Egorov, Nguyen Minh Chuong, Dang Anh Tuan, 2006, published in Doklady Akademii Nauk, 2006, Vol.. We consider a nonlinear boundary value problem for parabolic pseudodifferential equations
Trang 1ISSN 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 Req > 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) ( )2π –n/2
e–i x〈 ,ξ〉U x q( , )d x,
x n
∫
=
x,ξ
〈 〉 x iξi, [u x t( , )](x q, )
i= 1
n
∑
=
= 2( )π – 1/2
e–qt u x t( , )d t,
0
+ ∞
∫
n+ 1[u x t( , )] ξ τ( , )
= 2( )π – (n+ 1 ) /2
e–i x〈 ,ξ〉–itτu x t( , )d x t d
x t n, 1
∫
as the completion of P(n) = {u∈ (n× (–∞, +∞)):
suppu⊂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 Req = 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.
C0∞
u
P l p, λ µ n
0 + , ∞ )
× , ,
= 1( + ξ + µ+iτ1/λ)lp
ξ τn, 1
∫
⎝
⎜
⎜
⎛
× n+ 1[e–µt u x t( , )] ξ τ( , ) p
dξ dτ
⎠
⎟
⎞1/ p
C0∞
u
l p qn
n u(ξ,q) p
ξ
d
ξn
∫
;
=
U E
l p, λ µ n
, , ( ) U l p q1/ λ n
, , ,
p
τ
d
τ
∫
,
=
where q = µ+iτ
+
n
+
n
+
n
+
n
H l p q, , 1/ λ
∂k u
∂t k
-t= 0
MATHEMATICS
Nonclassical Semilinear Boundary Value Problem for Parabolic
Pseudodifferential Equations in Sobolev Spaces
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
Trang 2NONCLASSICAL SEMILINEAR BOUNDARY VALUE PROBLEM 875
Theorem 1 The Laplace transform induces an
iso-metric isomorphism of P l , p(λ, µ) and E l , p(λ, µ)
To prove this theorem, it is sufficient to note that
3 Nonclassical linear problem Let Ω be a
bounded domain with a smooth boundary ∂Ω in n
Assume that a field touches ∂Ω only at points of a
manifold Γ0 of dimension n – 2 on ∂Ω but does not
touch Γ0 The set of such manifolds Γ0 is naturally
divided into three classes: attracting, repelling, and
neutral (the first, second, or third classes in the
nomen-clature of [6]) In what follows, we assume that all
sub-manifolds Γ0 are either attracting or neutral
Consider the following elliptic problem with a
parameter q ∈ :
(1)
(2)
where j = 1, 2, …, s; |argq|≤ , and (x, D x , q1/λ), and
j(x, Dx, q1/λ) are defined as in [5, p 452] Specifically,
in a local coordinate system, (x, Dx, q1/λ) is a
pseudodifferential operator with the symbol
If Γ0 is attracting, then we add the conditions
(3)
Below, a function h(x) ∈ (Ω) is used such that
h (x) = 1 in the -neighborhood Γd/2 of Γ0 (d > 0 is
suf-ficiently small), h(x) = 0 outside the d-neighborhood Γd
of Γ0, and = d = {x ∈Γd|∃y∈Γ0, is a normal
n+ 1[e–µt u x t( , )] ξ τ( , ) = n[[u x t( , )](x q, )] ξ τ( , ),
ν
(x D, x,q1/λ)U x q( , ) = A x D( , x,q )U x q( , ) = F x q( , ),
x∈Ω,
j(x D, x,q1/λ)(DνU x q( , ))
= B j(x D, x,q)(DνU x q( , )) = G j(x q, ),
x∈∂Ω,
π
2
-σ(x, ,ξ q1/λ) qβgαβ(x,ξ)ξα:
α λβ +∑≤2s
=
(x D, x,q1/λ)u x( ,λ)
= 2( )π –n/2
e i x〈 ,ξ〉σ(x, ,ξ q1/λ)n u(ξ,q) ξd
ξn
∫
D n
k
U x q( , ) = U 0k(x q, ), x∈Γ0,
k = 0 1, , ,… s–1
C0∞ d
2
-xy
vector of ∂Ω} If Γ0 is attracting, then l , p(λ, µ, Ω) denotes the space
(4) with the norm
(5)
If Γ0 is neutral, then the terms with (hU)| are not nec-essary in (4) and (5) Denote by l , p(λ, µ, Ω, ∂Ω) the set (F, G j) ∈l – 2s, p(λ, µ, Ω) × (λ,
µ, ∂Ω) such that hGj ∈ (λ, µ, ∂Ω)
Theorem 2 Let ((x, D x , q1/λ), j (x, D x , q1/λ)|∂Ω)
be an elliptic operator for | |≤ , l ≥ l1 = max{2s,
m j + 2}, and 1 < p < +∞
Then, for sufficiently large µ = Req, problem (1), (2) (or problem (1), (2), (3) when Γ0 is attracting) has a unique solution U (x, q) ∈l , p(λ, µ, Ω) if (F, Gj) ∈l , p(λ,
µ, Ω, ∂Ω) When Γ0 is attracting, we have to add the condition U 0k∈ E l – k – 1 + 1/p, p(λ, µ, Γ0)
Proof This result is proved using Theorem 5.1 in [5].
4 Consider the following problem in Ω × (0, +∞):
(6)
(7)
If Γ0 is attracting, then we add the conditions
(8) Using the initial conditions
(9)
and formally applying the Laplace transform to (6) and (7), we obtain (1) and (2)
The spaces l , p(λ, µ, Ω × (0, +∞), ∂Ω × (0, +∞)) and
l , p(λ, µ, Ω × (0, +∞)) are defined in a similar manner
to l , p(λ, µ, Ω, ∂Ω) and l , p(λ, µ, Ω)
U∈E l p, (λ µ Ω, , ) Dν(hU)∈E l p, (λ µ Ω, , ),
{
hU
( ) ∈E l p, (λ µ, ,) }
U l p, ( λ µ Ω , , ) = U E l p, ( λ µ Ω , , )+ Dν(hU) E l p, ( λ µ Ω , , )
+ (hU) E
l p, ( λ µ , , )
⎩
⎨
⎧
E l m
j
– 2 +1/ p p,
j= 1
s
∏
E l–m j 1 +1/ p p,
⎭
⎬
⎫
q
2
-A x D x ∂
∂t
⎛ ⎞u x t( , ) = f x t( , ), x∈Ω, t>0,
B j x D x ∂
∂t
νu x t( , )
t>0, j = 1 2, , ,… s.
D n k u x t( , ) = u 0k(x t, ), x∈Γ0, t>0,
k = 0 1, , ,… s–1
∂k u
∂t k
-t= 0
0, k 0 1 … l0
λ
- ,
, , ,
l0 = max 2s m{ , j+1}
Trang 3876 EGOROV et al.
Applying Theorem 1, we obtain the following
result
Theorem 3 Let l ≥ l1 and 1 < p < +∞
Then the Laplace transform induces an isometric
isomorphism between the following pairs of spaces:
Using Theorems 2, 3, and Remark 1, we obtain the
following assertion
Theorem 4 Let ((x, Dx, q1/λ), j(x, Dx, q1/λ)|∂Ω) be
an elliptic operator for | |≤ , l≥ l1, and 1 < p < +∞
Then, for sufficiently large µ = Req, problem (6), (7)
(or problem (6), (7), (8) when Γ0 is attracting) has a
unique solution u ∈ l , p(λ, µ, Ω × (0, +∞)) for any
(f, g j ) from l , p(λ, µ, Ω× (0, +∞), ∂Ω× (0, +∞)) (if Γ0
is attracting, then u 0k∈ P l – k – 1 + 1/p, p(λ, µ, Γ0× (0, +∞))
5 Nonclassical semilinear problem Consider the
problem
(10)
(11)
where A and B j are the same operators as in (6) and (7),
(x, t) = (u(x, t), …, u (x, t)), and (x, t) =
(u(x, t), …, u (x, t)) Assume that conditions (8)
and (9) are satisfied Applying the Laplace transform
for Req > 0, we can reduce problem (10), (11) (or (10),
(11), (8) for an attracting Γ0) to the problem
(12)
l p, (λ µ Ω, , ×(0 +, ∞)) Ë l p, (λ µ Ω, , ),
l p, (λ µ Ω, , ×(0 +, ∞),∂Ω×(0 +, ∞)) and
l p, (λ µ Ω ∂Ω, , , ),
l p, (λ µ Ω, , ×(0 +, ∞),∂Ω×(0 +, ∞))
× P l–k 1 +1/ p,p(λ µ Γ, , 0×(0 +, ∞)) Ë
k= 0
s 1
∏
l p, (λ µ Ω ∂Ω, , , ) E l–k 1+1/ p,p(λ µ Γ, , 0)
k= 0
s 1
∏
×
q
2
-A x D x ∂
∂t
⎛ ⎞u x t( , ) = f x t u x t( , , ( , )), x∈Ω,
t>0,
B j x D x ∂
∂t
νu x t( , ) ( ) = g j(x t u, , j(x t, )),
x∈∂Ω, t>0, j = 1 2, , ,… s,
2s 1
u j
D x m j 1
1
λ
⎛ ⎞U x q( , ) A x D
x q
( )U x q( , )
=
= f x t 1
U x t( , ) , ,
(13) (14)
where –1 (x, t) = (–1U (x, t), …, –1 U (x, t))
and –1 (x, t) = (–1U (x, t), …, –1 U (x, t)).
Assume that the following conditions hold:
(i) The mappings (x, t, ) f (x, t, ) (from Ω ×
+× N to ) and (x, t, ) gj (x, t, ) (from ∂Ω×
+× to ) have Carathéodory property; i.e., they are continuous with respect to = (u1, u2, …, u N) and
= (u1, u2, …, ) for almost all (x, t) and are mea-surable with respect to (x, t) for every and
(ii) The mapping u(x, t) ( f (x, t, (x, t)), gi(x, t, (x, t))) (from l , p(λ, µ, Ω × (0, +∞)) to l , p(λ, µ, Ω ×
(0, +∞), ∂Ω × (0, +∞))) maps each bounded set into a relatively compact one
(iii) 1 > ||(A, B j)–1||||( f , g j))||M, µ, where
for all u such that ||u ≤ M}.
To prove Theorem 5, we need the following asser-tion
Lemma 1 The mapping
is a compact mapping from l , p(λ, µ, Ω) to l , p(λ, µ,
Ω, ∂Ω)
Theorem 5 Assume that the operator ((x, D x,
q1/λ), j (x, D x , q1/λ)|∂Ω) is elliptic for | | ≤ Let assumptions (i), (ii), and (iii) be satisfied Let l≥ l1, 1 <
p < +∞, and U 0k∈ El – k – 1 + 1/p, p(λ, µ, Γ0)
j x D x q
1
λ
νU x q( , )
= B j(x D, x,q)(DνU x q( , ))
= g j x t 1
U j(x t, )
, ,
[ ](x q, ), x∈∂Ω, j = 1 2, , ,… s,
D n k U x q( , ) = U 0k(x q, ), x∈Γ0,
k = 0 1, , ,… s–1,
2s 1
m j 1
N j
u
u
u j
inf
M→ + ∞
µ → + ∞
lim
f g, j
M
- (f x t u x t( , , ( , )),
⎩
⎨
⎧
=
g j(x t u, , j(x t, )) ) l p, ( λ µ Ω , , × 0 + , ∞ ) , ∂Ω × 0 + , ∞ ) )
||
l p, ( λ µ Ω , , × ( 0 + , ∞ ) )
U x q( , ) f x t 1
U x t( , ) , ,
(
g j x t 1
U j(x t, ) , ,
q
2
Trang 4-NONCLASSICAL SEMILINEAR BOUNDARY VALUE PROBLEM 877
Then, for sufficiently large µ = Req, problem (12),
(13) (or problem (12)–(14) when Γ0 is attracting) has a
solution U (x, q) ∈l , p(λ, µ, Ω)
Proof sketch We prove Theorem 5 for an attracting
manifold Γ0
For every W ∈l , p(λ, µ, Ω), we obtain
Therefore, for sufficiently large µ = Req, problem (1)–
(3) with the right-hand sides (f, g j , U 0k) has a unique
solution U ∈l , p(λ, µ, Ω) By Lemma 1, the mapping
W U is compact in l , p(λ, µ, Ω) Using (iii), we
obtain R > 0 such that this mapping is compact in the
closed ball B R = {U ∈ l , p(λ, µ, Ω)|||U ≤ R}
for sufficiently large µ Therefore, by the Schauder
the-orem [3, p 60], this mapping has a fixed point in B R,
which solves problem (12), (13), (4)
The following result is proved by applying Remark 1 and Theorems 3 and 5
Theorem 6 Under the assumptions of Theorem 5, if
Γ0 is neutral (or attracting), then, for sufficiently large
µ, problem (10), (11) (or problem (10), (11), (8) with additional data u 0k∈ P l – k – 1 + 1/p, p(λ, µ, Ω × (0, +∞)))
has a solution u∈l , p(λ, µ, Ω × (0, +∞))
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f x t 1
W x t( , ) , ,
(
g j x t 1
W j(x t, ) , ,
||l p, (λ µ Ω, , )