
1046 HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
and the general nonhomogeneous boundary conditions
Γ
(1)
m
[w] ≡
n−1
X
k=0
b
(1)
mk
(t)
∂
k
w
∂x
k
= g
(1)
m
(t) at x = x
1
(m = 1, . . . , s),
Γ
(2)
m
[w] ≡
n−1
X
k=0
b
(2)
mk
(t)
∂
k
w
∂x
k
= g
(2)
m
(t) at x = x
2
(m = s + 1, . . . , n),
(4)
where s ≥ 1 and n ≥ s + 1. We assume that both sets of the boundary forms Γ
(1)
m
[w]
(m = 1, . . . , s) and Γ
(2)
m
[w] (m = s + 1, . . . , n) are linearly independent, which means that
for any nonzero ψ
m
= ψ
m
(t) the following relations hold:
s
X
m=1
ψ
m
(t)Γ
(1)
m
[w] 6≡ 0,
n
X
m=s+1
ψ
m
(t)Γ
(2)
m
[w] 6≡ 0.
In what follows, we deal with the nonstationary boundary value problem (1)–(4). It is
assumed that there exist soluti