Regularity of Global Attractors for the Kirchhoff Wave Equation ()
1. Introduction
In this paper, we discuss the regularity of global attractors for the following Kirchhoff wave equation
(1.1)
(1.2)
where
is a bounded domain in
with the smooth boundary
,
,
and
are nonlinear functions and
is an external force term which is independent of time.
G. Kirchhoff [1] introduced the Equation (1.1) in
without dissipation
and nonlinear perturbations
and
, and described the oscillation of an elastic stretched string. Furthermore, if the string is made up of the viscoelastic material of rate-type, the equation with the strong damping
appeared [2]. Since
, the Equation (1.1) became the following strongly damped semi-linear wave equation
(1.3)
which described the thermal evolution and
denoted a source term depending nonlinearly on displacement,
denoted a nonlinearly temperature-dependent internal source term [3]. With different conditions about the growth exponents q and p of the nonlinearities
and
, some scholars [4] [5] analyzed the longtime behaviour of solutions of (1.3)-(1.2) by the global and exponential attractors in a bounded region of
. When the nonlinearities are of fully supercritical growth, which lead to that the weak solutions of the equation lose their uniqueness. Z. J. Yang and Z. M. Liu [6] established the existence of global attractor for the subclass of limit solutions of (1.3)-(1.2) by using J. Ball’s attractor theory on the generalized semiflow. Recently, I. Chueshov [7] founded that the Kirchhoff wave equation with strong nonlinear damping was still well-posed and the related evolution semigroup had a finite-dimensional global attractor in
in the sense of “partially strong topology”. Without “partially strong topology”, P. Y. Ding, Z. J. Yang [8] proved the existence of a finite-dimensional global attractor in the natural energy space. And H. L. Ma and C. K. Zhong [9] proved that global attractors for the Kirchhoff equations with strong nonlinear damping attracted
-bounded set with respect to the
norm.
Since
, the following quasi-linear wave equation of Kirchhoff type
(1.4)
was studied by M. Nakao, and the author proved the existence and absorbing properties of attractors in a local sense [10]. Replacing
with
, Y. H. Wang and C. K. Zhong [11] proved the upper semicontinuity of pullback attractors in non-autonomous case. Then Z. J. Yang and Y. Q. Wang [12] studied the longtime behavior of the Kirchhoff type equation with a strong dissipation and proved that the continuous semigroup
possessed global attractors in the phase spaces with low regularity. As for the Kirchhoff wave equation with strong damping and critical nonlinearities, Z. J. Yang and F. Da [13] also studied the stability for the Kirchhoff wave equation with strong damping and critical nonlinearities and proved the existence of global attractors and exponential attractors. Comparing with many researches about the longtime dynamic behavior of solutions for the Kirchhoff wave equation with different types of dissipations [14] - [23], there are few researches about problem of (1.1)-(1.2). And the attractor is a key point for studying these properties, we introduce readers to see the classical book [24].
Based on these, the purpose of this paper is to prove the global attractor of problem (1.1)-(1.2), which attracts every
-bounded set that is compacted in
by the way in ( [25], Theorem 3.1). And we also establish the asymptotic compactness of the global attractor by operator decomposition technique ( [24], Theorem 1.1). So these jobs provide a way to research the longtime dynamic behaviour of such Kirchhoff wave equations, and also reflect the strong damped properties of
to some extent.
The paper is arranged as follows. In Section 2, we verify some preliminaries. In Section 3, we prove the existence of the global attractor. In Section 4, we prove the regularity of the global attractor.
2. Preliminaries
Let
on
with
, and A strictly positive on
. We define the spaces
are Hilbert spaces with the following scalar products and the norms
(2.1)
Let
be the first eigenvalue of A, then
with
.
We define the phase space
with usual graph norm. Let
, then problem (1.1)-(1.2) becomes
(2.2)
(2.3)
For any
, we have the continuous embeddings
,
(2.4)
and the following inequalities hold true:
Interpolation inequality: if
, where
and
, then there exists a constant
such that
(2.5)
The Generalized Poincare inequality:
(2.6)
where
is the first eigenvalue of A.
The Young’s inequality with
: Let
, and
, then
(2.7)
especially, when
, then
(2.8)
The Gronwall inequality (differential form): let
is nonnegative continuous differentiable function (or nonnegative absolutely continuous function), and satisfy
(2.9)
here
are nonnegative integrable functions, then
(2.10)
Throughout this paper, we will denote by C a positive constant which is various in different line or even in the same line and use the following abbreviations:
with
.
Assumption 2.1.
1)
, and
(2.11)
where
if
2)
,
(2.12)
where
if
.
3)
(2.13)
Definition 2.2. Let
be a semigroup on a metric space
. A subset A of E is called a global attractor for the semigroup, if A is compact and enjoys the following properties:
1) A is invariant, i.e.
;
2) A attracts all bounded set of E. That is, for any bounded subset B of E,
Next we only formulate the following results, which is proved in [13] :
Lemma 2.3. Let (2.11)-(2.13) be valid. Then problem (2.2)-(2.3) admits a unique weak solution u, with
,
. Moreover, this solution possesses the following properties:
(Dissipativity)
(2.14)
where k denotes a small positive constant,
and
are positive constants.
Lemma 2.4. Let (2.11)-(2.13) be valid and when
. Then
(2.15)
Actually, by exploiting (2.11) and (2.14), we can get
are respectively bounded in
.
3. Existence of Global Attractors in
For every fixed
, we split the solution
into the sum
, where
and
solve the Cauchy problems
(3.1)
(3.2)
here
Having set
, and satisfying
(3.3)
(3.4)
From now on,
and
will denote generic constants and a generic function, respectively, depending only on
.
Theorem 3.1. Let (2.11)-(2.13) be valid, then the solution semigroup
possesses a global attractor
in X.
Proof. Estimate (2.14) shows
such that the ball
is an absorbing set of the semigroup
in X for
.
In order to prove the existence of the global attractors, now we need to prove the asymptotic compactness.
Multiplying the first equation of (3.1) by
and integrating over
, we get
By using
and the generalized Poincare inequality, then
By
, we know
where
is small enough such that
(3.5)
Actually, noting that
, and by exploiting (2.8) and (2.12), we deduce that
(3.6)
and
(3.7)
From (3.5)-(3.7), we get
where
is small enough such that (
) is negative. Furthermore, by the Gronwall inequality, we can get
(3.8)
Next multiplying the first equation of (3.2) by
and integrating over
, we get
(3.9)
where
is small enough. Then we define the energy functional
(3.10)
At the same time, by the interpolation inequality, we have
and by the embedding
, then
(3.11)
By exploiting (2.8) and the generalized Poincare inequality, from (3.9)-(3.11), we get
where
is small enough and by
, we get
are negative. Then from the Gronwall inequality and noting that
, we get
which provides the following estimate
(3.12)
From (3.8) and (3.12), we obtain that the evolution semigroup
is asymptotically compact in X, so the solution semigroup
possesses a global attractor
in
, which
where
is chosen such that
for
.
4. Regularity of Global Attractors
Now we are in a position to state and prove the main result:
Theorem 4.1. The attractor
of the semigroup
on X is bounded in
.
Proof. Having set
. For
, we split the solution into the sum
where
and
solve the following equations with initial data
,
(4.1)
and
(4.2)
where
.
Multiplying the first equation of (4.1) by
and integrating over
, by
we get
where
is small enough such that
By
and the generalized Poincare inequality, we deduce that
(4.3)
then by the Gronwall inequality, we get
(4.4)
Next multiplying the first equation of (4.2) by and integrating over, exploiting (2.8) and the Hölder’s inequality, the right side becomes
(4.5)
where is small enough, we know is bounded by (2.13) and lemma 2.3. At the same time, the left side becomes
(4.6)
then we define the energy functional
(4.7)
where is small enough such that. By combining (4.5)-(4.7) and the embedding, we get
where is small enough and by, we get are negative. From the Gronwall inequality, we get
which provides the estimate
(4.8)
From (4.4) and (4.8), for every bounded set, we get
so
Then we finish the proof.
5. Conclusion
In this paper, we first prove that the Kirchhoff wave equation with strong damping and critical nonlinearities possesses a global attractor in. Then we split the solution into two parts, one part decays exponentially and the other part satisfies asymptotic behaviour in spaces with higher regularity. By the operator decomposition technique, we get the global attractor which is compactly bounded in