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; cylindersIntroduction
The initial boundary value for the Schrödinger equation in cylinders with base containing conical points was established in [1]. Such a problem for parabolic systems was studied in Sobolev spaces with weights [2]. The boundary value problem without initial condition for parabolic equation was investigated in [3].
In the present paper, we consider the boundary value problem without initial condition
for Schrödinger systems in cylinders. Firstly, following the method in [1], we prove the existence of solutions
This paper is organized as follows. In the first section, we state the problem. In Section 2, we present the results on the unique solvability of problems with initial condition for Schrödinger systems in cylinders. The wellposedness of the problem without initial condition is dealt with in Sections 3 and 4.
1 Setting the problem
Let Ω be a bounded domain in
Denote
Let us introduce some functional spaces (see [4]) used in this paper.
We use
Denote by
and
In particular,
Especially, we set
Denote by the completion of infinitely differentiable vector functions vanishing near
Now we introduce a differential operator of order 2m
where
We assume further that the form
for all and a.e.
Now we consider the following problem in the cylinder
where ν is the unit vector of outer normal to the surrounding surface
Let
holds for all ,
2 The unique solvability of problems with initial condition
Firstly, for any
where ν is the unit vector of outer normal to the surrounding surface
The solution
for all ,
In [1], the unique solvability of problem (2.1)(2.3) is studied in the case
Theorem 2.1Assume that
(i)
(ii)
Then, for all
whereCis a nonnegative constant independent ofh, v, andf.
Proof The uniqueness is proved in a similar way as in [1]. We omit the details here. Now we prove the existence by the Galerkin approximating
method. Suppose that
So, multiplying both sides of (2.6) by
Adding this equation to its complex conjugate, integrating with respect to t from h to T, and then integrating by parts, we get
Using (1.1) and the Cauchy inequality, we receive from (2.7) that
Using the GronwallBellman inequality, put
Multiplying both sides of this equation by
We denote by I, II, III, IV the terms from the first, second, third, fourth, respectively, of the righthand sides of (2.9). We will give estimations for these terms. Firstly
and
Because of
The last term, IV, is equal to
Combining the above estimations, we get from (2.9) that
where the constant C is independent of h, N. □
From this inequality, by standard weakly convergent arguments (see [1]), we can conclude that the sequence
3 The uniqueness of generalized solution of problem (1.2)(1.3)
Theorem 3.1If
Proof Assume that
Then
From the definition of generalized solution, we obtain
Adding (3.1) to its complex conjugate, we discover
which leads to
Using the assumption of Theorem 3.1 and the Cauchy inequality, the lefthand sides of (3.2) can be estimated by
From (1.1) we have
which implies
Thus
Set
then
Replacing them into (3.3), noting that
Setting
we have
So
where the positive constant C depends only on μ and
Using the GronwallBellman inequality, we get
So
4 The existence of generalized solution
The generalized solution of problem (1.2)(1.3) can be approximated by a sequence of solutions of problems with initial condition (2.1)(2.3).
It is known that there is a smooth function
Moreover, if
where the constant C is independent of f, h.
Let us consider generalized solution
Define
Because
Because of the fact that
In conclusion, we have satisfying
for all
Because of the fact that
for all
For
for all
That means
Theorem 4.1Assume that:
(i)
(ii)
(iii)
Then, for all
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors studied, read and approved the final manuscript.
Acknowledgements
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.012011.30.
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