, t)dξ ds dt
0
≤
1
2
∂Ω
τ
Z
Ω
Z
0
2
(um (x, t)) ds dt +
∂Ω
1
2
τ
Z
0
next, using an inequality in [6, p.77],
Z
Z
v 2 (x, t)ds ≤
∂Ω
Z
∂Ω
Z
K(x, ξ, t)um (ξ, t)dξ
2
ds dt;
Ω
|∇v|2 + c()v 2 dx
(3.11)
Ω
we have
Z Z
Z Z Z
2
1 τ
1 τ
2
m
(u (x, t)) ds dt +
K(x, ξ, t)um (ξ, t)dξ ds dt
2 0 ∂Ω
2 0 ∂Ω
Ω
Z τZ
1
m
2
2
≤
|∇u (x, t)| + c()u dx dt
2 0 Ω
2
Z Z Z
1 τ
m
∇K(x, ξ, t)u (ξ, t)dξ
+
2 0 Ω
Ω
Z
2 + c()
K(x, ξ, t)um (ξ, t)dξ
dx dt.
Ω
Thus,
Z τZ
Z
um (x, t)
K(x, ξ, t)um (ξ, t)dξds dt
0
Ω
Z∂Ω Z
1 τ
m
≤
|∇u (x, t)|2 + c()u2 dx dt
2 0 Ω
Z Z
Z
1 τ
2
+
|Ω|
|∇K(x, ξ, t)um (ξ, t)| dξdx dt
2 0 Ω
Ω
Z Z
Z
1 τ
2
+
c()|Ω|
(K(x, ξ, t)um (ξ, t)) dξdx dt
2 0 Ω
Ω
Z Z
Z Z
Z
1 τ
1 τ
2
m
2
2
|∇u (x, t)| + c()u dx dt +
|Ω|K1
≤
(um (ξ, t)) dξdx dt
2 0 Ω
2 0 Ω
Ω
Z Z
Z
1 τ
2
m
c()|Ω|K0
(u (ξ, t)) dξdx dt,
+
2 0 Ω
Ω
and, finally,
Z τZ
Z
um (x, t)
K(x, ξ, t)um (ξ, t)dξds dt
0
∂Ω
Ω
Z Z
1 τ
|∇um |2 + c() + |Ω|2 K1 + c()|Ω|2 K0 dx dt.
≤
2 0 Ω
(3.12)
Applying the modified Cauchy inequality to the second integral in (3.10), we have
Z
Z τZ
um (x, τ )
K(x, ξ, η)um (ξ, η)dξ dηds
∂Ω
0
Ω
Z
Z Z τ Z
2
1
2
m
≤
(u (x, τ )) ds +
K(x, ξ, η)um (ξ, η)dξ dη ds.
2 ∂Ω
2 ∂Ω
0
Ω
8
S. A. BEILIN
EJDE-2006/103
Using again the inequality (3.11),
Z
Z Z τ Z
2
1
2
(um (x, τ )) ds +
K(x, ξ, η)um (ξ, η)dξ dη ds
2 ∂Ω
2 ∂Ω
0
Ω
Z
m
2
m
µ|∇u (x, τ )| + c(µ)(u (x, τ ))2 dx
≤
2 Ω
Z Z τZ
1
2
+
τ µ|Ω|
|∇K(x, ξ, η)|2 (um (ξ, η)) dξ dη
2 Ω
0
Ω
Z τZ
2
2
+ τ c(µ)|Ω|
(K(x, ξ, η)) (um (ξ, η)) dξ dη dx.
0
(3.13)
Ω
Choose and µ so that µ < 1, c(µ) < c1 . Inequalities (3.12) and (3.13), give us:
Z
2
m
2
m
2
(um
dx
t (x, τ )) + (1 − µ)|∇u (x, τ )| + (c1 − c(µ))(u (x, τ ))
Ω
Z
2
m
2
m
2
≤
(um
dx
t (x, 0)) + |∇u (x, 0)| + c2 (u (x, 0))
Ω
Z τZ
Z τZ
Z τZ
2
+
f 2 dx dt +
c3 (um )2 dx dt +
(um
t ) dx dt
0
Ω
0
Ω
0
Ω
Z τZ
+
|∇um |2 + h()(um )2 dx dt
0
Ω
Z τZ
Z τZ
c(µ)
µ
K0 τ |Ω|2
(um (x, t))2 dx dt +
(um (x, t))2 dx dt.
+ K1 τ |Ω|2
0
Ω
0
Ω
Thus, denoting
m = min{1, 1 − µ, c1 − c(µ)},
M = max{c3 + h() +
c(µ)
µ
K1 τ |Ω|2 +
K0 τ |Ω|2 }
we obtain the inequality
Z
2
m
2
m
2
m
(um
dx
t (x, τ )) + |∇u (x, τ )| + (u (x, τ ))
Ω
Z
2
m
2
m
2
≤
(um
dx
t (x, 0)) + |∇u (x, 0)| + (u (x, 0))
Ω
Z τZ
Z τZ
m 2
m 2
m 2
+M
(ut ) + |∇u | + (u ) dx dt +
f 2 (x, t)dx dt.
0
Ω
0
Ω
Applying the Gronwall inequality and integrating on τ from 0 to τ , we finally obtain
kum kW21 (Qτ ) ≤ C(T ) kf kL2 (Q) + kφkW21 (Ω) + kψkL2 (Ω) .
(3.14)
Since kf kL2 (Q) , kφkW21 (Ω) , kψkL2 (Ω) are bounded, the sequence {um } is bounded in
W21 (Q): kum kW21 (Q) ≤ D(T ); therefore, there exists a weakly converging subsequence (we will denote it um again for simplicity); now we shall show, that its limit
u ∈ W21 (Q) is a desired solution.
To prove this, we multiply (3.2) by the function hl (t) ∈ W21 (0, T ), hl (T ) = 0,
sum by l from 1 to m, and integrate on t from 0 to T ; denoting η m (x, t) =
EJDE-2006/103
Pm
l=1
ON A MIXED NONLOCAL PROBLEM
9
wl (x)hl (t), we obtain
Z TZ
m
m
m
m m
(−um
t ηt + ∇u ∇η + cu η ) dx dt
0
Ω
T Z
Z
+
η
0
Z
=
m
Z tZ
0
∂Ω
K(x, ξ, τ )um (ξ, τ )dξdτ ds dt
Ω
m
um
t (x, 0)η (x, 0)dx
Z
T
Z
+
0
Ω
f η m dx dt.
Ω
It is no a problem to go to the limit at m → ∞ in the latter expression but the one
addend
Z tZ
Z TZ
K(x, ξ, τ )um (ξ, τ )dξdτ ds dt.
ηm
(3.15)
0
0
∂Ω
Ω
Consider the integral
Z TZ
Z tZ
ηm
K(x, ξ, τ ) um (ξ, τ ) − u(ξ, τ ) dξdτ ds dt.
0
∂Ω
0
Ω
With the help of the Cauchy-Bunyakovski inequality, we have for the “internal”
integral
Z tZ
K(x, ξ, τ ) um (ξ, τ ) − u(ξ, τ ) dξdτ
0
≤
Ω
Z
0
T
Z
K 2 (x, ξ, τ )dξdτ
1/2 Z
Ω
0
T
Z
1/2
2
um (ξ, τ ) − u(ξ, τ ) dξdτ
Ω
= kKkL2 (Q) kum − ukL2 (Q) .
However, kKkL2 (Q) ≤ |Q|K0 , and kum − ukL2 (Q) → 0 (?!), therefore, we can go to
the limit in (3.15). Hence, the limit function u satisfies (3.2) for every η m (x, t) =
P
m
l=1 wl (x)hl (t).
Pm
Denote Nm a set of the functions of the form η m (x, t) = l=1 wl (x)hl (t), hl (t) ∈
S
∞
W21 (0, T ), hl (T ) = 0. While m=1 Nm is dense in Ŵ21 (Q), then (3.2) holds true for
every function from Ŵ21 (Q). This proves that u(x, t) is a generalized solution of
(1.1)-(1.2)-(1.3).
References
[1] Beilin S. Existence of solutions for one-dimensional wave equations with nonlocal conditions.
Electronic Journal of Differential Equations. 2001. Vol. 2001. N.76. Pp. 1–8.
[2] Bouziani A. Solution forte d’un problème mixte avec conditions non locales pour une classe
d’équations hyperboliques. Bull. Cl. Sci., Acad.Roy.Belg. 1997. N.8. Pp. 53–70.
[3] Cannon J. R. The solution of heat equation subject to the specification of energy. Quart.
Appl. Math. 1963. Vol. 21. N.2. Pp. 155–160.
[4] Gordeziani D. G., Avalishvili G. A. Solutions to nonlocal problems of one–dimensional oscillation of medium. Matem. Modelirovanie. 2000. Vol. 12. N.1. Pp. 94–103.
[5] Kozhanov A. I., Pulkina L. S. Boundary Value Problems with Integral Conditions for Multidimensional Hyperbolic Equations. Doklady Mathematics. 2005. Vol. 72. N.2. Pp.743–746.
Translated from Doklady Akademii Nauk, Volume 404, Nos. 16, 2005.
[6] Ladizhenskaja O. A. Boundary value problems of mathematical physics. M.: Nauka. 1973.
[7] Pulkina L. S. On a certain nonlocal problem for degenerate hyperbolic equation. Matem. Zametki. 1992. Vol. 51, N.3.. Pp. 91–96.
[8] Pulkina L. S. Mixed problem with integral conditions for a hyperbolic equation Matem. Zametki. 2003. Vol. 74, N.3. Pp. 435–445.
10
S. A. BEILIN
EJDE-2006/103
[9] Pulkina L. S. Nonlocal problem with integral conditions for a hyperbolic equation. Differenz. Uravnenija. 2004. Vol. 40. N.7. Pp. 887–892.
Sergei A. Beilin
Department of Mathematics, Samara State University, 1, Ac. Pavlov st. 443011 Samara,
Russia
E-mail address: sbeilin@narod.ru