In this paper we investigate the existence of uniform attractors for a nonlinear non-autonomous thermoviscoelastic equation with strong damping
where is a bounded domain with smooth boundary , u and are displacement and temperature difference, respectively. (the past history of u) is a given datum which has to be known for all , the function g represents the kernel of a memory, are non-autonomous terms, called symbols, and is a real number such that
Now let us recall the related results on nonlinear one-dimensional thermoviscoelasticity. Dafermos  , Dafermos and Hsiao  , proved the global existence of a classical solution to the thermoviscoelastic equations for a class of solid-like materials with the stress-free boundary conditions at one end of the rod. Hsiao and Jian  , Hsiao and Luo  obtained the large-time behavior of smooth solutions only for a special class of solid-like materials. Ducomet  proved the asymptotic behavior for a non-monotone fluid in one-dimension: the positive temperature case. Watson  investigated the unique global solvability of classical solutions to a one-dimensional nonlinear thermoviscoelastic system with the boundary conditions of pinned endpoints held at the constant temperature and where the pressure is not monotone with respect to u and may be of polynomial growth. Racke and Zheng  proved the global existence and asymptotic behavior of weak solutions to a model in shape memory alloys with a stress-free boundary conditions at least at one end of the rod. Qin   obtained the global existence, and asymptotic behavior of smooth solutions under more general constitutive assumptions, and more recently. Qin  has further improved these results and established the global existence, exponential stability and the existence of maximal attractors in . As for the existence of global (maximal) attractors, we refer to    . More recently, Qin and Lü  obtained the existence of (uniformly compact) global attractors for the models of viscoelasticity; Qin, Liu and Song  established the existence of global attractors for a nonlinear thermoviscoelastic system in shape memory alloys.
Our problem is derived from the form
which has several modeling features. The aim of this paper is to extend the decay results in  for a viscoelastic system to those for the thermoviscoelastic system (1.1-1.2) and then to establish the existence of the uniform attractor for this thermoviscoelastic systems. In the case is a constant, Equation (1.6) has been used to model extensional vibrations of thin rods (see Love  , Chapter 20). In the case is not a constant, Equation (1.6) can model materials whose density depends on the velocity . For instance, a thin rod which possesses a rigid surface and with an interior which can deforms slightly. We refer the reader to Fabrizio and Morro  for several other related models.
Let us recall some results concerning viscoelastic wave equations. In  , the author concerned with the quasilinear viscoelastic equation
he proved that the energy decays similarly with that of g. In  , Wu considered the nonlinear viscoleastic wave equation
with the same boundary and initial conditions as (1.7), the author proved that, for a class of kernels g which is singular at zero, the exponential decay rate of the solution energy. Later, Han and Wang  considered a similar system like:
with Dirichlet boundary condition, where are constants, they proved the energy decay for the viscoelastic equation with nonlinear damping. Then Park and Park  established the general decay for the viscoelastic problem with nonlinear weak damping
with the Dirichlet boundary condition, where is a constant. In  , Cavalcanti et al. studied the following equation with Dirichlet boundary conditions
where . They established a global existence result for and an exponential decay of energy for , and studied the interaction within the and the memory term . Messaoudi and Tatar  established, for small initial data, the global existence and uniform stability of solutions to the equation
with Dirichlet boundary condition, where are constants. In the case in (1.12), Messaoudi and Tatar  proved the exponential decay of global solutions to (1.12) without smallness of initial data, considering only the dissipation effect given by the memory. Considering nonlinear dissipation. Recently, Araújo et al.  studied the following equation
and proved the global existence, uniqueness and exponential stability, and the global attractor was also established, but they did not establish the uniform attractors for non-autonomous equation. Then, Qin et al.  established the existence of uniform attractors for a non-autonomous viscoelastic equation with a past history
Moreover, we would like to mention some results in      .
For problem (1.1)-(1.4) with , when was replaced by , Han and Wang  established the global existence of weak solutions and the uniform decay estimates for the energy by using the Faedo-Galerkin method and the perturbed energy method, respectively. To the best of our knowledge, there is no result on the existence of uniform attractors for non-autonomous thermoviscoelastic problem (1.1)-(1.4). Therefore in this paper, we shall establish the existence of uniform attractors for problem (1.1)-(1.4) by establishing uniformly asymptotic compactness of the semi-process generated by their global solutions. Noting that the symbol , which are dependent in t, so our estimates are more complicated than   and we must use new methods to deal with the symbol as the change of time. Therefore we improved the results in   . For more results concerning attractors, we can refer to  -  .
Motivated by    , we shall add a new variable to the system which corresponds to the relative displacement history. Let us define
A direct computation yields
and we can take as initial condition ( )
Thus, the original memory term can be written as
and we get a new system
with the boundary conditions
and initial conditions
The rest of our paper is organized as follows. In Section 2, we give some preparations for our consideration and our main result. The statements and the proofs of our main results will be given in Section 3 and Section 4, respectively.
For convenience, we denote the norm and scalar product in by and , respectively. denotes a general positive constant, which may be different in different estimates.
2. Preliminaries and Main Result
We assume the memory kernel is a bounded function such that
and suppose that there exists a positive constant verifying
In order to consider the relative displacement as a new variable, one introduces the weighted L2-space
which is a Hilbert space equipped with inner product and norm
Define the generalized energy of problem (1.17)-(1.21)
To present our main result, we need the following global existence and uniqueness results.
Theorem 2.1. Let , , and any fixed . Assume (2.1) and (2.2) hold. Then problem (1.17)-(1.21) admits a unique global solution such that
We now define the symbol space for (1.17)-(1.21).
Observe the following important fact: The properly defined (uniform) attractor A of problem (1.17)-(1.21) with the symbol must be simultaneously the attractor of each problem (1.17)-(1.21) with the symbol , which is called the hull of and defined as
where denotes the closure in Banach space .
We note that
where is a translation compact function in in the weak topology, which means that is compact in . We consider the Banach space of functions with values in a Banach space that are locally p-power integrable in the Bochner sense. In particular, for any time interval ,
Let , consider the quantity
Lemma 2.1. Let defined as before and , then
is a translation compact in
is also a translation compact in
, moreover, ;
2) The set is bounded in such that
Proof. See, e.g., Chepyzhov and Vishik  .
Lemma 2.2. For every , every non-negative locally summable function on and every , we have
for a.a. .
Proof. See, e.g., Chepyzhov, Pata and Vishik  .
Similar to Theorem 2.1, we have the following existence and uniqueness result.
Theorem 2.2. Let , where is an arbitrary but fixed symbol function. Assume (2.1) and (2.2) hold. Then for any and for any , problem (1.17)-(1.21) admits a unique global solution , which generates a unique semi-process on of a two-parameter family of operators such that for any ,
Our main result reads as follows.
Theorem 2.3. Assume that
is defined by (2.8), then the family of processes
corresponding to (1.17)-(1.21) has a uniformly (w.r.t.
) compact attractor .
3. The Well-Posedness
The global existence of solutions is the same as in    , so we omit the details here. Next we prove the uniqueness of solutions.
We consider two symbols and and the corresponding solutions and of problem (1.17)-(1.21) with initial data and respectively. Let , , .
with Dirichlet boundary conditions and initial conditions
The corresponding energy for (3.1)-(3.3) is defined
It is easy to see that
Noting that is differentiable since . Then
To simplify notations, let us say that the norm of the initial data is bounded by some . Then given we use to denote several positive constants which depend on R and T.
By Young’s inequality and the interpolation inequalities, we derive
which, together with (3.6)-(3.9), yields for some large
Integrating (3.10) from to t and using Hölder’s inequality, we have
then we get for any
Applying Gronwall’s inequality, we see that
Using , we know that is equivalent to the norm of in and we get
which, together with (3.13), gives for all
This shows that solutions of (1.17)-(1.21) depend continuously on the initial data. We complete the proof of Theorem 2.1.
4. Uniform Attractors
In this section, we shall establish the existence of uniform attractors for system (1.17)-(1.21). To this end, we shall introduce some basic conceptions and basic lemmas. For more results concerning uniform attractors, we can refer to      .
Let X be a Banach space, and be a parameter set. The operators are said to be a family of processes in X with symbol space if for any ,
Let be the translation semigroup on , we say that a family of processes satisfies the translation identity if
By we denote the collection of the bounded sets of X, and .
Definition 4.1. A bounded set is said to be a bounded uniformly (w.r.t ) absorbing set for if for any and , there exists a time such that
for all .
In the following, as usual, (w.r.t) will represent “with respect to”.
Definition 4.2. The family of semi-processes is said to be asymptotically compact in X if is precompact in X, whenever is bounded in X, , and as .
Definition 4.3. A set is said to be uniformly (w.r.t ) attracting for the family of semi-processes if for any fixed and any ,
here stands for the usual Hausdorff semidistance between two sets in X. In particular, a closed uniformly attracting set is said to be the uniform (w.r.t ) attractor of the family of the semi-process
if it is contained in any closed uniformly attracting set (minimality property).
Definition 4.4. Let X be a Banach space and B be a bounded subset of be a symbol (or parameter) space. We call a function , defined on to be a contractive function on if for any sequence and any , there is a subsequence and such that
We denote the set of all contractive functions on by .
Lemma 4.1. Let be a family of semi-processes satisfying the translation identities (4.3) and (4.4) on Banach space X and has a bounded uniformly (w.r.t ) absorbing set . Moreover, assuming that for any , there exist and such that
Then is uniformly (w.r.t ) asymptotically compact in X.
Proof. This lemma is a version for semi-processes of a result by Khanmamedov  . A proof can be found in Sun et al.  , Theorem 4.2.
Next, we will divide into two subsections to prove Theorem 2.3.
4.1. Uniformly (w.r.t. ) Absorbing Set in
In this subsection we shall establish the family of processes has a bounded uniformly absorbing set given in the following theorem.
Theorem 4.1. Assume that and is defined by (2.7), then the family of processes corresponding to (1.17)-(1.21) has a bounded uniformly (w.r.t. ) absorbing set B in .
Proof. We define
Using Young’s inequality, Poincaré’s inequality, we arrive at
Then (4.11) gives , whence from (4.9), for
Now we define
From (1.17), integration by parts and Young’s inequality, we derive for any ,
Using Young’s inequality, Hölder’s inequality and Poincaré’s inequality, we deduce
hereinafter we use to represent the Poincaré constant.
From the expression of , we get
which, together with (4.15)-(4.19), yields
Noting that and the embedding theorem , we have for any ,
which, together with (4.20) and Poincaré’s inequality, gives
Now we take so small that
Hence from (4.21)-(4.22), it follow
We define the functional
It follows from (1.17) that
From Young’s inequality, Hölder’s inequality and Poincaré’s inequality, we derive for any ,
which, together with (4.26)-(4.29), gives
then we have
By Young’s inequality, we derive
and for any
which, together with (4.30)-(4.32) and taking small enough, yields
Inserting (4.30) and (4.33) into (4.25), we arrive at
where M and are positive constants.
Then it follows from (4.10), (4.23), (4.34) and (2.2) that
Now we claim that there exist two constants such that
For any , we take so small that
For fixed , we choose small enough and M so large that
Then there exist a constant such that
which, together with (4.37), gives
Integrating (4.40) over with respect to t and using Lemmas 2.2-2.3, we obtain
Now for any bounded set , for any , there exists a constant such that . Taking
then for any , we have
is a uniform absorbing ball for any . The proof is now complete.
4.2. Uniformly (w.r.t. ) Asymptotic Compactness in
In this subsection, we will prove the uniformly (w.r.t. ) asymptotic compactness in , which is given in the following theorem.
Theorem 4.2. Assume that and is defined by (2.8), then the family of processes corresponding to (1.17)-(1.21) is uniformly (w.r.t. ) asymptotically compact in .
Proof. For any . We consider two symbols and and the corresponding solutions and of problem (1.17)-(1.21) with initial data , , respectively. Let , , .
with Dirichlet boundary conditions and initial conditions
The corresponding energy for (4.42)-(4.45) is defined
Using Hölder’s inequality, Poincaré’s inequality and Theorem 4.1, we derive
which, combined with (4.47)-(4.50), yields
It is very easy to verify
Taking the derivative of , it follows from (4.42)-(4.43) that
Applying Hölder’s inequality, Young’s inequality, Poinceré’s inequality and Theorem 4.1, we get
By virtue of (4.46), we have
Then from (4.54)-(4.59), we can conclude
Now we define
From (4.42)-(4.43) and integration by parts, we derive
Using Hölder’s inequality, Poincaré’s inequality and Theorem 4.1, we derive for any ,
then we see that
Plugging (4.64)-(4.72) into (4.63), we get
On the other hand, we can get
which, together with (4.53) and (4.74), yields
Now we take so small and M so large that
Then for any , we have
Now we take and so small that
For fixed and , we choose M so large that
Then there exist some constant such that
Integrating (4.79) over with respect to t, we derive
For any fixed , we choose so large that
which, together with (4.77) and (4.80), gives
It suffices to show for each fixed . From the proof of existence theorem, we can deduce that for any fixed , and the bound B depends on T,
is bounded in .
Let be the solutions corresponding to initial data with respect to symbol . Then from (4.84), we get
Taking , , , , , , , , noting that compact embedding , passing to a subsequence if necessary, we have
and converge strongly in .
Therefore we get
On the other hand, by , we see that
Hence it follows from (4.88)-(4.92)
that is, .
Therefore by Lemma 3.1, the semigroup is uniformly asymptotically compact and the proof is now complete.
Proof of Theorem 2.3. Combining Theorems 4.1-4.2, we can complete the proof of Theorem 2.3.
Shanghai Polytechnical University and the key discipline Applied Mathematics of Shanghai Polytechnic University with contract number XXKPY1604.
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