On the upper semicontinuity of global attractors for damped wave equations

Abstract: We provide a new proof of the upper-semicontinuity property for the global attractors admitted by the solution operators associated with some strongly damped wave equations. In particular, we demonstrate an explicit control over semidistances between trajectories in the weak energy phase space in terms of the perturbation parameter. This result strengthens the recent work by Y. Wang and C. Zhong [7].


Introduction
In this short article, we revisit the recent work of [7] who examine the upper-semicontinuity properties of the family of global attractors associated with the strong damping perturbation of weakly damped wave equations.Such equations are used in modeling non-Hookean viscoelastic materials.Here, the strong damping term −ε∆u t present in such equations indicates that we are accounting for the strain rate in the material, in addition to other forces.The upper-semicontinuity result in [7] shows that the global attractors do not "blow-up" as the perturbation parameter vanishes.Hence, the asymptotic behavior of the solutions is stable.What we offer here improves this result by communicating that the difference of trajectories corresponding to the perturbation problem and the limit problem, emanating from the same initial data, can be estimated in terms of the perturbation parameter ε in the topology associated with the weak energy phase space of the model problems.
Let Ω be a bounded domain in R 3 with boundary ∂Ω of class C 2 .We consider the semilinear strongly damped wave equation, where 0 ≤ ε ≤ 1 represents the diffusivity of the momentum.The equation is endowed with Dirichlet boundary condition, and with the initial conditions u(0, x) = u 0 (x), u t (0, x) = u 1 (x) at {0} × Ω. (1.3) For the nonlinear term, we assume f ∈ C 2 (R) satisfies the sign condition where λ 1 > 0 denotes the first eigenvalue of the Dirichlet-Laplacian, and we assume the growth assumption holds, for all s ∈ R, for some positive constant .We will refer to equations (1.1)-(1.3)under assumptions (1.4)-(1.5)as It is now well-known that the model problems admit globally defined weak-solutions in the (weak) energy phase space . Furthermore, when ε > 0, the operator associated with the linear part of the abstract Cauchy problem generates an analytic semigroup on H 0 .On these results we mention the following references [1][2][3][4][5].
The main result in this paper is the following: Theorem 1.1.The family of global attractors {A ε } ε∈[0,1] is upper-semicontinuous in the topology of H 0 in the following explicit sense: there is a constant C > 0 independent of ε in which A word about notation: we will often drop the dependence on x and even t from the unknown u(x, t) writing only u instead.The norm in the space L p (Ω) is denoted • p except in the common occurrence when p = 2 where we simply write the L 2 (Ω) norm as • .The L 2 (Ω) product is simply denoted (•, •).Other Sobolev norms are denoted by occurrence; in particular, since we are working with the homogeneous Dirichlet boundary conditions (1.2), in H 1 0 (Ω), we will use the equivalent norm u H 1 0 = ∇u .Given a subset B of a Banach space X, denote by B X the quantity sup x∈B x X .In many calculations C denotes a generic positive constant which may or may not depend on several of the parameters involved in the formulation of the problem.Finally, for each ε ∈ [0, 1], and t ≥ 0, we denote by S ε (t) the semigroup of solution operators acting on H 0 defined through the weak solution, where u ε here denotes the weak solution to Problem P ε .
The next section contains a proof of Theorem 1.1.

Continuity properties of the global attractors
Following [6, Section 10.8], the type of perturbation examined in this article is called regular because both classes of Problem P ε (ε > 0 and ε = 0) lie in the same phase space; in particular, the family of global attractors, {A ε } ε∈[0,1] , lies in H 0 .Hence, we will utilize [6,Theorem 10.16].
Proposition 2.1.Assume that for ε ∈ [0, ε 0 ) the semigroups S ε each admit a global attractor A ε and that there exists a bounded set X such that If in addition the semigroup S ε converges to S 0 in the sense that, for each t > 0, S ε (t)x → S 0 (t)x uniformly on bounded subsets Y of the phase space H, i.e., We now arrive at our first result.

Then ζ and ū satisfy the equations
After multiplying the equation (2.2) 1 by 2ū t in L 2 (Ω), we estimate the new product to arrive at the differential inequality, (2. 3) The constant C = C(L, Ω) > 0 is due to the local Lipschitz condition of f : H 1 0 → L 2 following assumptions (1.4) and (1.5), as well as the embedding H 1 0 → L 2 .It suffices to find an appropriate bound for ∇u 0 t (t) 2 .Indeed, since the global attractor for Problem P 0 consists of strong solutions (A 0 is bounded in H 1 ), we are allowed to test/multiply the weakly damped wave equation in L 2 (Ω) by −2∆u 0 t (t).To this end we obtain, Integrating this inequality over [0, T ] yields the desired bound, where the constant over [0, T ] and apply the bound (2.4) to the last term on the right-hand side to produce the claim (2.1).This completes the proof.
Remark 2.3.The above result (2.1) establishes that, on compact time intervals, the difference between trajectories of Problem P ε , ε ∈ (0, 1], and Problem P 0 , originating from the same initial data on A ε ⊂ H 1 , can be controlled, explicitly, in terms of the perturbation parameter ε in the topology of H 0 .
The well-known upper-semicontinuity result in Proposition 2.1 now follows for our family of global attractors.
depends on the bound on A 0 in H 1 (through the initial condition) and on T > 0.