^{1}

We study the long-time behavior of the Kirchhoff type equation with linear damping. We prove the existence of strong solution and the semigroup associated with the solution possesses a global attractor in the higher phase space.

We consider the following nonlinear Kirchhoff type equation with the initial-boundary conditions:

When

When the coefficient of

Similar models have been considered in [

Thus, to the best of our knowledge, the research about global attractors of the weak solutions for problem (

The paper is organized as follows. In Section

Throughout the paper we will denote

The norm and the scalar product in

For any given function

For convenience, the letters

We collect some basic concepts and general theorems, which are important for getting our main results. We refer to [

Let

A

Let X be a Banach space and

(i)

(ii)

Let X and Y be two Banach spaces such that

Let X, Y be two Banach spaces and

Assume that

(i)

(ii)

And assume furthermore that

Next we iterate some main results in [

For any initial data

Furthermore problem (

Choosing

Owing to [

By the Gronwall inequality, it follows that

Next, we show that

From (

Now if

Suppose

We prove the existence of strong solutions by using the Faedo-Galerkin schemes. Assume that there exists an orthonormal basis of

For this purpose, according to the basis theory of ordinary differential equations, we build the sequence of Galerkin approximate solutions. They are smooth functions of the form

Choosing

Like the estimates of (

It means that the sequence

So

It is easy to pass the limit in (

Furthermore, from Theorem

Finally, uniqueness is followed from [

Thus, the dynamical system generated by (

Suppose the conditions of Theorem

We first prove the following compactness results and the norm-to-weak continuity of semigroup.

Suppose that (

Suppose that

since there exists constant

Because

Similarly, we can prove the following Lemma.

Let

Since

The semigroup

Now we give our main results of the paper.

Suppose that

Applying Theorems

Let

Let

Since

Multiplying (

Applying the Young inequality,

Using the above estimates, we transform (

Choose

Denote

By Gronwall inequality

Next, we show that

Indeed, the right inequality is obtained using

Thus, combining (

Taking

Together with Theorems

If we transform the first equation of (

All data included in this study are available upon request by contact with the corresponding author.

The author declares that there are no conflicts of interest.

This work was supported by the National Science Foundation of China (61703181).