We study the asymptotic behavior of solutions to the stochastic strongly damped wave equation with additive noise defined on unbounded domains. We first prove the uniform estimates of solutions, and then establish the existence of a random attractor.
Let
the Borel
Thus,
Consider the following stochastic strongly damped wave equation with additive noise defined in the entire space
with the initial value conditions
where
function satisfying certain dissipative and growth conditions, and
valued Wiener processes on
Many works have been done regarding the dynamics of a variety of systems related to Equation (1). For example, the asymptotical behavior of solutions for deterministic strongly damped wave equation has been studied by many authors (see [
In general, the existence of global random attractor depends on some kind compactness (see, e.g., [
Throughout this paper, we use
In this section, we collect some basic knowledge about general random dynamical systems (see [
In the following, a property holds for
Definition 1 A continuous random dynamical system on X over
such that the following properties hold
・
・
・
Definition 2 (See [
・ A set-valued mapping
・ A random set
where
Let
・ A random set
・ A random set
where
・
・ A random compact set
Theorem 1 (See [
Moreover,
In this subsection, we outline some basic settings about (1)-(2) and show that it generates a random dynamical system.
Let
with the initial value conditions
where
Let
where
For our purpose, it is convenient to convert the problem (3)-(4) (or (1)-(2)) into a deterministic system with a random parameter, and then show that it generates a random dynamical system.
Let
Its unique stationary solution is given by
Note that the random variable
where
Then it follows from the above, for
Put
Now, let
with the initial value conditions
where
Let
where
By a standard method as in [
generates a continuous random dynamical system, where
Then, the transformation
also generates a random dynamical system associated with (3)-(4). Note that the two random dynamical systems are equivalent. By (13), it is easy to check that
In this subsection, we derive uniform estimates on the solutions of the stochastic strongly damped wave Equations (3)-(4) defined on
We assume that
Set
where
We define a new norm
for
The next lemma shows that
Lemma 1 Assume that (F1)-(F4),
dom ball
sorbing set for
Proof. Taking the inner product of the second equation of (9) with
By the first equation of (9), we have
Then substituting the above
From conditions (F1)-(F3) we get
Using the Cauchy-Schwartz inequality and the Young inequality, we have
By (19)-(24), it follows from (17) that
Recalling the new norm
Using the Gronwall lemma, we have
Substituting
By (5), we get
By assumption,
Note that
By (F3), we have that
Combining (28), (30), (31) and (32), there is a
where
To prove asymptotic compactness of the random dynamical system
Given
Choose a smooth function
and there exist constants
Lemma 2 Assume that (F1)-(F4),
lution
Proof. We first consider the random Equations (9)-(10). Then taking the inner product of the second equation
of (9) with
Substituting
By using conditions (F1), (F2) and (F3), we find
By the Cauchy-Schwartz inequality and the Young inequality, we obtain
Then it follows from (37)-(42) that
Letting
then, by (14) we have from (43) that
By using the Gronwall lemma, we get that
By replacing
By using (F3), there exists
In what follows, we estimate the terms on the right-hand side of (47). By (5),
Since
Note that
Next, we estimate the forth term on the right-hand side of (47). Using (F3), replacing t by s and then
it then follows that
Since
Letting
which implies
Then we complete the proof.
Let
Multiplying (9) by
Considering the eigenvalue problem
The problem has a family of eigenfunctions
such that
Lemma 3 Assume that (F1)-(F4),
Proof. Let
Then applying
Substituting
Using conditions (F1) and (F4), we have
it then follows that
By using the Cauchy-Schwartz inequality and the Young inequality, we have
From (63)-(73) we can obtain that
Since
Using the Gronwall lemma, we have
By substituting
We next estimate each term on the right-hand side of (77). Since
Since
term on the right-hand side of (77) satisfies
Next, we estimate the third term on the right-hand side of (77). By (6), (18) and (33),
which implies that there exists
Let
which completes the proof.
In this subsection, we prove the existence of a global random attractor for the random dynamical system generated by (9)-(10).
Theorem 2 Assume that (F1)-(F4),
Proof. Notice that the random dynamical system
Next, we will prove that the random dynamical system
Let
is a bounded in
By Lemma 2, we have that there are
In addition, it follows from Lemma 3 that there exist
Then, by (57) and (83),
implies that
dynamical system
Then, by Theorem 1, the random dynamical system
In the present article, we have discussed the existence of a random attractor to the stochastic strongly damped wave equation with additive noise defined on unbounded domains. It is also interesting to consider the the same
problem for stochastic strongly damped wave equation with multiplicative noise
coefficient
noise
in the future.
We thank the editor and the referee for their comments. The authors are supported by National Natural Science Foundation of China (Nos. 11326114, 11401244, 11071165 and 11471290); Natural Science Research Project of Ordinary Universities in Jiangsu Province (No. 14KJB110003); Zhejiang Natural Science Foundation under Grant No. LY14A010012 and Zhejiang Normal University Foundation under Grant No. ZC304014012. This support is greatly appreciated.