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).