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 of problems with initial conditions . Then, by letting , 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 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 () with boundary . For , set , . If , we use to refer to and to refer to . For each multiindex , set and .
Let us introduce some functional spaces (see [4]) used in this paper.
We use to be the space of sdimensional vector functions defined in Ω with the norm
Denote by the space consisting of all vector functions satisfying
and is the space of vector functions with the norm
In particular,
Denote by the completion of infinitely differentiable vector functions vanishing near with respect to norm.
Now we introduce a differential operator of order 2m
where are matrices with the bounded complexvalued components in , ( is the transposed conjugate matrix to ). Set
We assume further that the form is elliptic uniformly with respect to , which means there exists a constant independent of t and u such that
for all and a.e. , where is a subspace of , the dense subset of infinitely differentiable complex sdimensional vector functions with compact support in Ω.
Now we consider the following problem in the cylinder :
where ν is the unit vector of outer normal to the surrounding surface .
Let , a complex vectorvalued function is called a generalized solution of problem (1.2)(1.3) if and only if, for any , the equality
2 The unique solvability of problems with initial condition
Firstly, for any , we study the following problem in the cylinder :
where ν is the unit vector of outer normal to the surrounding surface .
The solution is surveyed in the generalized sense. That means is a generalized solution if and only if, for any , we have
In [1], the unique solvability of problem (2.1)(2.3) is studied in the case and . Now, by the same method, we consider that problem in the case and .
Theorem 2.1Assume that
Then, for all, , there exists a uniquely generalized solutionof problem (2.1)(2.3) satisfying
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 is an orthogonal basis of which is orthonormal in . For any , we consider the function , where is the solution of the ordinary differential system
So, multiplying both sides of (2.6) by and taking the sum with respect to k from 1 to N, we arrive at
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 , from (2.8) we obtain
Multiplying both sides of this equation by and integrating with respect to T from h to ∞, we get
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 , for any , we can choose satisfying . Next, the term III can be estimated by
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 possesses a subsequence convergent to a vector function , which is a generalized solution of problem (2.1)(2.3). Moreover, it follows from (2.10) that (2.5) holds.
3 The uniqueness of generalized solution of problem (1.2)(1.3)
Theorem 3.1Ifand, , , problem (1.2)(1.3) has no more than one solution.
Proof Assume that and are two generalized solutions of problem (1.2)(1.3), set . For any , , denote
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 , yields
Setting
we have
So
where the positive constant C depends only on μ and .
Using the GronwallBellman inequality, we get
So almost everywhere . Because of the uniqueness of the solution of the problem with initial condition (2.1)(2.3), we imply almost everywhere . □
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 which is equal to 1 on , is equal to 0 on and assumes value in on (see [[5], Th. 5.5] for more details). Moreover, we can suppose that all derivatives of are bounded. Let be an integer. Setting , we then get
where the constant C is independent of f, h.
Let us consider generalized solution and of problems (2.1)(2.3) in cylinders and with is replaced by and respectively. With , can be understood in with , .
Define , then is the generalized solution of problem (2.1)(2.3) in cylinder with is replaced by . According to (2.5),
Because
Because of the fact that , when . So when . Repeating this argument, we discover when . It follows that is a Cauchy sequence and is convergent to u in .
In conclusion, we have satisfying
Because of the fact that , , , (4.3) leads to
For , sending , (4.4) is written as
That means is a generalized solution of problem (1.2)(1.3). We obtain our main result.
Theorem 4.1Assume that:
Then, for all, , there exists a uniquely generalized solutionof problem (1.2)(1.3) satisfying
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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