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
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 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 well-posedness of the problem without initial condition is dealt with in Sections 3 and 4.
1 Setting the problem
Let us introduce some functional spaces (see ) used in this paper.
Now we introduce a differential operator of order 2m
2 The unique solvability of problems with initial condition
In , 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
whereCis a nonnegative constant independent ofh, v, andf.
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 is an orthogonal basis of which is orthonormal in . For any , we consider the function , where is the solution of the ordinary differential system
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
We denote by I, II, III, IV the terms from the first, second, third, fourth, respectively, of the right-hand sides of (2.9). We will give estimations for these terms. Firstly
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 ), 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)
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 left-hand sides of (3.2) can be estimated by
From (1.1) we have
Using the Gronwall-Bellman inequality, we get
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 [, 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.
In conclusion, we have satisfying
Theorem 4.1Assume that:
The authors declare that they have no competing interests.
All authors studied, read and approved the final manuscript.
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2011.30.
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