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In this paper, we consider the initial boudary value problem for modified Zakharov system in 3 dimensions with small initial condition. By using the continuity lemma and the linear interpolation theory, together with the properties of Sobolev spaces and the Galerkin method, we obtain the existence and uniqueness of the global solution.

In this paper, we study the global existence and uniqueness of solutions for a modified Zakharov system with initial boundary value conditions as follows.

where

The classical Zakharov system was derived by Zakharov to describe the propagation of Langmuir waves in a plasma [

However, some important effects, such as quantum effects, have been ignored in the classical Zakharov system. The importance of quantum effects in ultrasmall electronic devices, dense astrophysical plasma systems and laser plasmas has produced an increasing interest on the investigation of the quantum counterpart of some of plasma physics phenomena [

Recently, S. You and B. Guo have considered the existence and uniqueness of the global solution to the initial boundary value problem for the above system in 1 dimension and 2 dimensions, respectively [

Now we give some notations:

-For

-Let

-We denote by C a positive constant which may change from one line to the next line.

To study the smooth solution of the modified Zakharov system, we introduce function

with initial condition

and boundary condition

Lemma 2.1. Assume that

Proof. Multiplying Equation (2.1) by

Since

Taking the imaginary part of Equation (2.7), then we have

Lemma 2.2. [

If

Lemma 2.3. [

where

Lemma 2.4. Assume that

Proof. Multiplying Equation (2.1) by

Since

then taking the real part of Equation (2.8), we have

where we have used the fact

Similarly, multiplying Equation (2.2) by

i.e.

Adding Equation (2.9) to Equation (2.11), we deduce

Set

therefore, using Young’s inequality, we have

Choosing

Set

if the initial condition is small enough. Substituting it into Equation (2.12), we have

Lemma 2.5. Assume that

Proof. Differentiating Equation (2.1) with respect to t, and then multiplying it by

Since

Taking the imaginary part of Equation (2.13) yields

therefore, using Hölder’s inequality and Sobolev imbedding, we have

Differentiating Equation (2.2) with respect to t, and then multiplying it by

i.e.

therefore

Adding Equation (2.14) to Equation (2.15), we have

Using Gronwall’s inequality, we have

From Equation (2.1), Equation (2.2), and Equation (2.3), it easily get

Theorem 3.1. Assume that

Proof. We first give the proof of the uniqueness of the solution. Suppose

therefore

with initial condition

Multiplying Equation (3.1) by

taking the imaginary part yields

therefore

Multiplying Equation (3.3) by

Since

therefore

Adding Equation (3.5) to Equation (3.6), we have

Using Gronwall’s inequality and the initial condition Equation (3.4), we can obtain

Next we show the existence of the solution.

By using the Garlerkin method, choose basic functions

where the undetermined coefficients

with initial conditions

According to the basic theory of ordinary differential equations, the above equations have a unique local solution.

Similarly to the proof of Lemma 2.1 and Lemma 2.4, we have

By compactness argument, we can choose subsequences, still denoted by

Indeed

By using the density of

This work was supported by the National Natural Science Foundation of China (No. 11271141, No. 11426069 and No.61375006).