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In present paper we prove the local well-posedness for Von Neumann-Landau wave equation by the T. Kato’s method.

For the stationary Von Neumann-Landau wave equation, Chen investigated the Dirichlet problems [

where

If the plus “+” is replaced by the minus “−” on right hand in Equation (1), then the resulted equation is the Schrödinger equation. For the Schrödinger equation, the well-posedness problem is investigated for various nonlinear terms

The paper is organized as follows. Section 2 contains the list of assumptions on the interaction term

In this section we list the assumptions on the interaction term

Definition 2.1. Fix

and

Remark 2.1. The pairs

Secondly, let

and

for all

where

for all measurable function

Finally, let us make the notion of solution more precise.

Definition 2.2. Let

for all

The main result is the following theorem:

Theorem 1. Suppose

(i) For any

where

(ii) The map

(iii) For every

(iv) There is the blowup alternative: If

Remark 2.2. It follows from Strichartz estimates that

for any admissible pair

Remark 2.3. For the Schrödinger equations, the similar results hold [

In this subsection, we recall that the Strichartz estimates. Let

for any

for any

The following result is the fundamental estimate for

Lemma 1. If

where

Proof. For the proof please see [

The following estimates, known as Strichartz estimates, are key points in the method introduced by T. Kato [

Lemma 2. Let

the dual homogeneous Strichartz estimate

and the inhomogeneous Strichartz estimate

for any interval

Proof. For the proof please see [

Proof. Let

one easily verifies that for any

Set

And it follows from Remark 1.3.1 (vii) in [

We now proceed in four steps.

Step 1. Proof of (i). Fix

equipped with the distance

We claim that

and that

thus,

Taking up any

1.2.2 (iii) in [

and

Using the embedding

and

Given

It follows from (22) and Strichartz estimates (lemma 2) that

and

Also, we deduce from (23) that

Finally, note that

It then follows from (26) and (28) that for any

Thus,

In particular,

point

For uniqueness, assume that

For simplicity, we set

for

Similarly, for

Note that

where the constant

Step 2. Proof of (ii). Suppose that

the unique solution of (1) corresponding to the initial value

and the estimate (29) which implies that (27) holds for

Hence, we have

Next, we need to estimate

A similar identity holds for

Note that

By choosing

There, if we prove that

as

as

By using (37) and possibly extracting a subsequence, we may assume that

and

we obtain from the dominated convergence a contradiction with (44).

Step 3. Proof of (iii). Consider

It follows from part (i) there exists a solution

of (1).

Step 4. Proof of (iv). Suppose now that

One shows by the same argument that if

This completes the proof. □

We are grateful to the anonymous referee for many helpful comments and suggestions, which have been incorporated into this version of the paper. C. Liu was supported in part by the NSFC under Grants No. 11101171, 11071095 and the Fundamental Research Funds for the Central Universities. And M. Liu was supported by science research foundation of Wuhan Institute of Technology under grants No. k201422.