874
ISSN 1064–5624, Doklady Mathematics, 2006, Vol. 74, No. 3, pp. 874–877. © Pleiades Publishing, Inc., 2006.
Original Russian Text © Yu.V. Egorov, Nguyen Minh Chuong, Dang Anh Tuan, 2006, published in Doklady Akademii Nauk, 2006, Vol. 411, No. 6, pp. 732–735.
We consider a nonlinear boundary value problem
for parabolic pseudodifferential equations of an arbi-
trary order. By applying the Laplace transform, the
problem is reduced to a boundary value problem for an
elliptic equation. The existence of a solution is proved
by using the Schauder theorem.
1. A nonclassical nonlinear boundary value problem
for elliptic pseudodifferential equations in the Sobolev
spaces H
l, p
, 1 < p < ∞ was considered in [5]. In this
paper, we use the Laplace transform to study a similar
problem for parabolic pseudodifferential equations in
Sobolev spaces. The proof is substantially simplified by
applying the Schauder theorem (instead of the Leray–
Schauder one).
2. Suppose that q ∈ and Re q > 0. Let
where
We use the following spaces. Let l ≥ 0; 1 < p < +∞; and
0 < λ, μ. The space P
l, p
(λ, μ,
n
× (0, +∞)) is defined
n
U ξ q , ( ) 2 π ( )
n /2 –
e
i x ξ , 〈 〉 –
Uxq , ( ) x , d
x
n
∫
=
x ξ , 〈 〉 x
i
ξ
i
, uxt , ( ) [ ] xq , ( )
i 1 =
n
∑
=
= 2 π ( )
1/2 –
e
qt –
uxt , ( ) t , d
0
+∞
∫
n 1 +
uxt , ( ) [ ]ξτ , ( )
= 2 π ( )
n 1 + ( ) /2 –
e
i x ξ , 〈 〉 – it τ –
uxt , ( ) x t . d d
xt ,
n 1 +
∫
as the completion of P(
n
) = {u ∈ (
n
× (–∞, +∞)):
supp u ⊂
n
× (0, +∞)} in the norm
H
l, p, q
(
n
) is the completion of (
n
) in the norm
E
l, p
(λ, μ,
n
) is the completion of P(
n
) in the norm
Using an extension of functions from and Ω to
n
, where Ω is a bounded domain in
n
, we define the
spaces P
l, p
(λ, μ, × (0, +∞)), P
l, p
(λ, μ, Ω × (0, +∞)),
H
l, p
( ), H
l, p, q
(Ω), E
l, p
(λ, μ, ), and E
l, p
(λ, μ, Ω).
Remark 1. (i) If U(x, q) ∈ E
l, p
(λ, μ) and Re q = μ,
then U(x, q) ∈ for almost all q.
(ii) If u ∈ P
l, p
(λ, μ), then = 0 for λ ≥ 0 and
k ≤ l.
C
0
∞
u
P
lp ,
λμ
n
0+∞ , ( ) × , , ( )
= 1 ξ μ i τ +
1/ λ
+ + ( )
lp
ξτ ,
n 1 +
∫
⎝
⎜
⎜
⎛
×
n 1 +
e
μt –
uxt , ( ) [ ]ξτ , ( )
p
d ξ τ d
⎠
⎟
⎞
1/ p
.
C
0
∞
u
lpq
n
, , ,
1 ξ q + + ( )
lp
n
u ξ q , ( )
p
ξ d
ξ
n
∫
⎝ ⎠
⎜ ⎟
⎜ ⎟
⎛ ⎞
1/ p
; =
U
E
lp ,
λμ
n
, , ( )
U
lpq
1/ λ
n
, , ,
p
τ d
τ
∫
⎝ ⎠
⎜ ⎟
⎛ ⎞
1/ p
, =
where q μ i τ . + =
+
n
+
n
+
n
+
n
H
lpq
1/ λ
, ,
∂
k
u
∂ t
k
-------
t 0 =
MATHEMATICS
Nonclassical Semilinear Boundary Value Problem for Parabolic
Pseudodifferential Equations in Sobolev Spaces
Yu. V. Egorov
a
, Nguyen Minh Chuong
b
, and Dang Anh Tuan
c
Presented by Academician V.S. Vladimirov March 23, 2006
Received March 24, 2006
DOI: 10.1134/S1064562406060226
a
Université Paul Sabatier, Toulouse, France
b
Institute of Mathematics, Hanoi, Vietnam
c
Hanoi National University, Hanoi, Vietnam