1. Introduction
The initial-boundary value problems for parabolic equations in domains with
conical points were considered in [3, 4], where some important results on the unique
existence of solutions for these problem were given. The problem without initial
condition for second-order parabolic equations in cylinders with smooth base was
considered in [1, 2]. In this paper, we will prove the unique solvability of boundary problem for second-order parabolic equations without an initial condition in
cylinders with non-smooth base
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JOURNAL OF SCIENCE OF HNUE
Natural Sci., 2011, Vol. 56, No. 7, pp. 18-22
ON THE SOLVABILITY OF THE BOUNDARY PROBLEM
FOR SECOND-ORDER PARABOLIC EQUATIONS
WITHOUT AN INITIAL CONDITION IN CYLINDERS
WITH NON-SMOOTH BASE
Nguyen Manh Hung and Le Thi Duyen(∗)
Hanoi National University of Education
(∗)E-mail: Linhlinh041@gmail.com
Abstract. The goal of this paper is to establish the unique existence of gen-
eralized solutions of boundary problem for second-order parabolic equations
without an initial condition in cylinders with non-smooth base.
Keywords: Generalized solutions, without an initial condition.
1. Introduction
The initial-boundary value problems for parabolic equations in domains with
conical points were considered in [3, 4], where some important results on the unique
existence of solutions for these problem were given. The problem without initial
condition for second-order parabolic equations in cylinders with smooth base was
considered in [1, 2]. In this paper, we will prove the unique solvability of bound-
ary problem for second-order parabolic equations without an initial condition in
cylinders with non-smooth base.
2. Formulation of the problem
Let G be a bounded domain in Rn, n ≥ 2, with the boundary ∂G. We
suppose that S = ∂G \ 0 is an infinitely differentiable surface everywhere except
the origin. Denote G∞ = G × (−∞, T ), S∞ = S × (−∞, T ), Gh = G × (h, T ),
Sh = S × (h, T ), for each T, 0 < T ≤ ∞.
We use the following notation: for each multi-index α = (α1, ....αn) ∈ Nn,
|α| = α1 + .... + αn, the symbol Dαu = ∂|α|u/∂α1x1 .....∂αnxn = uxα11 ...xαnn denotes the
generalized derivative of order α with respect to x = (x1, ..., xn).
We begin with recalling some functional spaces which will be used frequently
in this paper.
18
On the solvability of the boundary problem...
Wm2 (G) is the space consisting of all functions u(x) ∈ L2(G), such that
Dαu(x) ∈ L2(G) for almost |α| ≤ m with the norm
‖u‖Wm
2
(G) =
( m∑
|α|=0
∫
G
|Dαu|2dx
)1/2
.
Let X, Y be Banach spaces.
L2((0, T );X) is the space consisting of all measurable functions u : (0, T )→
X with the norm
‖u‖L2((0,T );X) =
( T∫
0
‖u(t)‖2Xdt
) 1
2
.
W 12 ((0, T );X) is the space consisting of all u ∈ L2((0, T );X) such that the
generalized derivative ut = u
′
exists and belongs to L2((0, T );X). The norm in
W 12 ((0, T );X) is defined by
‖u‖W 1
2
((0,T );X) =
( T∫
0
‖u(t)‖2X + ‖ut‖2Xdt
) 1
2
.
W 12 ((0, T );X, Y ) is the space consisting of all u ∈ L2((0, T );X) such that the
generalized derivative ut = u
′
exists and belongs to L2((0, T ); Y ). The norm in
W 12 ((0, T );X, Y ) is defined by
‖u‖W 1
2
((0,T );X,Y ) = ‖u(t)‖2L2((0,T );X) + ‖ut‖2L2((0,T );Y )dt
) 1
2
.
Now we introduce a differential operator of order 2m
L = L(x, t,Dx) =
m∑
|α|,|β|=0
(−1)|α|Dαx (aαβ(x, t)Dβx),
in GT with smooth coefficients in GT , aαβ = aβα for |α| , |β| ≤ m.
We introduce also a system of boundary operators
Bj = Bj(x, t,Dx) =
∑
|α|≤µj
bj,α(x, t)D
α, j = 1, · · · , m,
on ST with smooth coefficients in GT . Suppose that
ordBj = µj ≤ m− 1 for j = 1, · · · , λ,
19
Nguyen Manh Hung and Le Thi Duyen
m ≤ ordBj = µj ≤ 2m− 1 for j = λ+ 1, · · · , m,
and coefficients of Bj are independent of t if ordBj < m.
Let HmB (G) = {u ∈ Wm2 (G) : Bju = 0 on S for j = 1, · · · , λ} with the same
norm in Wm2 (G).
By H−mB (G) we denote the dual space to H
m
B (G)
We consider the following problem without an initial condition in the cylinder
G∞
ut + Lu = f in G∞, (2.1)
Bju = 0 on S∞, j = 1, · · · , m, (2.2)
where f is defined in G∞.
Let χ ∈ C∞(R) such that
χ(t) = 0, t ≤ 0
χ(t) = 1, t ≥ 1
0 ≤ χ(t) ≤ 1, ∀t ∈ R.
Let h be a nonnegative integer. We set χh(t) = χ(t−h).We define the function
fh(x, t) by the equality
fh = χh(t)f. (2.3)
For any h, we have
‖fh‖2L2((h,T );H−mB (G)) ≤ ‖f‖
2
L2((h,T );H
−m
B
(G))
. (2.4)
Now we consider the following problem for an equation of parabolic type in
the cylinder Gh
ut + Lu = fh in Gh, (2.5)
Bju = 0 on Sh, j = 1, · · · , m, (2.6)
u|t=h = 0 on G. (2.7)
Let fh ∈ L2((h, T );H−mB (G)). Using the results in [3], we get a unique gener-
alized solution uh ∈ W12 ((h, T );HmB (G), H−mB (G)) of problem (2.5) - (2.7) satisfing
‖uh‖2W1
2
((h,T );Hm
B
(G),H−m
B
(G))
≤ C‖fh‖2L2((h,T );H−mB (G)), (2.8)
where C is a constant independent of fh and uh.
Let f ∈ L2((−∞, T );H−mB (G)). A function u ∈ W12 ((−∞, T );HmB (G), H−mB (G))
is called a generalized solution of problem (2.1) - (2.2) if
uh −→ u in W12 ((−∞, T );HmB (G), H−mB (G)) as h→ −∞,
here uh is a generalized solution of problem (2.5) - (2.7).
20
On the solvability of the boundary problem...
3. The unique solvability of the generalized solution
Theorem 3.1. If f ∈ L2((−∞, T );H−mB (G)), then there exists a unique general-
ized solution u ∈ W12 ((−∞, T );HmB (G), H−mB (G)) of the problem (2.1) - (2.2) which
satisfies
‖u‖2
W1
2
((−∞,T );Hm
B
(G),H−m
B
(G))
≤ C‖f‖2
L2((−∞,T );H
−m
B
(G))
, (3.1)
where C is the constant independent of f and u.
Proof. Let fh be defined by (2.3). Ref. [3] shows that the problem (2.5) - (2.7) has a
unique generalized solution uh ∈ W12 ((h, T );HmB (G), H−mB (G)) which satisfies (2.8).
From (2.4) and (2.8), we obtain
‖uh‖2W1
2
((h,T );Hm
B
(G),H−m
B
(G))
≤ C‖f‖2
L2((h,T );H
−m
B
(G))
. (3.2)
Suppose that h < h
′
. We define uh′ in the cylinder Gh for h ≤ t < h
′
by setting
uh′ (x, t) = 0 for h ≤ t < h
′
.
We set v = uh − uh′ , then v ∈ W12 ((h, T );HmB (G), H−mB (G)) is a solution of the
following problem, similar to (2.5) - (2.7)
vt + Lv = fh − fh′ in Gh, (3.3)
Bjv = 0 on Sh, j = 1, · · · , m, (3.4)
v|t=h = uh(h)− uh′ (h) = 0 on G. (3.5)
Using also these results in [3] for the problem (3.3) - (3.5), we get
‖v‖2
W1
2
((h,T );Hm
B
(G),H−m
B
(G))
≤ C‖fh − fh′‖2L2((h,T );H−mB (G)). (3.6)
We have
‖fh − fh′‖2L2((h,T );H−mB (G)) =
T∫
h
‖fh(t)− fh′ (t)‖2H−m
B
(G))
dt
≤
h+1∫
h′
‖fh(t)− fh′ (t)‖2H−m
B
(G))
dt −→ 0 (h, h′ → −∞).
From (3.6) we see that uh is a Cauchy sequence in W12 ((h, T );HmB (G), H−mB (G)).
Thus, uh tends to a function u ∈ W12 ((−∞, T );HmB (G), H−mB (G)) which is a gener-
alized solution of the problem (2.1) - (2.2).
21
Nguyen Manh Hung and Le Thi Duyen
Now, we will prove the uniqueness of the solution. Let uˆ be also a generalized
solution of the problem (2.1) - (2.2). This means that exists the function uˆh such
that
uˆh −→ uˆ in W12 ((−∞, T );HmB (G), H−mB (G)) as h→ −∞,
where uˆh is a generalized solution of the problem (2.5) - (2.7) with fh is replaced by
fˆh.
We have
‖uh−uˆh‖2W1
2
((h,T );Hm
B
(G),H−m
B
(G))
≤ C‖fh−fˆh‖2L2((h,T );H−mB (G)) ≤ C‖f‖
2
L2((h,h+1);H
−m
B
(G))
,
where C depends on χh and χˆh. We remark that
‖f‖L2((h,h+1);H−mB (G)) =
h+1∫
h
‖f‖H−m
B
(G)dt→ 0 as h→ −∞.
For simplicity we will write W :=W12 ((−∞, T );HmB (G), H−mB (G)). We have
‖u− uˆ‖W ≤ ‖u− uh‖W + ‖uh − uˆh‖W + ‖uˆh − uˆ‖W −→ 0 as h→ −∞.
This implies u = uˆ in W. The proof is completed.
Acknowledgement. This work was supported by the National Foundation for
Science and Technology Development (NAFOSTED:101.01.58.09), Vietnam.
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