Asymptotics
of
viscoelastic
materials
with
nonlinear
density
and
memory
effects
✩M. Conti
a,
T.F.
Ma
b,∗,
E.M. Marchini
a,
P.N. Seminario Huertas
b a PolitecnicodiMilano,DipartimentodiMatematica,ViaBonardi9,20133Milano,Italyb DepartamentodeMatemática,InstitutodeCiênciasMatemáticasedeComputação,UniversidadedeSãoPaulo,13566-590 SãoCarlos,SP,Brazil
Abstract
Thispaperisconcernedwiththenonlinearviscoelasticequation
|∂tu|ρ∂t tu− ∂t tu− u +
∞ 0
μ(s)u(t− s) ds + f (u) = h,
suitabletomodelingextensionalvibrationsofthinrodswithnonlinearmaterialdensity(∂tu)= |∂tu|ρ, andpresenceofmemoryeffects.Thisclassofequationswasstudiedbymanyauthors,butwell-posedness inthewholeadmissiblerangeρ∈ [0,4] andforf growinguptothecriticalexponentwereestablished onlyrecently.Theexistenceofglobalattractorswasprovedinpresenceofanadditionalviscousor fric-tionaldamping.Inthepresentworkweshowthatthesoleweakdissipationgivenbythememorytermis enoughtoensureexistenceandoptimalregularityoftheglobalattractorAρforρ <4 andcritical nonlin-earity f .
✩ The first and the third authors are partially supported by the research project GNAMPA-INdAM 2015 “Proprietà asintotiche di sistemi differenziali con memoria degenere”, the second author by CNPq grant 310041/2015-5, and the fourth by CAPES/PROEX grant 8477445/D.
* Corresponding author.
E-mailaddresses:monica.conti@polimi.it(M. Conti), matofu@icmc.usp.br(T.F. Ma), elsa.marchini@polimi.it (E.M. Marchini), pseminar@icmc.usp.br (P.N. Seminario Huertas).
Keywords: Viscoelastic equation; Memory; Nonlinear density; Global attractors
© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/ by-nc-nd/4.0/
Published Journal Article available at: https://doi.org/10.1016/j.jde.2017.12.010
Received 27 January 2017; revised 7 August 2017
1. Introduction
In the recent literature the nonlinear equation
|∂tu|ρ∂t tu− ∂t tu− γ ∂tu− u + t
0
μ(t− s)u(s) ds + F = 0 in × R+, (1.1)
has been considered by many authors. The problem is usually set in a bounded domain of RN
with
0 < ρ≤ 2
N− 2 if N≥ 3 and ρ > 0 if N = 1, 2, (1.2)
where γ ≥ 0 accounts for the structural mechanical damping, μ is a nonnegative nonincreasing function describing memory effects in the material, and F represents forcing terms. Its motiva-tion comes from the prototype equamotiva-tion
∂t tu− ∂t tu− u = 0, (1.3)
which models several applications, among them, extensional vibrations of thin roads [19, Chap. 20]and ion-sound waves[3, Sec. 6]. However, as observed in[6], in certain cases the material density may depend on small variations of the velocity, that is, = (∂tu). Then a
natural assumption would be
(∂tu)= 1 + |∂tu|ρ, ρ >0,
where ∈ R is a small parameter. By simplicity, without loss of mathematical complexity, we shall assume (∂tu) = |∂tu|ρ, ρ > 0. From this understanding, we obtain equation (1.1)by
adding memory effects and damping to the model (1.3).
Equation (1.1)was firstly studied in[5]with F = 0, where the existence of global solutions is proved; besides, in the case γ > 0, which corresponds to the strongly damped problem, the decay of the energy as t→ ∞ is established. Later, many authors have contributed to this problem in several directions, cf. [6,15–18,20–23,25,32], always assuming condition (1.2).
On the other hand, the uniqueness of solutions to this class of problem was not considered until the recent paper[1], in the more general context with infinite delay
|∂tu|ρ∂t tu− ∂t tu− γ ∂tu− u +
∞ 0
μ(s)u(t− s) ds + f (u) = h, (1.4)
complemented by the Dirichlet boundary condition
Here, u(t) = u0(t)for t≤ 0 is a prescribed “past history”, and the positive kernel μ ∈ C1(R+) ∩ L1(R+)with μ(0) <∞, satisfies the classical assumption
μ + δμ ≤ 0 for some δ > 0. (1.6)
With dissipation provided by the memory kernel μ only (that is γ= 0), the authors proved the well-posedness of the problem and the exponential decay of the energy, for a class of nonlinear-ities f (u) growing at most as |u|NN−2. In addition, by assuming γ > 0, the existence of a global attractor was also proved. Nonetheless, their argument for uniqueness rely on the differentiability of s→ |s|ρsat s= 0, and it is therefore valid only for ρ > 1.
Subsequently, in the same context of past history, the existence of uniform and pullback attrac-tors were proved in [28,29]by assuming f (u) = 0 and the restriction ρ > 1 in (1.2). However, the dissipation assumption is weakened by taking a frictional damping ∂tuinstead of −∂tu.
The well-posedness of the problem was revisited in[8]in the case of N= 3, where the previ-ous literature was significantly improved by allowing
ρ∈ [0, 4] and |f (u)| ≤ C(1 + |u|p), p= 5,
both the exponents ρ= 4 and p = 5 being critical in space dimension three. Besides, the kernel
μis required to satisfy much weaker assumptions than (1.6). Then, in[9], in the case when the strong damping −∂tuis present, the same authors proved the existence and optimal regularity
of the global attractor for ρ < 4.
The aim of the present paper is to contribute with some new results for the class of prob-lems (1.4)–(1.5)without additional frictional or structural damping. They are summarized in the
following.
(a) Letting γ = 0 in (1.4), we prove the existence of the global attractor of optimal regularity for 0 ≤ ρ < 4 and critical nonlinearity f . Besides, when h ≡ 0 and f is essentially positive in the sense of (2.2), we prove the exponential decay of the energy associated to the system, including the critical case ρ= 4. This is done by defining a new multiplier which exploits the energy relation between |∂tu|ρ∂t tuand the memory term in order to overcome the lack
of instantaneous mechanical damping (see Lemma 4.2and the proof of Lemma 6.3).
(b) With respect to the equation (1.3), with = 1, it was shown in[27]that a weak frictional damping ∂tuis not sufficient for the exponential stabilization of the energy. In this direction,
the existence of global attractors for the corresponding semilinear system was established in[4,30]by adding a strong damping −∂tu. Therefore our results, even with = 1, also
complement the ones in[4,27,30]since we replace the structural damping by memory dissi-pation.
1.1. Assumptions
Throughout the paper we always let the following assumptions to be in place. • ⊂ R3is a smooth bounded domain, γ = 0 and ρ ∈ [0, 4].
• The nonlinearity f is locally Lipschitz, with f (0) = 0, and fulfills the critical growth restriction |f (u) − f (v)| ≤ c|u − v|(1 + |u|4+ |v|4),
(1.7) along with the dissipation conditions1
f (u)u≥ F (u) −λ1 2 (1− ν)|u| 2− m f (1.8) F (u)≥ −λ1 2 (1− ν)|u| 2− m f (1.9)
for some ν > 0 and mf ≥ 0. Here λ1>0 denotes the first eigenvalue of the Dirichlet operator
− and F (u)= u 0 f (y)dy.
• The convolution (or memory) kernel μ ∈ C1(R+)is a nonnegative, nonincreasing function on R+of finite total mass
∞ 0 μ(s)ds= κ ∈ (0, 1), satisfying μ (s)+ δμ(s) ≤ 0, ∀s ∈ R+, (1.10)
for some δ > 0. Besides, μ(0) <∞.
1.2. Functional setting
We define A = −, the Dirichlet operator on L2()with domain H2() ∩ H1
0()and we introduce the scale of compactly nested Hilbert spaces
Hr= Dom(Ar2), u, vr= A2ru, A2rv
L2(), ur= A
r 2u
L2(). The index r∈ R is omitted whenever zero. In particular,
H−1= H−1(), H= L2(), H1= H01(), H2= H2()∩ H01().
Further, we define the history spaces
1 Assumption (1.8)–(1.9)are satisfied, for instance, if f∈ C1(R) with lim inf |u|→∞f
(u)>−λ 1.
M. Conti et al. / J. Differential Equations 264 (2018) 4235– 4259
Mr= L2
μ(R+; H1+r )
endowed with the inner products
η, ξMr = ∞ 0
μ(s)η(s), ξ(s)1+rds.
We consider the infinitesimal generator T of the right-translation semigroup on M defined as
T η= −η , Dom(T )=η∈ M : η ∈ M, η(0) = 0.
Here the prime stands for weak derivative. Since μ is decreasing, the following inequality holds for every η∈ Dom(T ) (see e.g.[13])
−T η, ηM= − ∞ 0 μ(s)T η(s), η(s)1ds= −1 2 ∞ 0 μ (s)η(s)21ds≥ 0. (1.11)
At last, the extended history spaces are defined as
Hr= H1+r× H1+r× Mr.
Notation. Along the paper we will use diffusively the usual Sobolev embeddings, the Young,
Hölder and Poincaré inequalities, often without explicit mention. The generic positive constants will be denoted as c, while Q(·) will stand for a generic increasing positive function.
2. The dynamical system: old and new results
Following the Dafermos approach[12]as in [8,9], problem (1.4)–(1.5)can be rewritten in the equivalent form ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ |∂tu|ρ∂t tu+ A∂t tu+ Au + ∞ 0 μ(s)Aη(s)ds+ f (u) = h, ∂tη= T η + ∂tu. (2.1)
The model is subject to the initial conditions
u(0)= u0, ∂tu(0)= v0, η0= η0,
where u0, v0: → R and η0: × R+→ R are prescribed functions. Then, according to[8], we have the following existence and uniqueness result.
Theorem 2.1. System(2.1)generates a solution semigroup S(t) : H → H that satisfies the joint continuity
(t, z)→ S(t)z ∈ C([0, ∞) × H, H).
Besides, given any initial data z= (u0, v0, η0) ∈ H and denoting the corresponding solution by
(u(t), ∂tu(t ), ηt)= S(t)z,
we have the explicit representation formula
ηt(s)=
u(t )− u(t − s) 0 < s≤ t,
η0(s− t) + u(t) − u0 s > t.
We also recall the following uniform energy estimate (see [8, Proposition 4.2]).
Proposition 2.2. Let B ⊂ H be any bounded set of H and let z ∈ B. Then, S(t)zH+ ∂t tu(t )1≤ Q(BH), ∀t ≥ 0,
whereQ is independent of the particular choice of z in B.
Aim of this paper is studying the long-term behavior of the semigroup S(t).
2.1. Asymptotic behavior: new results
Before presenting our results we recall some well-known definitions related to attractors of a dynamical system (X, S(t)), where S(t) is a strongly continuous semigroup acting on a Banach space X. We refer the reader to the classical textbooks [2,7,14,31](see also[24]).
• A global attractor of a dynamical system (X, S(t)) is a compact set A ⊂ X that is fully invariant and uniformly attracting, namely,
S(t)A = A, ∀ t ≥ 0 and lim
t→∞distX(S(t)B,A) = 0,
for any bounded set B⊂ X, where dXdenotes the Hausdorff semidistance
dX(A, B)= sup x∈A
inf
y∈Bx − yX.
• Given a set B ⊂ X, its unstable manifold Mu(B)is the set of points z∈ X that belong to
some complete trajectory {y(t)}t∈Rand satisfy
y(0)= z and lim
• S(t) is called a gradient system on X if there exists a Lyapunov functional, according to the following definition.
Definition 2.3. A function L ∈ C(X, R) is called a Lyapunov functional if
(i) L(ζ ) → ∞ if and only if ζ X→ ∞;
(ii) t→ L(S(t)z) is nonincreasing for any z ∈ X;
(iii) if L(S(t)z) = L(z) for all t > 0, then z is a stationary point for S(t). Our main result is the following.
Theorem 2.4. Let Assumptions 1.1be in play, h ∈ H and ρ < 4. Then, the dynamical system
(H, S(t)) corresponding to problem (2.1)is a gradient system and possesses a global attractor
A characterized as the unstable manifold Mu(N ) of the set N of stationary solutions of (2.1).
Furthermore,A is bounded in H1.
Moreover, when the nonlinearity satisfies the further assumption
f (s)s≥ F (s) ≥ 0, for every s∈ R, (2.2)
and no external force is acting, we can prove the uniform exponential decay of the associated energy.
Theorem 2.5. Let Assumptions 1.1be in play with h ≡ 0, and let f satisfy (2.2). Then, there
exists ω > 0 such that, for any B ⊂ H bounded, the following decay estimate holds S(t)zH≤ Q(BH)e−ωt, ∀t ≥ 0,
for every z∈ B and some Q independent of z. The rest of the paper is devoted to the proofs.
3. The gradient system structure
Here we show the existence of a gradient system structure. As a preliminary step, we note that the dissipation conditions (1.8)–(1.9)imply
f (u), u ≥ F (u), 1 −1 2(1− ν)u 2 1− Mf, (3.1) F (u), 1 ≥ −1 2(1− ν)u 2 1− Mf, (3.2) where Mf = mf||.
Proposition 3.1. S(t) is a gradient system on H. Moreover, the set N of stationary points of S(t)
Proof. For ζ = (u, v, η), let us define L(ζ )=1 2ζ 2 H+ρ+ 21 |v|ρ+2dx+ F (u), 1 − h, u.
Owing to (3.2)and to the growth assumption (1.7), we can prove in a standard way that
ν 4ζ 2 H− cf,h≤ L(ζ ) ≤ cζ2H 1+ ζ 4H + h2−1, (3.3) with cf,h= Mf+ 1 νh 2 −1.
This immediately establishes (i) of Definition 2.3. Now, in light of (1.11), a multiplication of
(2.1)by (∂tu, η)in H × M gives d dtL(S(t )z)= T η t, ηt M=12 ∞ 0 μ (s)ηt(s)21ds. (3.4)
Since μ is decreasing, the right-hand-side is negative, this yields (ii). In order to prove (iii), let us suppose that for some initial data z0= (u0, u1, η0) ∈ H the map t → L(S(t)z0)is constant. Then, from (3.4)and exploiting (1.10), we have
0=1 2 ∞ 0 μ (s)ηt(s)21ds≤ −δ 2η t2 M, ∀ t > 0.
Therefore ηt(x, s) = 0 for a.e. x ∈ and t, s > 0 yielding in particular η0= 0. Besides, from the
second equation in (2.1)we infer that ∂tu(t ) = 0, so that u(t) = u0for all t . In conclusion,
S(t)z0= z0= (u0,0, 0),
meaning that z0is a stationary point. This proves that S(t) is a gradient system on H.
We are left to show that the set of stationary points is bounded. To this aim, note that any stationary solution (u, 0, 0) of S(t) satisfy
−u + f (u) = h. Then, a multiplication by u in H yields
u2 1+ f (u)udx= hudx,
νu21≤ 2Mf + hu ≤
ν
2u 2
1+ 2Mf + ch2.
We thus conclude that u1≤ c, as claimed. 2
4. Exponential decay
In this section we restrict the attention to a class of nonlinearities satisfying the further as-sumption (2.2). In which case, we can prove the uniform exponential decay of the associated energy, as stated in Theorem 2.5. We start by showing a weaker form of the theorem, where the rate of the exponential decay of the energy depends on the size of the initial data.
Proposition 4.1. Let h ≡ 0 and f satisfy (2.2). Then, for any B ⊂ H bounded, there exists
σ= σBH>0 such that
S(t)zH≤ Q(BH)e−σ t, ∀t ≥ 0, for every z∈ B.
To this aim, the following technical tool plays an essential role.
Lemma 4.2. Let h ≡ 0 and let S(t)z = (u(t), ∂tu(t ), ηt) be the solution at time t >0 to problem (2.1), with zH≤ R. We define the functional
(t )= − ∞ 0 μ(s)∂tu(t ), ηt(s)1ds− 1 ρ+ 1 ∞ 0 μ(s)|∂tu(t )|ρ∂tu(t ), ηt(s)ds.
Then, there exists Q(R) > 0, independent of z, such that, for every ε > 0, d dt+ κ 2∂tu 2 1+ κ ρ+ 1 |∂tu|ρ+2dx ≤ εu2 1+ Q(R) ε η 2 M+ Q(R) ∞ 0 − μ (s)η(s)2 1 ds.
Proof. Taking the time derivative of , we get
d dt= − ∞ 0 μ(s)A∂t tu+ |∂tu|ρ∂t tu, η(s)ds − ∞ 0 μ(s)∂tu, ∂tη(s)1ds− 1 ρ+ 1 ∞ 0 μ(s)|∂tu|ρ∂tu, ∂tη(s)ds.
−
M. Conti et al. / J. Differential Equations 264 (2018) 4235– 4259 ∞ 0 μ(s)A∂t tu+ |∂tu|ρ∂t tu, η(s)ds = ∞ 0 μ(s)Au + f (u), η(s)ds + ∞ 0 μ(s)A1/2η(s)ds 2 where ∞ 0 μ(s)A1/2η(s)ds 2 ≤ ∞ 0 μ(s)A1/2η(s)ds 2 ≤ κη2 M.
Besides, since f (0) = 0 and |f (s)| ≤ c|s|(1 + s4)by (1.7), and taking into account that u1is bounded (with a bound depending on R), we have
f (u)L6/5≤ Q(R)u1 so we obtain
∞ 0
μ(s)Au + f (u), η(s)ds ≤ c(u1+ f (u)L6/5) ∞ 0 μ(s)η(s)1ds ≤Q(R) ε η 2 M+ εu21. Using the second equation of (2.1), we have
−∂tu, ∂tηM= −κ∂tu21− ∂tu, T ηM.
Integrating by parts with respect to s, in light of the decay of μ and of the equality η(0) = 0, we obtain −∂tu, T ηM= − ∞ 0 μ (s)∂tu, η(s)1ds ≤ ∂tu1 − ∞ 0 μ (s)η(s)1ds ≤ κ 4∂tu 2 1− cμ(0) ∞ 0 μ (s)η(s)21ds. Besides,
− ∞ 0 μ(s)|∂tu|ρ∂tu, ∂tη(s) ds = −κ∂tuρL+2ρ+2− ∞ 0 μ(s)|∂tu|ρ∂tu, T η(s) ds.
Exploiting the uniform bound ∂tu1≤ Q(R)
− 1 ρ+ 1 ∞ 0 μ(s)|∂tu|ρ∂tu, T η(s)ds ≤ −c∂tuρ1∂tu1 ∞ 0 μ (s)η(s)1ds ≤κ 4∂tu 2 1− Q(R)μ(0) ∞ 0 μ (s)η(s)21ds.
Collecting all the above inequalities the proof is done. 2
Proof of Proposition 4.1. Take B ⊂ H bounded and fix z ∈ B. Let
L(t )= L(S(t)z) =1 2S(t)z 2 H+ρ+ 21 |∂tu(t )|ρ+2dx+ F (u(t)), 1
be the Lyapunov functional of Proposition 3.1, where now h ≡ 0 and f satisfies (2.2). As a consequence, the two-sides control (3.3)becomes
ν 4S(t)z 2 H≤ L(t) ≤ cS(t)z2H 1+ S(t)z4H , ∀t ≥ 0. (4.1) Besides, L satisfies the equality
d dtL(t )+ 1 2 ∞ 0 − μ (s)ηt(s)21 ds= 0, hence, by assumption (1.10)on μ, d dtL(t )+ δ 4η t2 M+14 ∞ 0 − μ (s)ηt(s)2 1 ds≤ 0. (4.2)
We introduce the further functional
(t)= ∂tu(t ), u(t )1+
1
ρ+ 1|∂tu(t )|
ρ∂
tu(t ), u(t ).
d dt+ u 2 1+ f (u), u = − ∞ 0 μ(s)η(s), u1ds + ∂tu21+ 1 ρ+ 1 |∂tu|ρ+2dx,
hence from (2.2)we obtain d dt+ 1 2u 2 1+ F (u), 1 ≤ κ 2η 2 M+ ∂tu21+ 1 ρ+ 1 |∂tu|ρ+2dx. (4.3)
Now, for α, β > 0 to be properly chosen later, we define
E(t) = L(t) + α(t) + β(t),
where is given in Lemma 4.2and satisfies, for some c= c(BH) >0 and for any ε > 0,
the inequality d dt+ κ 2∂tu 2 1+ κ ρ+ 1 |∂tu(t )|ρ+2dx (4.4) ≤ εu2 1+ c ε η 2 M+ c ∞ 0 − μ (s)η(s)2 1 ds.
Collecting (4.2), (4.3)and (4.4), we end up with d dtE + δ 4η t2 M+ 1 4 ∞ 0 − μ (s)ηt(s)21 ds+ ακ 2∂tu 2 1 + α κ ρ+ 1 |∂tu(t )|ρ+2+ β 2u 2 1+ βF (u), 1 ≤ αεu2 1+ α c εη 2 M+ αc ∞ 0 − μ (s)η(s)21 ds + βκ 2η 2 M+ β∂tu21+ β ρ+ 1 |∂tu|ρ+2dx.
This implies that taking
β=ακ 4 , ε= κ 16, and α≤ min δ 8 1 c16 κ + κ2 8 , 1 4c ,
we get the inequality d dtE + ακ 16u 2 1+ α κ 4∂tu 2 1+ δ 8η t2 M+ (4.5) + α κ 2(ρ+ 1) |∂tu(t )|ρ+2dx+ ακ 4 F (u), 1 ≤ 0.
Now, by the embedding H1⊂ L65(ρ+1), which holds since ρ≤ 4, we have || ≤1 2u 2 1+ 1 2∂tu 2 1+ 1 ρ+ 1 |∂tu|ρ+1|u| dx ≤1 2u 2 1+ 1 2∂tv 2 1+ 1 ρ+ 1∂tu ρ+1 L65(ρ+1)uL 6 ≤1 2u 2 1+ 1 2∂tu 2 1+ c∂tuρ1 ∂tu21+ u21 ≤ Q(BH)S(t)z2H, and, similarly, || ≤1 2∂tu 2 1+ κ 2η 2 M+ρ+ 11 ∞ 0 μ(s) |∂tu|ρ+1|η(s)| dx ds ≤ Q(BH)S(t)z2H.
Hence, for any sufficiently small α > 0 (depending on BH), we obtain by (4.1)
ν
8S(t)z 2
H≤ E(t) ≤ Q(BH)S(t)z2H. (4.6)
Combining (4.5)and (4.6), we end up with the differential inequality d
dtE + 2σ E ≤ 0,
for some σ= σ (BH) >0. An application of the Gronwall lemma in light of (4.6)yields
ν
8S(t)z 2
H≤ E(t) ≤ e−2σ tE(0) ≤ Q(BH)e−2σ t,
ending the proof. 2
We are now ready to prove the full decay result.
Proof of Theorem 2.5. Let B a bounded subset of H and let z ∈ B. From Proposition 4.1there exists T=T(BH) ≥ 0 such that
S(T)zH≤ 1.
S(t)zH= S(t −T)S(T)zH≤ Q(BH)eωTe−ωt, ∀t >T.
Further, since there exists Q(BH)such that
S(t)zH≤ Q(BH), ∀t ≤T,
joining the last two inequalities the thesis follows. 2
Remark 4.3. It is clear from the proof that the conclusion of Theorem 2.5holds for any nonlin-earity f satisfying (3.1)–(3.2)with Mf = 0.
5. Existence of the global attractor In what follows we assume that
ρ∈ [0, 4)
and we prove the first part of Theorem 2.4. Namely, in this section we establish the existence of the global attractor A, as well as its characterization as the unstable manifold Mu(N ) of the
stationary solutions of (2.1). Both the claims will be obtained by a general result of the theory of dynamical systems in presence of a gradient structure, see Theorem A.3 in[10]. Accordingly, in light of Proposition 3.1, it will suffice to find, for every bounded set B ⊂ H, a decomposition
S(t) = L(t) + N(t) with the following properties:
lim t→∞ sup x∈B L(t)xH= 0, (5.1) and N (t )B ⊂ K(t), (5.2)
where K(t) is a compact set depending on t.
To this aim, we need a suitable decomposition of f , envisaged in Lemma 5.1 of[9].
Lemma 5.1. The nonlinearity f admits the decomposition
f (s)= f0(s)+ f1(s) for some f0, f1with the following properties:
• f1is Lipschitz continuous with f1(0) = 0;
• f0vanishes inside [−1, 1] and fulfills the critical growth restriction |f0(u)− f0(v)| ≤ c|u − v|(|u| + |v|)4;
• f0fulfills for every s∈ R the bounds
f0(s)s≥ F0(s)≥ 0,
where F0(s) =0sf0(y) dy.
Now let B ⊂ H be any given bounded set, and let z ∈ B be fixed. We introduce the operators
L(t )z= (ˆv(t), ∂tˆv(t), ˆξt) and N (t )z= ( ˆw(t), ∂t ˆw(t), ˆψt)
by considering the solutions at time t > 0 to the problems ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ |∂tˆv|ρ∂t tˆv + A∂t tˆv + Aˆv + ∞ 0 μ(s)A ˆξ (s)ds+ f0(ˆv) = 0, ∂tˆξ = T ˆξ + ∂tˆv, (ˆv(0), ∂tˆv(0), ˆξ0)= z, (5.3) and ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ |∂tu|ρ∂t tu− |∂tˆv|ρ∂t tˆv + A∂t t ˆw + A ˆw + ∞ 0 μ(s)A ˆψ (s)ds= g ∂t ˆψ = T ˆψ + ∂tˆw, (ˆw(0), ∂t ˆw(0), ˆψ0)= 0, (5.4) where g= h − f0(u)+ f0(ˆv) − f1(u).
Notice that by Theorem 2.5 we immediately have that L(t)z is exponentially decaying, namely, there exists ω > 0 (independent of z and B) such that
ˆv(t ), ∂tˆv(t), ˆξt H≤ ce−ωt, (5.5)
where along this section the generic constant c≥ 0 is allowed to depend on BHbut not on the particular choice of the initial data z∈ B.
Concerning the operator N (t)z, we can prove that it is bounded in Hσ, having defined
σ = min 1 3, 4− ρ 2 >0. (5.6)
Lemma 5.2. For every t≥ 0, we have
ˆw(t), ∂tˆw(t), ˆψt Hσ ≤ ect.
Proof. The proof goes exactly as in[9], see estimate (6.7) therein, where in fact the presence of mechanical damping does not play any role. Indeed, one defines the higher-order energy
ˆEσ(t)=
1
2 ˆw(t), ∂tˆw(t), ˆψ
t 2
Hσ, and tests (5.4)by (∂tˆw, ˆψ) in H1+σ× Mσ, so obtaining
d dt ˆEσ− T A σ 2 ˆψ, A σ 2 ˆψM≤ g, Aσ∂t ˆw + |∂tˆv|ρ∂t tˆv − |∂tu|ρ∂t tu, Aσ∂t ˆw.
The right-hand side is the same as in[9, Lemma 6.3](see also the subsequent Lemma 6.3for similar estimates), where it is proved by delicate estimates that
g, Aσ∂
t ˆw + |∂tˆv|ρ∂t tˆv − |∂tu|ρ∂t tu, Aσ∂tˆw ≤ c ˆEσ+ c.
On the other hand, since μ is negative, by (1.11)we have
−T Aσ2ˆη, Aσ2 ˆψM= −1 2 ∞ 0 μ (s) ˆψ(s)21+σ ≥ 0.
We thus obtain the differential inequality d
dt ˆEσ≤ c ˆEσ+ c,
and an application of the Gronwall lemma, taking into account that ˆEσ(0) = 0, proves the
claim. 2
Lemma 5.2ensures that N (t)B is bounded in Hσ, but since the embedding Hσ ⊂ H is not compact (see[26]), we need a further step to complete the proof of(5.2). To this aim we introduce the set
t=
z∈B
ˆψt.
By Lemma 5.2we immediately see that
N (t )B ⊂ K(t) := B(t) × t, (5.7)
where t denotes the closure of tin M, and B(t) is a closed ball of H1+σ× H1+σ centered at
Then, it is enough to show that t⊂ Mσ is pre-compact in M in order to establish the
com-pactness of K(t) in H. But this follows by a nowadays classical argument based on a well-known compactness result in[26], and we refer the reader to[9]for the details.
In light of (5.5)and (5.7), the sufficient conditions (5.1)and (5.2)are verified, thus concluding the proof of the existence of the global attractor and its characterization as unstable manifold of the stationary points of the system.
6. Regularity of the attractor
This section is devoted to complete the proof of Theorem 2.4, by investigating the regularity of the global attractor A of S(t). We first establish an intermediate result, namely
Proposition 6.1. The global attractor A is bounded in Hσ, with σ as in (5.6).
The proof is based on the higher order estimates in Hσobtained in the previous section, which are not uniform in time and thus, not enough to conclude that the claimed regularity holds true. To this aim, we will apply Theorem 3.1 in[11], a tool specifically developed to establish regularity of the global attractor when uniform-in-time estimates are not available. The crucial ingredient is a further decomposition of S(t) as follows.
6.1. A further decomposition
First of all, notice that, as a consequence of the existence of a global attractor, the semigroup
S(t)also possesses a bounded absorbing set B. Now, let z ∈ B be fixed. For any couple y ∈ B
and x∈ Hσ satisfying y+ x = z, define the operators
Vz(t)y= (v(t), ∂tv(t ), ξt) and Uz(t)x= (w(t), ∂tw(t), ψt),
with (v(t), ∂tv(t ), ξt)and (w(t), ∂tw(t), ψt)solutions respectively of systems (5.3)and (5.4)
without the hats and initial data
(v(0), ∂tv(0), ξ0)= y and (w(0), ∂tw(0), ψ0)= x.
In light of the exponential decay (5.5)and Proposition 2.2, we immediately have
Lemma 6.2. The operator Vz(t) satisfies
(v(t), ∂tv(t ), ξt)H= Vz(t)yH≤ cyHe−ωt. (6.1)
Besides,
∂t tv1≤ c. (6.2)
Lemma 6.3. We have sup z∈B Uz(t)xHσ ≤ ce− νt 4xHσ+ Q(t),
for some ν > 0 and Q(·) independent of x.
Proof. We preliminarily recall that, since the initial data are in the absorbing set B, we have
u1+ ∂tu1+ ∂t tu1≤ c,
where here and in the following c > 0 denotes a generic constant that may depend on B. We first consider the energy
Eσ(t)=
1
2Uz(t)x 2
Hσ.
We multiply system (5.4)(without the hats) by (∂tw, ψ )in Hσ× Mσ, so obtaining
d dtEσ − T A σ+1 2 ψ, A σ+1 2 ψM= γ, ∂twσ where γ is defined by γ= g − |∂tu|ρ∂t tu+ |∂tv|ρ∂t tv, and g= h − f0(u)+ f0(v)− f1(u).
Due to (1.10)and (1.11), we have
−T Aσ+1 2 ψ, Aσ+12 ψM≥ −1 4 ∞ 0 μ (s)ψ(s)21+σds+δ 4ψ 2 Mσ.
The nonlinearity g can be handled as in [9, Section 7], but we detail the estimates for a further use. We write
|g| ≤ |h| + c|w|(|u| + |v|)4+ c(1 + |u|)
(6.3) ≤ |h| + c|w|(|ˆv| + |v|)4+ c(|u| + |v|)| ˆw|4+ c(1 + |u|),
where ˆv and ˆw are defined in the previous section by (5.3)and (5.4). Then, by standard compu-tations we obtain
g, Aσ∂ tw ≤hAσ∂tw + c(vL6+ ˆvL6)4wL6/(1−2σ)Aσ∂twL6/(1+2σ) + c(uL6+ vL6) ˆw4 L6/(1−2σ)Aσ∂twL6/(1+2σ)+ c(1 + u)Aσ∂tw ≤c∂tw1+σ+ c(v1+ ˆv1)4w1+σ∂tw1+σ + c(u1+ v1) ˆw41+σ∂tw1+σ+ c(1 + u1)∂tw1+σ,
where we are using the Sobolev embeddings
H1+σ⊂ L1−2σ6 () and H1−σ⊂ L1+2σ6 ().
Estimates (5.5), (6.1), and Lemma 5.2imply that g(t), Aσ∂ tw(t) ≤ α 2∂tw(t) 2 1+σ+ ce−4ωt w(t)2 1+σ+ ∂tw(t)21+σ + Q(t), (6.4) for any α > 0 and some Q(·) independent of x but depending on α. Besides, we can prove that
−|∂tu|ρ∂t tu+ |∂tv|ρ∂t tv, Aσ∂tw ≤ α 2∂tw 2 1+σ+ c α. (6.5)
Indeed, by the definition of σ , we have 23ρ−σ ≤ 6. Hence, the embedding H1⊂ L2−σ3ρ ()and
Proposition 2.2ensure −|∂tu|ρ∂t tu, Aσ∂tw ≤ ∂tuLρ3ρ/(2−σ)∂t tuL6Aσ∂twL6/(1+2σ) ≤ ∂tuρ1∂t tu1∂tw1+σ ≤α 4∂tw 2 1+σ+ c α.
Recalling (6.1)and (6.2), we control analogously the term |∂tv|ρ∂t tv, Aσ∂tw.
Collecting all the above estimates we have that Eσ satisfies the differential inequality
d dtEσ(t)− 1 4 ∞ 0 μ (s)ψt(s)21+σds+δ 4ψ t2 Mσ (6.6) ≤ α∂tw(t)21+σ+ ce−4ωt w(t)2 1+σ+ ∂tw(t)21+σ + Q(t), for any α > 0 and some Q(·) independent of x, depending on α.
The functional σ. We now define
σ(t)= ∂tw(t), w(t)1+σ.
d dtσ+ w 2 1+σ= ∂tw21+σ− ∞ 0 μ(s)ψ(s), w1+σds+ γ, wσ ≤ ∂tw21+σ+ 1 2w 2 1+σ+ κ 2ψ 2 Mσ+ γ, wσ,
with γ defined as above. By analogous computations to those leading to (6.4)and (6.5), we draw γ (t), Aσw(t) ≤ ε+ e−4ωtw(t)2
1+σ+ Q(t), thus proving the differential inequality
d dtσ(t)+ 1 2w(t) 2 1+σ≤ ∂tw(t)21+σ+ κ 2ψ t2 Mσ (6.7) + ε+ e−4ωt w(t)2 1+σ+ Q(t), for any ε > 0 and some Q(·) independent of x (depending on ε).
The functional σ. With the aim of overcoming the lack of mechanical damping, we now
intro-duce the further functional
σ(t)= − ∞ 0 μ(s)∂tw(t), ψt(s)1+σds − 1 ρ+ 1 ∞ 0 μ(s)|∂tu(t )|ρ∂tu(t )− |∂tv(t )|ρ∂tv(t ), ψt(s)σds.
We claim that, for every ε > 0 d dtσ+ κ 4∂tw 2 1+σ≤ c εψ 2 Mσ − c ∞ 0 μ (s)ψ21+σds+ εw21+σ+ c, (6.8)
with Q(·) as above. Indeed, taking the time derivative of σ we find
d dtσ = − ∞ 0 μ(s)A∂t tw+ |∂tu|ρ∂t tu− |∂tv|ρ∂t tv, ψ (s)σds − ∂tw, ∂tψMσ− 1 ρ+ 1 ∞ 0 μ(s)|∂tu|ρ∂tu− |∂tv|ρ∂tv, ∂tψ (s)σds= I1+ I2+ I3.
I2= −∂tw, ∂tψMσ= −κ∂tw21+σ− ∂tw, T ψMσ.
Integrating by parts with respect to s, in light of the decay of μ and of the equality ψ(0) = 0, we obtain −∂tw, T ψMσ = − ∞ 0 μ (s)∂tw, ψ (s)1+σds ≤ ∂tw1+σ − ∞ 0 μ (s)ψ(s)1+σds ≤κ 4∂tw 2 1+σ− cμ(0) ∞ 0 μ (s)ψ(s)21+σds.
To control the first term in dtdσ, notice that, in light of the first equation in the problem for
Uz(t)x, it becomes I1= ∞ 0 μ(s)A(1+σ )/2ψ (s)ds 2 + ∞ 0 μ(s)Aw, ψ(s)σds− ∞ 0 μ(s)g, ψ(s)σds,
where, by standard computations, ∞ 0 μ(s)A(1+σ )/2ψ (s)ds 2≤ ∞ 0 μ(s)A(1+σ )/2ψ (s)ds 2 ≤ κψ2 Mσ, and ∞ 0 μ(s)Aw, ψ(s)σds≤ c εψ 2 Mσ+ ε 2w 2 1+σ.
Concerning the term 0∞μ(s)g, ψ(s)σds, by using (6.3)and reasoning as above for the proof
of (6.4), we have g, Aσψ ≤ hAσψ + c(u L6+ vL6)4wL6/(1−2σ)AσψL6/(1+2σ)+ c(1 + u)Aσψ ≤ cψ1+σ+ c(u1+ v1)4w1+σψ1+σ+ c(1 + u1)ψ1+σ ≤ c εψ 2 1+σ+ ε 2κw 2 1+σ+ c,
for any ε > 0. As a consequence, we obtain the estimate ∞ 0 μ(s)g, ψ(s)σds≤ c εψ 2 Mσ+ ε 2w 2 1+σ + c.
We are left to control the last term I3in dtdσ. Using the equation ruling the evolution of ψ ,
we have − ∞ 0 μ(s)|∂tu|ρ∂tu, Aσ∂tψds = − ∞ 0 μ(s)|∂tu|ρ∂tu, Aσ∂tw+ T Aσψds.
With analogous computations as those proving (6.5), we get
−|∂tu|ρ∂tu, Aσ∂tw ≤ ∂tuρ1∂tu1∂tw1+σ
and we end up with
− 1 ρ+ 1 ∞ 0 μ(s)|∂tu|ρ∂tu, Aσ∂twds ≤ κ 4∂tw 2 1+σ+ c.
Reasoning analogously, we have
− 1 ρ+ 1 ∞ 0 μ(s)|∂tu|ρ∂tu, T Aσψds ≤ c∂tuρ1∂tu1 − ∞ 0 μ (s)ψ(s)1+σds ≤ c − ∞ 0 μ (s)ψ(s)21+σds.
Since the very same bounds hold true for the terms involving |∂tv|ρ∂tv, collecting all the above
inequalities the proof of (6.8)is done. We now define, for ε∈ (0, 1) to be chosen,
σ= Eσ + ε2σ+
8
κε
2
σ. (6.9)
Owing to (6.6)with α= ε3, (6.7), (6.8), and exploiting (1.10), we get d dtσ(t)+ ε 2 1 2− ε − 8 κε w2 σ+1+ ε2 2− 1 − ε ∂tw2σ+1+ δ 4− κ 2ε 2− cεψ2 Mσ + 1 4 − cε 2 ∞ 0 −μ (s)ψ(s)21+σds≤ ce−4ωtEσ(t)+ Q(t)
(here c and Q depend on ε). It is now clear that, for any fixed ε small enough, we end up with d
dtσ(t)+ νEσ(t)≤ ce −4ωtE
σ(t)+ Q(t), (6.10)
for some ν > 0 (depending on δ). Note that we have the straightforward control |σ| ≤ Eσ,
while arguing as in the proof of (6.5),
|σ| ≤ 1 2w 2 1+σ+ κ 2ψ 2 Mσ+ 1 ρ+ 1 ∂tuρ1∂tu1+ ∂tvρ1∂tv1 ∞ 0 μ(s)ψ(s)1+σds ≤1 2w 2 1+σ+ κ 2ψ 2 Mσ+ ψ2Mσ+ c ≤ cEσ+ c.
Hence, up to choosing a smaller ε, the following two side control holds 1
2Eσ− c ≤ σ ≤ 2Eσ + c. (6.11)
Finally, in light of (6.10)and (6.11), we infer from the Gronwall lemma that Uz(t)xHσ ≤ ce−
νt
4xHσ+ Q(t), completing the proof. 2
Proof of Proposition 6.1. In light of the above decomposition we are in the position to apply
Theorem 3.1 from[11]. Accordingly, we deduce the existence of a (closed) ball Bσ⊂ Hσ,
and constants c, > 0 such that
distH(S(t)B, Bσ)≤ ce−t, ∀t ≥ 0,
meaning that Bσis an attracting set. This allows to conclude that the global attractor A is bounded
in Hσ. Indeed, since A is fully invariant, it is contained in every closed attracting set, hence
A ⊂ Bσ,
Remark 6.4. At this point it is easy to obtain, by comparison in the equation, that ∂t tuundergoes
a regularization effect, namely, for initial data z∈ A, we have ∂t tu1+σ≤ c,
for some c > 0 (depending only on A).
6.2. Optimal regularity of the attractor
In order to complete the proof of Theorem 2.4we are left to show that
A is bounded in H1.
Indeed, the optimal regularity of the attractor now follows by a bootstrap result.
Lemma 6.5. Let σ as in (5.6). Then, given any r∈ [σ, 1 − σ ] the following holds:
A ⊂ Hr ⇒ A ⊂ Hr+σ.
Accordingly, in a finite number of steps with reiterated application of Lemma 6.5, one easily achieves the boundedness of A in Hr+σ with r+ σ = 1.
The proof of the lemma can be obtained recasting the argument in[9], through a new decom-position of the semigroup for initial data z∈ A into the sum
S(t)z= L(t)z + K(t)z,
where L(t)z and K(t)z solve suitable systems, see [9, Lemma 8.1].
We omit the details, limiting ourselves to point out that the only meaningful difference appears in the proof of the boundedness of K(t)z in Hr+σ. There, a suitable energy functional r+σ is
needed: having in mind Lemma 6.3above, the appropriate functional can be easily constructed by generalizing in higher-space dimensions the definition of σ in (6.9).
Acknowledgments
We are grateful to the anonymous referee for stimulating suggestions.
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