R E S E A R C H
Open Access
On the solvability of the boundary value
problem without initial condition for
Schrödinger systems in infinite cylinders
Nguyen Manh Hung
1and Nguyen Thi Lien
2**Correspondence: [email protected] 2Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam Full list of author information is available at the end of the article
Abstract
In this paper, we deal with the boundary value problems without initial condition for Schrödinger systems in cylinders. We establish several results on the existence and uniqueness of solutions.
Keywords: generalized solution; problems without initial condition; cylinders
Introduction
The initial boundary value for the Schrödinger equation in cylinders with base containing conical points was established in []. Such a problem for parabolic systems was studied in Sobolev spaces with weights []. The boundary value problem without initial condition for parabolic equation was investigated in [].
In the present paper, we consider the boundary value problem without initial condition for Schrödinger systems in cylinders. Firstly, following the method in [], we prove the existence of solutionsuh of problems with initial conditionst=h. Then, by lettingh→ –∞, the solvability of a problem without initial condition is obtained.
This paper is organized as follows. In the first section, we state the problem. In Sec-tion , we present the results on the unique solvability of problems with initial condiSec-tion for Schrödinger systems in cylinders. The well-posedness of the problem without initial condition is dealt with in Sections and .
1 Setting the problem
Letbe a bounded domain inRn(n≥) with boundaryS=∂. Fora<b, setb a=
×(a,b),Sab=S×(a,b). If (a,b) =R, we useRto refer to∞–∞andSRto refer toS∞–∞. For each multi-indexα= (α, . . . ,αn)∈Nn, set|α|=α+· · ·+αnand∂α=∂xα· · ·∂
αn
xn.
Denoteu(x,t) = (u(x,t), . . . ,us(x,t)),Dαu= (Dαu, . . . ,Dαus),|Dαu|=
s
i=|Dαui|and
utj= (∂ ju
∂tj , . . . ,
∂jus
∂tj ) ,|utj|=si=|∂ ju
i ∂tj |.
Let us introduce some functional spaces (see []) used in this paper.
We useHk() to be the space ofs-dimensional vector functions defined inwith the
norm
uHk()=
k
|α|=
Dαudx
.
Denote byHk,l(b
a) the space consisting of all vector functionsu:ba−→Cssatisfying
uHk,l(b a)=
ba
k
|α|=
Dαu+
l
j=
|utj|dx dt
,
andHk,l(–γ,b
a) is the space of vector functions with the norm
uHk,l(–γ,ba)=
ba
k
|α|=
Dα
u+
l
j=
|utj|
e–γtdx dt
.
In particular,
uHk,(–γ,b a)=
k
|α|=
ba
Dαue–γtdx dt
.
Especially, we setL(–γ,ba) =H,(–γ,ba).
Denote byH◦k,l(–γ,ba) the completion of infinitely differentiable vector functions van-ishing nearSb
awith respect toHl,k(–γ,ba) norm.
Now we introduce a differential operator of order m
L(x,t,D) =
m
|p|,|q|=
(–)|p|Dpapq(x,t)Dq ,
whereapqares×smatrices with the bounded complex-valued components inR,apq=
a∗qp(a∗qpis the transposed conjugate matrix toapq). Set
B(t,u,v) =
m
|p|,|q|=
apqDquDpv dx, t∈R.
We assume further that the formB(t,·,·) isH◦m-elliptic uniformly with respect tot∈R, which means there exists a constantμ> independent oftandusuch that
B(t,u,u)≥μu(·,t)Hm() (.)
for allu∈H◦m() and a.e.t∈R, whereH◦m() is a subspace ofHm(), the dense subset
of infinitely differentiable complexs-dimensional vector functions with compact support in.
Now we consider the following problem in the cylinderR:
(–)m–iL(x,t,D)u–ut=f(x,t) inR, (.)
∂ju ∂νj
SR= , j= , . . . ,m– , (.)
Letf ∈L(–γ,R), a complex vector-valued functionu∈ ◦
Hm,(–γ,
R) is called a gen-eralized solution of problem (.)-(.) if and only if, for anyT> , the equality
(–)m–i
T
–∞
B(t,u,η)dt+
T–∞
uηtdx dt=
T–∞
fηdx dt (.)
holds for allη∈H◦m,(γ,R),η(x,t) = witht≥T.
2 The unique solvability of problems with initial condition
Firstly, for anyh∈R, we study the following problem in the cylinder∞h :
(–)m–iL(x,t,D)v–vt=f(x,t) in∞h , (.)
v|t=h= , x∈, (.)
∂jv
∂νj
S∞h = , j= , . . . ,m– , (.)
whereνis the unit vector of outer normal to the surrounding surfaceS∞h .
The solutionv(x,t) is surveyed in the generalized sense. That meansv∈H◦m,(–γ,∞h ) is a generalized solution if and only if, for anyT> , we have
(–)m–i
T h
B(t,v,η)dt+
Th
vηtdx dt=
Th
fηdx dt (.)
for allη∈H◦m,(γ,∞
h ),η(x,t) = for allt≥T.
In [], the unique solvability of problem (.)-(.) is studied in the caseh= andf,ft∈
L∞(,∞;L()). Now, by the same method, we consider that problem in the caseh∈R
andf,ft∈L(–γ,∞h ).
Theorem . Assume that
(i) sup{|∂a∂pqt |: (x,t)∈∞h , ≤ |p|,|q| ≤m}=μ<∞; (ii) ft,f ∈L(–γ,∞h ).
Then,for allγ >γ=m μ μ,m
=
|α|≤m,there exists a uniquely generalized solution v∈
◦
Hm,(–γ,∞h )of problem(.)-(.)satisfying vHm,(–γ,∞
h)≤C
fL
(–γ,∞h)+ft
L(–γ,∞h)
, (.)
where C is a nonnegative constant independent of h,v,and f.
Proof The uniqueness is proved in a similar way as in []. We omit the details here. Now we prove the existence by the Galerkin approximating method. Suppose that{ϕk}∞k=is an
orthogonal basis ofHm() which is orthonormal inL(). For anyN∈N, we consider
the functionvN(x,t) =N
k=CNk(t)ϕk(x), where (CNk(t))Nk= is the solution of the ordinary
differential system
(–)m–iBt,vN,ϕk –
vNt ϕkdx=
fϕkdx, (.)
So, multiplying both sides of (.) bydtd(CkN(t)) and taking the sum with respect tokfrom toN, we arrive at
(–)m
m
|p|,|q|=
apqDqvNDpvNt dx–i
vNt vN t dx=i
f vN t dx.
Adding this equation to its complex conjugate, integrating with respect totfromhtoT, and then integrating by parts, we get
(–)m
m
|p|,|q|=
apq(x,T)DqvN(x,T)DpvN(x,T)dx
= (–)m
m
|p|,|q|=
Th
∂apq
∂t D
qvNDpvNdx dt
– Im
f(x,T)vN(x,T)dx–
Th
ftvNdx dt
. (.)
Using (.) and the Cauchy inequality, we receive from (.) that
vN(·,T)Hm()≤
mμ+
μ–
T h
vN(·,t)Hm()dt
+
(μ–)
f(·,T)L()+
T h
ft(·,t)L()dt
. (.)
Using the Gronwall-Bellman inequality, put α=mμ–μ+, from (.) we obtain
vN(·,T)Hm()≤α
T h
eα(T–t)f(·,t)L()+
T h
fs(·,s)L
()ds
dt
+
(μ–)
f(·,T)L()+
T h
ft(·,t)
L()dt
.
Multiplying both sides of this equation bye–γTand integrating with respect toTfromh
to∞, we get
vNHm,(–γ,∞ h)≤
(μ–)
∞
h
e–γTf(·,T)L()dT
+
(μ–)
∞
h
e–γT
T h
ft(·,t)
L()dt dT
+ α
∞
h
e–γT
T h
eα(T–t)f(·,t)L()dt dT
+ α
∞
h
e–γT
T h
eα(T–t)
t h
fs(·,s)
L()ds dt dT. (.)
We denote byI,II,III,IV the terms from the first, second, third, fourth, respectively, of the right-hand sides of (.). We will give estimations for these terms. Firstly
I=
(μ–)
f
and
II=
(μ–)
∞
h
ft(·,t)L()
∞
t
e–γTdT dt= γ (μ–)
ftL(–γ,∞h).
Because ofinf<<μ
mμ+ μ– =
mμ
μ = γ, for anyγ >γ, we can choose> satisfying α=
mμ+
μ– < γ. Next, the termIIIcan be estimated by
III= α
∞
h
e–αtf(·,t)
L()
∞
t
e(α–γ)TdT dt= α
γ–αf
L(–γ,∞h).
The last term,IV, is equal to
IV= α
∞
h
fs(·,s)
L()
∞
s
e–αt
∞
t
e(α–γ)TdT dt ds= α
γ(γ –α)ft
L(–γ,∞h).
Combining the above estimations, we get from (.) that
vNHm,(–γ,∞ h)≤C
f
L(–γ,∞h)+ftL(–γ,∞h)
, (.)
where the constantCis independent ofh,N.
From this inequality, by standard weakly convergent arguments (see []), we can con-clude that the sequence{vN}∞
N=possesses a subsequence convergent to a vector function
v∈H◦m,(–γ,∞
h ), which is a generalized solution of problem (.)-(.). Moreover, it
fol-lows from (.) that (.) holds.
3 The uniqueness of generalized solution of problem (1.2)-(1.3)
Theorem . Ifγ > and|∂a∂pqt |<μeγt,∀t∈R,∀|p|,|q| ≤m,problem(.)-(.)has no
more than one solution.
Proof Assume thatu(x,t) andu(x,t) are two generalized solutions of problem (.)-(.),
setu(x,t) =u(x,t) –u(x,t). For anyT> ,b≤T, denote
η(x,t) =
⎧ ⎨ ⎩ t
bu(x,τ)dτ, –∞ ≤t≤b,
, b≤t≤T.
Thenη(x,T) = ,η∈H◦m,(γ,T
–∞) andηt(x,t) =u(x,t), –∞ ≤t≤b.
From the definition of generalized solution, we obtain
(–)m
m
|p|,|q|=
b–∞
apqDpηtDqηdx dt+i
b–∞
|ηt|dx dt= . (.)
Adding (.) to its complex conjugate, we discover
(–)m
m
|p|,|q|=
b–∞ apq
∂(DpηDpη)
which leads to
–
m
|p|,|q|=
b–∞ ∂apq
∂t D
pηDqηdx dt
= lim
τ→–∞
m
|p|,|q|=
τ
b
apq(x,τ)Dpu(x,τ)Dqu(x,τ)dx dτ. (.)
Using the assumption of Theorem . and the Cauchy inequality, the left-hand sides of (.) can be estimated by
m
|p|,|q|=
b–∞ ∂apq
∂t D
pηDqηdx dt≤μ
m
|p|,|q|=
b–∞
DqηDpeγtdx dt
≤μ
m
|p|,|q|=
b–∞
Dqηeγt+Dpηeγt dx dt
≤μmηHm,(γ,b –∞).
From (.) we have
B(τ,η,η) =
m
|p|,|q|=
apq(x,τ)Dpη(x,τ)Dqη(x,τ)dx≥μη(·,τ)
Hm(),
which implies
lim
τ→–∞
m
|p|,|q|=
apq(x,τ)Dpη(x,τ)Dqη(x,τ)dx≥μ lim
τ→–∞η(·,τ)
Hm().
Thus
lim
τ→–∞η(·,τ)
Hm()≤CηHm,(γ,b
–∞). (.)
Set
vp(x,t) =
–∞
t
Dpu(x,τ)dτ, –∞ ≤t≤b,
then
Dpη(x,t) =
t b
Dpu(x,τ)dτ=vp(x,b) –vp(x,t).
Replacing them into (.), noting thatlimτ→–∞vp(x,t) = , yields m
|p|=
vp(x,b)dx
≤C
m
|p|=
b–∞
≤C
m
|p|=
b–∞
eγtvp(x,b)
+vp(x,t)
dx dt
≤Ceγb
m
|p|=
vp(x,b)dx+C
m
|p|=
b–∞
eγtvp(x,t)
dx dt.
Setting
J(t) =
m
|p|=
vp(x,t)dx,
we have
–Ceγb J(b)≤C
b
–∞e
γtJ(t)dt.
So
J(b)≤C
b
–∞e
γtJ(t)dt, ∀b∈
–∞, γ ln
C
,
where the positive constantCdepends only onμandμ.
Using the Gronwall-Bellman inequality, we get
J(t)≡ on
–∞, γ ln
C
.
Sou(x,t) = almost everywheret∈(–∞,γ lnC]. Because of the uniqueness of the so-lution of the problem with initial condition (.)-(.), we implyu(x,t) =u(x,t) almost
everywheret∈R.
4 The existence of generalized solution
The generalized solution of problem (.)-(.) can be approximated by a sequence of so-lutions of problems with initial condition (.)-(.).
It is known that there is a smooth functionχ(t) which is equal to on [,∞), is equal to on (–∞, ] and assumes value in [, ] on [; ] (see [, Th. .] for more details). Moreover, we can suppose that all derivatives ofχ(t) are bounded. Leth∈(–∞, ] be an integer. Settingfh(x,t) =χ(t–h)f(x,t), we then get
fh(x,t) =
⎧ ⎨ ⎩
f(x,t) ift≥h+ ,
ift≤h.
Moreover, iff,ft∈L(–γ,R),fh,fth∈L(–γ,R) and
fhL
(–γ,R)≤ f
L(–γ,R), (.)
fthL
(–γ,R)≤C
ftL(–γ,R)+fL(–γ,R) , (.)
Let us consider generalized solutionuhandukof problems (.)-(.) in cylinders∞ h
and∞k withf(x,t) is replaced byfh(x,t) andfk(x,t) respectively. Withh>k,uh can be
understood inH◦m,(–γ,∞
k ) withuh(x,t) = ,∀k≤t≤h.
Defineukh(x,t) =uk(x,t) –uh(x,t), thenukh(x,t) is the generalized solution of problem
(.)-(.) in cylinder∞k withf(x,t) is replaced byfkh(x,t) =fk(x,t) –fh(x,t). According to (.),
ukh◦
Hm,(–γ,R)=ukh
◦
Hm,(–γ,∞ k)
≤Cfh–fkL(–γ,∞
k)+
fth–ftkL(–γ,∞
k )
.
Because
fh–fkL(–γ, R)=f
h–fk
L(–γ,∞k)=
h+
k
e–γtfh–fkL
()dt
=
h+
k
e–γtχ(t–h) –χ(t–k)· fL()dt
≤
h+
k
e–γtfL()dt.
Because of the fact thatf∈L(–γ,R),lim
h+
k e–
γtf
L()dt= whenh,k→–∞. So
limfh–fkL(–γ,R)= whenh,k→–∞. Repeating this argument, we discoverlimfth– fk
t L(–γ,R)= whenh,k→–∞. It follows that{uh}–h∞=is a Cauchy sequence anduhis
convergent touinH◦m,(–γ,
R).
In conclusion, we haveuh∈H◦m,(–γ,
R) satisfying
(–)m–i
T h
Bt,uh,η dt+
Th
uhηtdx dt=
Th
fhηdx dt (.)
for allT> ,η∈H◦m,(γ,∞
h ),η(x,t) = witht≥T.
Because of the fact thatuh(x,t) = ,fh(x,t) = ,∀t≤h, (.) leads to
(–)m–i
T
–∞
Bt,uh,η dt+
T–∞
uhηtdx dt=
T–∞
fηdx dt (.)
for allT> ,η∈H◦m,(γ,R),η(x,t) = for allt≥T.
Forf(x,t)∈L(–γ,R), sendingh→–∞, (.) is written as
(–)m–i
T
–∞
B(t,u,η)dt+
T–∞
uηtdx dt=
T–∞
fηdx dt (.)
for allT> ,η∈H◦m,(γ,
R),η(x,t) = for allt≥T.
That meansu(x,t) is a generalized solution of problem (.)-(.). We obtain our main result.
Theorem . Assume that:
(ii) |∂a∂pqt | ≤μ·eγt,for all(x,t)∈R,≤ |p|,|q| ≤m;
(iii) f,ft∈L(–γ,R). Then, for allγ >γ= m
μ μ, m
=
|α|≤m, there exists a uniquely generalized solution
u(x,t)∈H◦m,(–γ,
R)of problem(.)-(.)satisfying
u◦
Hm,(–γ,R)≤C
fL(–γ,R)+ftL(–γ,R)
.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors studied, read and approved the final manuscript.
Author details
1National Institute of Education Management, Hanoi, Vietnam.2Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam.
Acknowledgements
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2011.30.
Received: 29 January 2013 Accepted: 16 June 2013 Published: 1 July 2013 References
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doi:10.1186/1687-2770-2013-156