Attractors for a Class of Abstract Evolution Equations with Fading Memory

In this paper, we study the dynamics of an abstract evolution equation with fading memory with a critical growing nonlinearity. By use of some new methods and asymptotic estimate techniques, we first verify the asymptotic compact of solution semigroup and then prove the existence of global attractors in weak topological space and strong topological space, while the forcing term only belongs toH−1(Ω) or L2(Ω), respectively. The results are new and appear to be optimal.

And there exists a constant  > 0, such that and assumption (6) about  will be used to prove the existence of absorbing set in the strong topological space and that it has nothing to do with the testing of asymptotic compactness.
For the semilinear hyperbolic equation in viscoelasticity, the asymptotic behaviors of the solution have been studied in [1,[9][10][11][12][13].In [1,9,10], the authors investigated the asymptotic behaviors of solutions when nonlinearity is subcritical growth.In [11], Sun et al. considered large-time behaviors of the solutions and gained the existence of global attractors for nonautonomous strongly damped wave-type evolution equation with the critical nonlinearity and linear memory.In [12], Cavalcanti et al. studied the long-time dynamics of a semilinear wave equation with degenerate viscoelasticity defined in a bounded domain Ω of R 3 , with Dirichlet boundary condition and nonlinear forcing term with critical growth.In [13], Zhou and Zhao proved the existence of random attractors for the continuous random dynamical systems generated by stochastic damped nonautonomous wave equations with linear memory and additive white noise when the nonlinearity has a critically growing exponent and studied the upper semicontinuity of random attractors.
For the floating beam equation, the authors considered the dynamics and obtained the existence of a global attractor for the deterministic floating beam in [8] and deeply studied the existence of a compact random attractor for the random dynamical system generated by a model for nonlinear oscillations in a floating beam equation with strong damping and white noise in [14].
The above equations are all special cases of an abstract evolution equation with fading memory that we will study in this paper.Until now, we find that no one else has studied the long-time behavior of the solutions about the problem with weak damping and critical nonlinearity, which just causes our strong research motivation and interest.Based on this, we will study the asymptotic behaviors and regularity of solutions for this problem in this paper.
As we know, if we want to prove the existence of global attractors, the key point is to obtain the compactness of the semigroup in some sense.However, there exist some essential difficulties to test the continuity and compactness (asymptotic compactness) of the semigroup of solutions for (1).First, due to the memory terms, we cannot use ( −   ) or ( −   )   as test function to verify the asymptotic compactness of the solution semigroup; second, the critical nonlinearity leads to the fact that some obstacles are hard to avoid in the process of energy estimation.Furthermore, since   is an abstract operator, it brings about more substantive barriers in the process of proof.Despite all this, we still overcome the above bottlenecks in the process of estimation and proof by applying semigroup theory and decomposition technology.Finally, we test the continuity and compactness (asymptotic compactness) of the solution semigroup of solutions and obtain the existence of global attractors of (1) in the weak topological space   ×  2 (Ω) ×  2  (R + ,   ) and the strong topological space  2 ×   ×  2  (R + ,  2 ).This paper is organized as follows.Some preliminaries, including some notations that we will use, the assumption on nonlinearity, and some general abstract results about dynamical system are presented in Section 2.Then, the proof of our main result about compactness testing and the existence of global attractor for the dynamical system generated by the solution of (1) are given in Sections 3 and 4. The main results are Theorems 11 and 18.

Preliminaries
Let  =  2 (Ω) and (, V) be a bilinear continuous form on  which is symmetric and coercive.With this form, we associate the linear operator  in  by setting (, V) = (, V), ∀, V ∈ , and  can be considered as a self-adjoint unbounded operator in  with domain () ⊂ .
In the remainder of this section, we denote by {()} ⩾0 the semigroup associated with the solution of ( 17)- (18).
Similar to the proof of Theorem 5, we can gain the corresponding existence and uniqueness of solutions for (55) and (56); furthermore, we know that the solutions of (55) also form a semigroup.For the sake of convenience, we denote the solution operators of ( 55) and ( 56) by { 1 ()} ⩾0 and { 2 (, ⋅)} ⩾0 , respectively.Then, for every  0 ∈ H 1 , we have Hereafter, we will test the necessary condition of asymptotic smoothness.
At first, akin to the proof of Theorem 6, we can obtain the following result about the solution  1 () of (55) in H 1 .
Proof.Taking the  2 inner product of (70) with V   + V  , we have Mathematical Problems in Engineering 7 For the forcing term, we know About the nonlinearity, from (48), we have where  0 (V  ) = ∫ V  0 ().In view of (49), ( 13), (14), and (44), we know (62) here the constant  ,, 0 being very small depends not only on  and , but also on the bound  0 in (44).Choosing  to be sufficiently small, we have Thanks to Lemma 1 and using (7), we get that Then, Then, combining the above estimates with (59), we see that where  1 = min{/2, /2, /2,  2 }.
After that, we can easily know that By virtue of (88), we get that (ii) holds.
We complete the proof.
Moreover, applying Lemma 2, we can obtain that K   is relatively compact in  2  (R + ;   ).Then, using the compact embedding  + →   once again, we can get the following.Lemma 10.Let  2 (, ⋅) be the corresponding solution operator of (56).Under the assumption of Lemma 8, for any  > 0,  2 (,  * 0 ) is relatively compact in H 1 .

The Existence of Bounded Absorbing Set.
At first, we will prove the existence of bounded absorbing set in H 2 .
For the forcing term, we see easily and since  ∈  2 (Ω), then Taking  small enough, we then obtain that In combination with the above estimates, we can deduce that, for all  ⩾  0 , where  = 2‖‖ 2 and  3 = min{, /2, /2}.From (102), there exist constants  1 ,  2 , such that therefore, applying Gronwall lemma, we have Presuming that ‖(0 The proof is complete.

The Existence of Global
Attractor in H 2 .According to the general theorem about the existence of global attractors of infinite-dimensional dynamical systems (see [6,15]), we also need to test the asymptotic compactness of {()} ⩾0 in H 2 .
The proof is complete.
Let  * 1 be the bounded absorbing set obtained from Theorem 14, and Thanks to Lemma 2, we can obtain that K  *  is relatively compact in  2  (R + ;  2 ).Thus, using the compact embedding  2+ →  2 once again, we can get the following.Lemma 17.Let  2 (, ⋅) be the corresponding solution operator of (56).Under the assumption of Lemma 16, for any  > 0,  2 (,  * 1 ) is relatively compact in H 2 .