# Hyperbolic Relaxation of a Fourth Order Evolution Equation

Authors: Renato Colucci; Gerardo R. Chacón
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We propose a hyperbolic relaxation of a fourth order evolution equation,
with an inertial termηutt, where η∈(0,1]. We prove the existence of several absorbing sets having different regularities and the existence of a global attractor that is bounded in H4(I)×{H2(I)∩H01(I)}.


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## Body

## 1. Introduction

LetingI⊂ℝ be an open interval, with |I|≤1, we consider the following initial-boundary value problem for u:I×ℝ+→ℝ:
(1)ηutt+ut=-ε2uxxxx+12W′′(ux)uxx,(x,t)∈I×ℝ+,u(x,0)=u0(x),inI,ut(x,0)=u1(x),inI,u=uxx=0,in∂I,t≥0,
where the function W(p)=(p2-1)2 is the so-called double-well potential, 0<ε≪1 and η∈(0,1] are positive parameters.Problem (1), with η=0, was proposed in [1] where the global dynamics was studied. In particular, the dynamical behavior of the solutions for small values of the parameter ε was studied by means of numerical experiments.  The existence of three well-differentiated time scales with peculiar dynamical behavior was showen. In the first time scale of order O(ε2) there is the formation of microstructure (see [2]) in the region where the gradient of the initial datum falls in the nonconvex region of W; this phenomenon produces a drastic reduction of the energy of the initial datum. In the second time scale of order O(1) the equation exhibits a heat equation-like behavior in the convex regions while slow motion in the nonconvex ones. In the last time scale of order O(ε-2) the equation shows a finite-dimensional behavior: the solution is approximately the union of consecutive segments and the dynamic is slow.In [3], the third time scale was studied; the authors proved the existence of a global attractor 𝒜ε⊂L2(I) (see [4]) that is bounded in H2(I). The time for which the solutions enter the absorbing set ℬε is of order O(ε-2) and it is consistent with the estimates found in [1]. Moreover the authors proved the existence of an exponential attractor ℐε with finite fractal dimension of order O(ε-10). In [5] the authors proved the existence of an inertial manifold ℳε (see [6]) whose dimension is of order O(ε-19), and by the n-dimensional volume elements methods (see [7]) an estimate of the dimension of the global attractor of order O(ε-1) was found. This estimate is also consistent with the numerical experiments developed in [1]; in fact it was found that the wave length of the microstructure is of order O(ε-1).In the last years the viscous and no viscous hyperbolic relaxation of the Cahn-Hilliard equation has been extensively investigated. The model was proposed in [8] while in [9] the existence of a family of exponential attractors was proved. The viscous and nonviscous perturbation has been studied in [10] where the existence of a family of global attractors that are upper semicontinuous with respect to the vanishing of perturbations parameters was proved. These results have been extended in 2 and 3 dimensions; see for example [11, 12] and the references therein.Due to the similarity of problem (1) to the Cahn-Hilliard equation we consider it interesting to study the hyperbolic relaxation of the fourth order evolution equation proposed in [1]. In particular if v is the solution of the Cahn-Hilliard equation
(2)vt+Δ[ε2Δv-W′(v)]=0,
with Neumann boundary conditions:
(3)∂∂nv=∂∂nΔv=0,x∈{0,1},
then
(4)u(x)=∫0x‍v(s)ds,
is the solution of (1), with η=0, with the corresponding boundary conditions:
(5)u=uxx=0,x∈{0,1}.
In the present work we put the problem in the correct mathematical framework and prove the existence of a global attractor 𝒜η,ε while we have left the proof of the existence of exponential attractors for a forthcoming paper. In Sections 2 and 3 we define the solution semigroup in the appropriate spaces and present some important energy estimates. In Section 4 we prove the existence of several absorbing sets with different regularities while in the last section we prove the existence of the global attractor.


## 2. Preliminaries

We begin this section by defining the following Hilbert spaces that will be helpful for our analysis:(6)ℋη,ε=[H2(I)∩H01(I)]×L2(I),(7)𝒱η,ε=D(A)×H2(I),
where
(8)D(A)={u∈H4(I):u=uxx=0in∂I}
is the domain of the differential operator A=∂4/∂x4. The above spaces are equipped with the norms
(9)∥(u,v)∥ℋη,ε2=ε22∥uxx∥2+η2∥v∥2,(10)∥(u,v)∥𝒱η,ε2=ε22∥uxxxx∥2+η2∥vxx∥2,
where ∥·∥ represents the L2 norm. We will denote by 〈·,·〉 the inner product in L2(I). We recall that for all u∈H2(I)∩H01(I) with |I|≤1, we have
(11)∥u∥≤∥ux∥≤∥uxx∥.
Throughout the paper we will use two norms that are equivalent to (9) and (10) in order to simplify the computation. Given functions u∈H2(I)∩H01(I) and v∈L2(I) we define the function E~ on ℋη,ε by
(12)E~(u,v)=η∥v∥2+12ε2∥uxx∥2+βη〈v,u〉,
where β∈[0,ε2].Proposition 1.
For allβ∈[0,ε2] the function E~(·,·) induces a norm equivalent to the norm on ℋη,ε.Proof.
By Schwartz inequality (11) and using the fact that β≤ε2<1, η∈(0,1], we get
(13)βη〈v,u〉≤βη∥v∥∥u∥≤12η2∥v∥2+12ε4∥uxx∥2≤η2∥v∥2+ε22∥uxx∥2=∥(u,v)∥ℋε,η2.
From the previous inequality and by definition of E~(u,v) we get
(14)E~(u,v)≤3∥(u,v)∥ℋε,η2.
By a different application of Schwartz inequality and from (11) we get
(15)βη〈v,u〉≤βη∥v∥∥u∥≤η2∥v∥2+β24∥u∥2≤η∥v∥2+ε44∥u∥2≤η∥v∥2+ε24∥uxx∥2.
Combining (13) and (15) we get
(16)E~(u,v)≥max{η2∥v∥2,14ε2∥uxx∥2},
and as a consequence
(17)E~(u,v)≥13∥(u,v)∥ℋε,η2.The proof of the following theorem follows from classical applications of the Faedo-Galerkin method. We will only show a Lipschitz estimate that will be needed for further computations.Theorem 22.
For every(u0,u1)∈ℋε,η there exists a unique solution u(t) for the initial value problem (1) such that
(18)u∈Cb(ℝ+;H2(I)∩H01(I))∩Cb1(ℝ+;L2(I)).
If, moreover, (u0,u1)∈𝒱ε,η, then:
(19)u∈Cb(ℝ+;D(A))∩Cb1(ℝ+;H2(I)∩H01(I))∩Cb2(ℝ+;L2(I)).Proposition 33.
For any constantR≥0 there exists a positive constant K=K(R) such that, for any initial data u1(0), u2(0) with ∥ui(0)∥Hη,ε≤R, i=1,2 one has
(20)∥Sε,η(t)u1(0)-Sε,η(t)u2(0)∥Hη,ε≤e(K2/ε2)t∥u1(0)-u2(0)∥Hη,ε,
where Sε,η(t) is the solution semigroup of the problem (1).Proof.
Letu1, u2, two solutions of (1) with initial data u1(0) and u2(0). Let, w=u1-u2 then we write
(21)ηwtt+wt+ε2wxxxx=12ddx{W′(u1x)-W′(u2x)}.
We multiply the above equation by wt in L2(I); then
(22)η2ddt∥wt∥2+ε22ddt∥wxx∥2+∥wt∥2=-2∫I‍wxxwtdx+6∫I‍{u1x2wxx+u2xxwx[u1x+u2x]}wtdx≤2∥wxx∥∥wt∥+6∥u1x∥∞2∥wxx∥∥wt∥+12∥u2xx∥(∫I‍wx2wt2[u1x2+u2x2]dx)1/2≤2{(∥u1xx∥2+∥u2xx∥2)1/21+18C2∥u1xx∥2+12∥u2xx∥×(∥u1xx∥2+∥u2xx∥2)1/2}∥wxx∥∥wt∥≤2{1+(36C2+242)R2ε2}∥wxx∥∥wt∥≤2K∥wxx∥∥wt∥≤K2∥wxx∥2+∥wt∥2,
where we have used the following inequality (see [13] and inequality (11)):
(23)∥ux∥∞2≤C2∥ux∥H1(I)2≤2C2∥uxx∥2.
The constant K can be explicitly computed using the estimates of the absorbing set presented in the next section.
From the last inequality we get(24)ddt∥(w,wt)∥ℋε,η2≤2K2ε2∥(w,wt)∥ℋε,η2,
and then using Gronwall's lemma we get
(25)∥(w,wt)∥ℋε,η≤e(K2/ε2)t∥(w(0),wt(0))∥ℋε,η.


## 3. A Priori Estimates

In this section we provide useful a priori estimates of energy type. Equation (1) admits a Liapunov functional of the form:
(26)E(u,ut)=η∥ut∥2+12ε2∥uxx∥2+12∫I‍W(ux)dx,
that is not increasing along the solutions; in fact if we multiply (1) by ut and integrate over I we obtain
(27)ddtE(u,ut)=-∥ut∥2.
Moreover, integrating the previous inequality on (0,t) with respect to time we get
(28)∫0t‍∥ut∥2ds+E(u,ut)=E(u0,u1).
Then for any fixed initial data (u0,u1)∈ℋε,η we have that the corresponding solution satisfies
(29)supt≥0{η2∥ut∥2+ε22∥uxx∥2}=supt≥0∥(u,ut)∥ℋε,η2≤E(u0,u1).
Moreover if we integrate over ℝ+ we get the integral control
(30)∫0∞‍∥ut∥2dt≤E(u0,u1).
We consider an important estimate that will be useful later. Let β>0 be a parameter to be determined later; if we multiply (1) by βu we obtain
(31)βη∫I‍uttudx+β2ddt∥u∥2+βε2∥uxx∥2+2β∥ux∥44=2β∥ux∥2.
By summing (27) and (31) we get
(32)ddtE+βηddt〈ut,u〉+(1-βη)∥ut∥2+β2ddt∥u∥2+βε2∥uxx∥2+2β∥ux∥44=2β∥ux∥2.
Using the expression of the energy we can rewrite the previous in the following way:
(33)ddtE+βηddt〈ut,u〉+(1-3βη)∥ut∥2+β〈ut,u〉+2β(E+∥ux∥442-|I|2)=0.
We will estimate some of the terms of the previous inequality in the following lemma.Lemma 4.
Fixη≤1 then for all β∈(0,1/3) one has
(34)2β2η〈ut,u〉≤(1-3βη)∥ut∥2+β∥ux∥44+β|I|+β〈ut,u〉.Proof.
By Holder's inequality we get(35)2∥ux∥2≤2∥ux∥42|I|1/2≤∥ux∥44+|I|,
then we use Poincaré's inequality and the fact that |I|≤1 to conclude that
(36)∥u∥2≤∥ux∥442+|I|2.
Therefore, for a positive constant c1 to be determined later, we have that
(37)〈ut,u〉≤∥ut∥∥u∥≤c12∥ut∥2+12c1∥u∥2≤c12∥ut∥2+12c1(∥ux∥442+|I|2).
Consequently, by choosing the parameters c1=1/12 and β<1/3 we have that
(38)(2β2η-β)〈ut,u〉≤|2β2η-β|c12∥ut∥2+|2β2η-β|4c1∥ux∥44+|2β2η-β|4c1|I|≤(1-3βη)∥ut∥2+β∥ux∥44+β|I|.Now, using the previous lemma and (33) we conclude that
(39)ddtE+βηddt〈ut,u〉+2β2η〈ut,u〉+2βE≤2β|I|.
If we set
(40)Φ(u,ut)=Φ(t)=E(u,ut)+βη〈ut,u〉,
then from (39) we get
(41)ddtΦ(t)+2βΦ(t)≤2β.
Now, integrating we have that
(42)Φ(t)≤[Φ(u0,u1)-1]e-2βt+1.
The previous inequality will be used in the next section to show the existence of absorbing sets for the problem (1).We remark that the following inequality holds for the functionsE, E~, and Φ:
(43)Φ(t)=E~(t)+12∫I‍W(ux)dx≥E~≥0.


## 4. Absorbing Sets

In this section we will show the existence of several absorbing sets for the solution semigroup of (1) in the space ℋε,η. Also, assuming further regularity of the initial data, the existence of a more regular absorbing sets is also shown.First, notice the following estimate of the nonlinear term of the energy functional defined in (26). Using (23) we get
(44)12∫I‍W(ux)dx≤12∥ux∥2∥ux∥L∞(I)2-∥ux∥2+|I|2≤12∥ux∥2∥uxx∥2+|I|2≤12∥uxx∥4+|I|2.
We will use this inequality several times in what follows.


### 4.1. Absorbing Set inℋε,η

Proposition 5.
For allR0>1 the set
(45)BΦ∶={(u,ut)∈ℋε,η:Φ(u,ut)≤R0}
is bounded, absorbing, and positively invariant for the semigroup Sε,η(t) in ℋε,η.Proof.
The set is positively invariant, in fact if(u0,u1)∈ℋε,η with Φ(u0,u1)≤R0 we have, from (42), that
(46)Φ(u,ut)≤[R0-1]e-2βt+1≤R0,∀t>0.
The boundness of BΦ follows directly from (42). Now suppose that (u0,u1) are such that Φ(u0,u1)≤R, with R>R0 then again from (42) we get
(47)Φ(u,ut)≤R0,∀t≥τΦ,
where
(48)τΦ∶=12βlog{R-1R0-1}.
Then BΦ is absorbing for the semigroup Sε,η(t).Proposition 6.
For allR1>1 one has that the set
(49)BE~∶={(u,ut)∈ℋε,η:E~(u,ut)≤R1}
is absorbing for the semigroup Sε,η(t).Proof.
We have from (42) and (43) that
(50)E~(u,ut)≤[Φ(u0,u1)-1]e-2βt+1.
Now, take (u0,u1) such that E~(u0,u1)≤R, with R>R1. Then, by (42) and (44) we get that
(51)E~(t)≤Φ(t)≤E~(t)+12∥ux∥44+|I|2≤E~(t)+12∥uxx∥4+|I|2≤E~(t)+4ε2E~(t)2+|I|2.
Therefore,
(52)E~(u,ut)≤{E~(u0,u1)+4ε2E~(u0,u1)2+|I|2-1}e-2βt+1≤{R+4ε2R2+|I|2-1}e-2βt+1.
Consequently, we have that
(53)E~(u,ut)≤R1,∀t≥τE~,
where
(54)τE~=12βlog{R+(4/ε4)R2+(|I|/2)-1R1-1}.Now using the equivalence of the norm onℋε,η and (52) we get
(55)∥(u,ut)∥ℋε,η2≤3{{|I|2}3∥(u0,u1)∥ℋε,η2+36ε2∥(u0,u1)∥ℋε,η4+|I|2-1}e-2βt+3.
Let R2>3. If (u0,u1)∈ℋε,η is such that
(56)∥(u0,u1)∥ℋε,η≤R,
then we have that
(57)∥(u,ut)∥ℋε,η≤R2,∀t≥τH,
where
(58)τH=12βlog3{3R2+(36/ε2)R4+|I|/2-1}R22-3.
Thus, we have proved the following.Proposition 77.
For allR2>3 one has that the ball
(59)Bℋε,η∶={(u,ut)∈ℋε,η:∥(u,ut)∥ℋε,η≤R2}
is an absorbing set for the semigroup Sε,η(t) in ℋε,η.


### 4.2. Absorbing Set in𝒰ε,η

Now suppose that the initial data has some additional regularity. Then we can prove the existence of more regular absorbing sets. Let us define the following space:(60)𝒰ε,η={H3(I)∩H2(I)∩H01(I)}×H1(I),
equipped with the norm
(61)∥(u,ut)∥𝒰ε,η2=η2∥utx∥2+ε22∥uxxx∥2.
Then we have the following.Proposition 88.
There existsR3>0 such that the closed ball
(62)ℬ3={(u,ut)∈𝒰ε,η:∥(u,ut)∥𝒰ε,η≤R3},
is a bounded absorbing set for Sε,η(t) in 𝒰ε,η.Proof.
If we multiply (1) by uxxt+uxx and integrate over I we obtain
(63)ddt{η2∥utx∥2+ε22∥uxxx∥2+12∥ux∥2+η〈utx,ux〉}+(1-η)∥utx∥2+ε2∥uxxx∥2=-12〈[W′(ux)]x,uxxt〉-12〈[W′(ux)]x,uxx〉.
Let us denote the differential term of the previous inequality as
(64)Ψ0(t)=η2∥utx∥2+ε22∥uxxx∥2+η〈utx,ux〉+12∥ux∥2.
We will estimate the right hand side of (63). The first term can be estimated as follows:
(65)-12〈[W′(ux)]x,uxx〉≤12∥W′′(ux)∥∞∥uxx∥2≤2(3∥ux∥∞2+1)∥uxx∥2≤2(3∥uxx∥2+1)∥uxx∥2≤4ε2R22(6R22ε2+1)∶=C1.
Let us define
(66)Ψ1=14∫I‍W′′(ux)uxx2dx;
then we can rewrite the the second term of r.h.s of (63)
(67)-12〈[W′(ux)]x,uxxt〉=-ddtΨ1+14∫I‍W′′′(ux)uxtuxx2,
and estimate
(68)14∫I‍W′′′(ux)uxtuxx2≤6∥uxx∥∞∫I‍uxuxtuxxdx≤6∥uxx∥∞∥uxt∥∥uxuxx∥≤6∥uxx∥∞∥uxt∥∥uxx∥2≤6∥uxx∥5/2∥uxxx∥1/2∥uxt∥≤6(2εR2)5/2∥uxxx∥1/2∥uxt∥≤η2∥uxt∥2+18η(2εR2)5∥uxxx∥≤η2∥uxt∥2+12[18ηε(2εR2)5]2+ε22∥uxxx∥2∶=η2∥uxt∥2+ε22∥uxxx∥2+C2.
Then if η∈(0,1/2) and setting Ψ∶=Ψ0+Ψ1, C∶=C1+C2 we have
(69)ddtΨ+η2∥uxt∥2+ε22∥uxxx∥2≤C.
To conclude the proof we note that
(70)Ψ≥∥(u,ut)∥𝒰ε,η2+12∥uxx∥2-η∥utx∥∥ux∥-∥uxx∥2≥∥(u,ut)∥𝒰ε,η2-η4∥utx∥2-32∥uxx∥2≥12∥(u,ut)∥𝒰ε,η2-3ε2R22∶=12∥(u,ut)∥𝒰ε,η2-c1,
and that
(71)Ψ≤∥(u,ut)∥𝒰ε,η2+η2∥utx∥2+∥uxx∥2+∥uxx∥2(3∥ux∥∞2+1)≤2∥(u,ut)∥𝒰ε,η2+2ε2R22(6ε2R22+2)∶=2∥(u,ut)∥𝒰ε,η2+c2.
From the previous inequalities we get
(72)12∥(u,ut)∥𝒰ε,η2-c1≤Ψ≤2∥(u,ut)∥𝒰ε,η2+c2.
Then from (72) we get
(73)ddtΨ+12Ψ≤C+c2∶=C~,
and by Gronwall's lemma we obtain
(74)Ψ(t)≤(Ψ(0)-2C~)e-(1/2)t+2C~,∥(u,ut)∥𝒰ε,η2≤2(∥(u0,u1)∥𝒰ε,η2+c2-2C~)e-(1/2)t+4C~+2c1.
Then if (u0,u1)∈𝒰ε,η such that
(75)∥(u0,u1)∥𝒰ε,η≤R,
we have that
(76)∥(u,ut)∥𝒰ε,η≤R3,∀t≥τU,
where
(77)τU∶=2log(2R2+2c2-4C~R32-4C~-2c1),
and this concludes the proof.


### 4.3. Absorbing Set in𝒱ε,η

We consider the space𝒱ε,η as defined in (7), equipped with the norm defined in (10). By considering more regular initial conditions we can prove the following result.Proposition 99.
There existsR4>0 such that the closed ball
(78)ℬ4={(u,ut)∈𝒱ε,η:∥(u,ut)∥𝒱ε,η≤R4}
is a bounded absorbing set for the semigroup Sε,η(t) in 𝒱ε,η.Proof.
Multiply (1) by uxxxxt+uxxxx; then we obtain
(79)ddt{η2∥utxx∥2+ε22∥uxxxx∥2+η〈utxx,uxx〉+12∥uxxx∥2{ε22}}+(1-η)∥uxxt∥+ε2∥uxxxx∥2=12〈[W′(ux)]x,uxxxxt+uxxxx〉.
We call θ0(t) the differential term of the previous inequality and we estimate the r.h.s.:
(80)12〈[W′(ux)]x,uxxxx〉=-12〈[W′(ux)]xx,uxxx〉=-12〈W′′′(ux)uxx2,uxxx〉-12〈[W′′(ux)]xx,uxxx2〉=-12∫I‍uxuxx2uxxxdx-2∫I‍(3ux2-1)uxxx2dx≤2∥uxxx∥2+12∥uxx∥∞2∥ux∥∥uxxx∥≤2∥uxxx∥2(6∥uxx∥2+1)≤ε2R32(12ε2R22+1)∶=A1.
Let us define
(81)θ1(t)∶=14∫I‍W′′(ux)uxxx2dx;
then
(82)12〈[W′(ux)]x,uxxxxt〉=-12〈[W′(ux)]xx,uxxxt〉=-12〈W′′′(ux)uxx2,uxxxt〉-12〈W′′(ux)uxxx,uxxxt〉=-ddtθ1+12〈[W′′′(ux)uxx2]x,uxxt〉+14∫I‍W′′′(ux)uxtuxxx2dx.
We estimate the last two terms of the previous equality:
(83)14∫I‍W′′′(ux)uxtuxxx2dx=6∫I‍uxuxtuxxx2dx≤6∥uxxx∥∞∥ux∥∞∥uxt∥∥uxxx∥≤6∥uxxx∥3/2∥uxxxx∥1/2∥uxt∥∥uxx∥2≤(12·23/4R22R33/2ε7/2)∥uxxxx∥1/2∥uxt∥≤(12·23/4R22R33/2ε7/2)2∥uxxxx∥+∥uxt∥2≤12ε2(12·23/4R22R33/2ε7/2)4+2ηR32+ε22∥uxxxx∥2∶=ε22∥uxxxx∥2+A2,12〈[W′′′(ux)uxx2]x,uxxt〉=12〈Wiv(ux)uxx3+W′′′(ux)2uxxuxxx,uxxt〉=12∫I‍uxx3uxxtdx+24∫I‍uxuxxuxxxuxxtdx≤12(∥uxx∥∞2∥uxx∥+2∥ux∥∞∥uxx∥∞∥uxxx∥)∥uxxt∥≤12(∥uxxx∥3/2∥uxx∥2∥uxxx∥+2∥uxx∥3/2∥uxxx∥3/2)∥uxxt∥≤η2∥uxxt∥2+72({∥uxxx∥3/2}∥uxx∥2∥uxxx∥+2∥uxx∥3/2∥uxxx∥3/2)2≤η2∥uxxt∥2+72[22R22R3ε3+2(2R2R3ε2)3/2]2∶=η2∥uxxt∥2+A3.
Then if we set θ(t)=θ0(t)+θ1(t) we get
(84)ddtθ(t)+∥(u,ut)∥𝒱ε,η2≤A1+A2+A3∶=A.
Moreover we have
(85)θ(t)≥∥(u,ut)∥𝒱ε,η2-η4∥uxxt∥2-12∥uxx∥2≥12∥(u,ut)∥𝒱ε,η2-12R22∶=12∥(u,ut)∥𝒱ε,η2-a1,θ(t)≤∥(u,ut)∥𝒱ε,η2+η∥uxxt∥∥uxx∥+12∥uxx∥2+14∥W′′(ux)∥∞∥uxxx∥3≤2∥(u,ut)∥𝒱ε,η2+∥uxx∥2+(3∥ux∥∞2+1)∥uxxx∥2≤2∥(u,ut)∥𝒱ε,η2+2ε2R22+(6ε2R22+1)2ε2R32∶=2∥(u,ut)∥𝒱ε,η2+a2.
Then putting all together we get
(86)12∥(u,ut)∥𝒱ε,η2-a1≤θ(t)≤2∥(u,ut)∥𝒱ε,η2+a2.
From the previous inequality we get
(87)ddtθ(t)+12θ(t)≤A+a22∶=A~
and by Gronwall's lemma
(88)θ(t)≤[θ(0)-2A~]e-(1/2)t+2A~.
Using again (86) we get
(89)∥(u,ut)∥𝒱ε,η2≤2[2∥(u0,u1)∥𝒱ε,η2+a2-2A~]e-(1/2)t+4A~+2a1.
Then if (u0,u1)∈𝒱ε,η such that
(90)∥(u0,u1)∥𝒱ε,η≤R,
then there exists R4>0:
(91)∥(u,ut)∥𝒱ε,η≤R4,∀t≥τV,
where
(92)τV∶=2log[2(2R2+a2-2A~)R42-4A~-2a1],
and this concludes the proof.


## 4.1. Absorbing Set inℋε,η

Proposition 5.
For allR0>1 the set
(45)BΦ∶={(u,ut)∈ℋε,η:Φ(u,ut)≤R0}
is bounded, absorbing, and positively invariant for the semigroup Sε,η(t) in ℋε,η.Proof.
The set is positively invariant, in fact if(u0,u1)∈ℋε,η with Φ(u0,u1)≤R0 we have, from (42), that
(46)Φ(u,ut)≤[R0-1]e-2βt+1≤R0,∀t>0.
The boundness of BΦ follows directly from (42). Now suppose that (u0,u1) are such that Φ(u0,u1)≤R, with R>R0 then again from (42) we get
(47)Φ(u,ut)≤R0,∀t≥τΦ,
where
(48)τΦ∶=12βlog{R-1R0-1}.
Then BΦ is absorbing for the semigroup Sε,η(t).Proposition 6.
For allR1>1 one has that the set
(49)BE~∶={(u,ut)∈ℋε,η:E~(u,ut)≤R1}
is absorbing for the semigroup Sε,η(t).Proof.
We have from (42) and (43) that
(50)E~(u,ut)≤[Φ(u0,u1)-1]e-2βt+1.
Now, take (u0,u1) such that E~(u0,u1)≤R, with R>R1. Then, by (42) and (44) we get that
(51)E~(t)≤Φ(t)≤E~(t)+12∥ux∥44+|I|2≤E~(t)+12∥uxx∥4+|I|2≤E~(t)+4ε2E~(t)2+|I|2.
Therefore,
(52)E~(u,ut)≤{E~(u0,u1)+4ε2E~(u0,u1)2+|I|2-1}e-2βt+1≤{R+4ε2R2+|I|2-1}e-2βt+1.
Consequently, we have that
(53)E~(u,ut)≤R1,∀t≥τE~,
where
(54)τE~=12βlog{R+(4/ε4)R2+(|I|/2)-1R1-1}.Now using the equivalence of the norm onℋε,η and (52) we get
(55)∥(u,ut)∥ℋε,η2≤3{{|I|2}3∥(u0,u1)∥ℋε,η2+36ε2∥(u0,u1)∥ℋε,η4+|I|2-1}e-2βt+3.
Let R2>3. If (u0,u1)∈ℋε,η is such that
(56)∥(u0,u1)∥ℋε,η≤R,
then we have that
(57)∥(u,ut)∥ℋε,η≤R2,∀t≥τH,
where
(58)τH=12βlog3{3R2+(36/ε2)R4+|I|/2-1}R22-3.
Thus, we have proved the following.Proposition 77.
For allR2>3 one has that the ball
(59)Bℋε,η∶={(u,ut)∈ℋε,η:∥(u,ut)∥ℋε,η≤R2}
is an absorbing set for the semigroup Sε,η(t) in ℋε,η.


## 4.2. Absorbing Set in𝒰ε,η

Now suppose that the initial data has some additional regularity. Then we can prove the existence of more regular absorbing sets. Let us define the following space:(60)𝒰ε,η={H3(I)∩H2(I)∩H01(I)}×H1(I),
equipped with the norm
(61)∥(u,ut)∥𝒰ε,η2=η2∥utx∥2+ε22∥uxxx∥2.
Then we have the following.Proposition 88.
There existsR3>0 such that the closed ball
(62)ℬ3={(u,ut)∈𝒰ε,η:∥(u,ut)∥𝒰ε,η≤R3},
is a bounded absorbing set for Sε,η(t) in 𝒰ε,η.Proof.
If we multiply (1) by uxxt+uxx and integrate over I we obtain
(63)ddt{η2∥utx∥2+ε22∥uxxx∥2+12∥ux∥2+η〈utx,ux〉}+(1-η)∥utx∥2+ε2∥uxxx∥2=-12〈[W′(ux)]x,uxxt〉-12〈[W′(ux)]x,uxx〉.
Let us denote the differential term of the previous inequality as
(64)Ψ0(t)=η2∥utx∥2+ε22∥uxxx∥2+η〈utx,ux〉+12∥ux∥2.
We will estimate the right hand side of (63). The first term can be estimated as follows:
(65)-12〈[W′(ux)]x,uxx〉≤12∥W′′(ux)∥∞∥uxx∥2≤2(3∥ux∥∞2+1)∥uxx∥2≤2(3∥uxx∥2+1)∥uxx∥2≤4ε2R22(6R22ε2+1)∶=C1.
Let us define
(66)Ψ1=14∫I‍W′′(ux)uxx2dx;
then we can rewrite the the second term of r.h.s of (63)
(67)-12〈[W′(ux)]x,uxxt〉=-ddtΨ1+14∫I‍W′′′(ux)uxtuxx2,
and estimate
(68)14∫I‍W′′′(ux)uxtuxx2≤6∥uxx∥∞∫I‍uxuxtuxxdx≤6∥uxx∥∞∥uxt∥∥uxuxx∥≤6∥uxx∥∞∥uxt∥∥uxx∥2≤6∥uxx∥5/2∥uxxx∥1/2∥uxt∥≤6(2εR2)5/2∥uxxx∥1/2∥uxt∥≤η2∥uxt∥2+18η(2εR2)5∥uxxx∥≤η2∥uxt∥2+12[18ηε(2εR2)5]2+ε22∥uxxx∥2∶=η2∥uxt∥2+ε22∥uxxx∥2+C2.
Then if η∈(0,1/2) and setting Ψ∶=Ψ0+Ψ1, C∶=C1+C2 we have
(69)ddtΨ+η2∥uxt∥2+ε22∥uxxx∥2≤C.
To conclude the proof we note that
(70)Ψ≥∥(u,ut)∥𝒰ε,η2+12∥uxx∥2-η∥utx∥∥ux∥-∥uxx∥2≥∥(u,ut)∥𝒰ε,η2-η4∥utx∥2-32∥uxx∥2≥12∥(u,ut)∥𝒰ε,η2-3ε2R22∶=12∥(u,ut)∥𝒰ε,η2-c1,
and that
(71)Ψ≤∥(u,ut)∥𝒰ε,η2+η2∥utx∥2+∥uxx∥2+∥uxx∥2(3∥ux∥∞2+1)≤2∥(u,ut)∥𝒰ε,η2+2ε2R22(6ε2R22+2)∶=2∥(u,ut)∥𝒰ε,η2+c2.
From the previous inequalities we get
(72)12∥(u,ut)∥𝒰ε,η2-c1≤Ψ≤2∥(u,ut)∥𝒰ε,η2+c2.
Then from (72) we get
(73)ddtΨ+12Ψ≤C+c2∶=C~,
and by Gronwall's lemma we obtain
(74)Ψ(t)≤(Ψ(0)-2C~)e-(1/2)t+2C~,∥(u,ut)∥𝒰ε,η2≤2(∥(u0,u1)∥𝒰ε,η2+c2-2C~)e-(1/2)t+4C~+2c1.
Then if (u0,u1)∈𝒰ε,η such that
(75)∥(u0,u1)∥𝒰ε,η≤R,
we have that
(76)∥(u,ut)∥𝒰ε,η≤R3,∀t≥τU,
where
(77)τU∶=2log(2R2+2c2-4C~R32-4C~-2c1),
and this concludes the proof.


## 4.3. Absorbing Set in𝒱ε,η

We consider the space𝒱ε,η as defined in (7), equipped with the norm defined in (10). By considering more regular initial conditions we can prove the following result.Proposition 99.
There existsR4>0 such that the closed ball
(78)ℬ4={(u,ut)∈𝒱ε,η:∥(u,ut)∥𝒱ε,η≤R4}
is a bounded absorbing set for the semigroup Sε,η(t) in 𝒱ε,η.Proof.
Multiply (1) by uxxxxt+uxxxx; then we obtain
(79)ddt{η2∥utxx∥2+ε22∥uxxxx∥2+η〈utxx,uxx〉+12∥uxxx∥2{ε22}}+(1-η)∥uxxt∥+ε2∥uxxxx∥2=12〈[W′(ux)]x,uxxxxt+uxxxx〉.
We call θ0(t) the differential term of the previous inequality and we estimate the r.h.s.:
(80)12〈[W′(ux)]x,uxxxx〉=-12〈[W′(ux)]xx,uxxx〉=-12〈W′′′(ux)uxx2,uxxx〉-12〈[W′′(ux)]xx,uxxx2〉=-12∫I‍uxuxx2uxxxdx-2∫I‍(3ux2-1)uxxx2dx≤2∥uxxx∥2+12∥uxx∥∞2∥ux∥∥uxxx∥≤2∥uxxx∥2(6∥uxx∥2+1)≤ε2R32(12ε2R22+1)∶=A1.
Let us define
(81)θ1(t)∶=14∫I‍W′′(ux)uxxx2dx;
then
(82)12〈[W′(ux)]x,uxxxxt〉=-12〈[W′(ux)]xx,uxxxt〉=-12〈W′′′(ux)uxx2,uxxxt〉-12〈W′′(ux)uxxx,uxxxt〉=-ddtθ1+12〈[W′′′(ux)uxx2]x,uxxt〉+14∫I‍W′′′(ux)uxtuxxx2dx.
We estimate the last two terms of the previous equality:
(83)14∫I‍W′′′(ux)uxtuxxx2dx=6∫I‍uxuxtuxxx2dx≤6∥uxxx∥∞∥ux∥∞∥uxt∥∥uxxx∥≤6∥uxxx∥3/2∥uxxxx∥1/2∥uxt∥∥uxx∥2≤(12·23/4R22R33/2ε7/2)∥uxxxx∥1/2∥uxt∥≤(12·23/4R22R33/2ε7/2)2∥uxxxx∥+∥uxt∥2≤12ε2(12·23/4R22R33/2ε7/2)4+2ηR32+ε22∥uxxxx∥2∶=ε22∥uxxxx∥2+A2,12〈[W′′′(ux)uxx2]x,uxxt〉=12〈Wiv(ux)uxx3+W′′′(ux)2uxxuxxx,uxxt〉=12∫I‍uxx3uxxtdx+24∫I‍uxuxxuxxxuxxtdx≤12(∥uxx∥∞2∥uxx∥+2∥ux∥∞∥uxx∥∞∥uxxx∥)∥uxxt∥≤12(∥uxxx∥3/2∥uxx∥2∥uxxx∥+2∥uxx∥3/2∥uxxx∥3/2)∥uxxt∥≤η2∥uxxt∥2+72({∥uxxx∥3/2}∥uxx∥2∥uxxx∥+2∥uxx∥3/2∥uxxx∥3/2)2≤η2∥uxxt∥2+72[22R22R3ε3+2(2R2R3ε2)3/2]2∶=η2∥uxxt∥2+A3.
Then if we set θ(t)=θ0(t)+θ1(t) we get
(84)ddtθ(t)+∥(u,ut)∥𝒱ε,η2≤A1+A2+A3∶=A.
Moreover we have
(85)θ(t)≥∥(u,ut)∥𝒱ε,η2-η4∥uxxt∥2-12∥uxx∥2≥12∥(u,ut)∥𝒱ε,η2-12R22∶=12∥(u,ut)∥𝒱ε,η2-a1,θ(t)≤∥(u,ut)∥𝒱ε,η2+η∥uxxt∥∥uxx∥+12∥uxx∥2+14∥W′′(ux)∥∞∥uxxx∥3≤2∥(u,ut)∥𝒱ε,η2+∥uxx∥2+(3∥ux∥∞2+1)∥uxxx∥2≤2∥(u,ut)∥𝒱ε,η2+2ε2R22+(6ε2R22+1)2ε2R32∶=2∥(u,ut)∥𝒱ε,η2+a2.
Then putting all together we get
(86)12∥(u,ut)∥𝒱ε,η2-a1≤θ(t)≤2∥(u,ut)∥𝒱ε,η2+a2.
From the previous inequality we get
(87)ddtθ(t)+12θ(t)≤A+a22∶=A~
and by Gronwall's lemma
(88)θ(t)≤[θ(0)-2A~]e-(1/2)t+2A~.
Using again (86) we get
(89)∥(u,ut)∥𝒱ε,η2≤2[2∥(u0,u1)∥𝒱ε,η2+a2-2A~]e-(1/2)t+4A~+2a1.
Then if (u0,u1)∈𝒱ε,η such that
(90)∥(u0,u1)∥𝒱ε,η≤R,
then there exists R4>0:
(91)∥(u,ut)∥𝒱ε,η≤R4,∀t≥τV,
where
(92)τV∶=2log[2(2R2+a2-2A~)R42-4A~-2a1],
and this concludes the proof.


## 5. Global Attractor

In this section we will show the existence of a global attractor for the semigroupSε,η(t) in ℋε,η. Since we have already proved the existence of the absorbing set in ℋε,η, then it is sufficient (see, e.g., [4] or [14] for general results or [15] for a recent application on a weakly damped wave equation) to prove that, for any fixed bounded set ℬ⊂ℋε,η, the solution semigroup Sε,η(t) admits the decomposition:
(93)Sε,η(t)=Lε,η(t)+Nε,η(t),
such that
(94)(C1)supt≥0supz∈ℋε,η∥Lε,η(t)z∥𝒱ε,η<∞,(95)(C2)limt→∞{supz∈ℋε,η∥Nε,η(t)z∥ℋε,η}=0.Letℬ⊂ℋε,η be a fixed bounded set and let (u0,u1)∈ℬ⊂ℋε,η. We will define the decomposition of Sε,η as follows:
(96)Sε,η(t)(u0,u1)=(u,ut),Lε,η(t)(u0,u1)=(h,ht),Nε,η(t)(u0,u1)=(v,vt),
where h and v are solutions of the following problems:
(97)ηhtt+ht+ε2hxxxx-αhxx=12W′′(hx)hxx-αuxx,h(x,0)=0,ht(x,0)=0,h=hxx=0,in∂I,(98)ηvtt+vt+ε2vxxxx-αvxx=12W′′(ux)uxx-12W′′(hx)hxx,v(0)=u0,vt(0)=u1,v=vxx=0,in∂I,
where
(99)α≥62εR2>66.
Before showing that the semigroups Lε,η(t) and Nε,η(t) satisfy the conditions (94) and (95), respectively, we consider the following lemma (see [9]) that will be useful for the sequel.Lemma 1010.
Letψ:ℝ+→ℝ be an absolutely continuous function which fulfills, for some ν>0 and almost every t≥0, the differential inequality
(100)ddtψ(t)+2νψ(t)≤f(t)ψ(t),
where f is a positive function satisfying
(101)∫0t‍f(y)dy≤νt+c,∀t≥0.
Then
(102)ψ(t)≤cψ(0)e-νt,t≥0.


### 5.1. The SemigroupLε,η(t)

Due to the similarity of problem (97) to (1), Proposition 9 still holds in this setting. We will omit the proof. Consequently we get that condition (94) holds by noting that (h(0),ht(0))=(0,0).Moreover by (97), multiplying the equation by ht and integrating over I, we get that there exists a constant c>0 such that
(103)ddt{η2∥ht∥2+ε22∥hxx∥2+12∫I‍W(hx)dx+α2∥hx∥2+α〈uxx,h〉}+∥ht∥2=α〈ut,hxx〉≤c∥ut∥.
Then integrating the above equation on (0,t) and using (30) we get that there exist constants A>0 and B>0 such that
(104)∫0t‍∥ht∥2ds≤A+c∫0t‍∥ut∥ds≤A+Bt1/2.
Consequently, for any γ>0 there exists a constant Cγ≥0 such that
(105)∫0t‍∥ht∥2ds≤γt+Cγ.


### 5.2. The SemigroupNε,η(t)

Letβ>0 be a parameter to be determined later and let us multiply, in L2(I), (98) by 2vt+βv. Then we get
(106)ddt{{W′(hx)vx,}η∥vt∥2+ηβ〈vt,v〉+ε2∥vxx∥2+β2∥v∥2+α∥vx∥2+〈W(ux)-W(hx)-W′(hx)vx,1〉}+(2-ηβ)∥vt∥2+ε2β∥vxx∥2-β2〈W′(ux)-W′(hx),vx〉=〈htx,W′(ux)-W′(hx)-W′′(hx)vx〉.
We estimate the term involving W and its derivatives:
(107)-β2〈W′(ux)-W′(hx),vx〉=-β2〈W′′(kx)vx,vx〉=2β∫I‍(1-3kx2)vx2dx≥-6β∥vx∥∞∥vx∥2≥-6β2εR2∥vx∥2,
where kx(x) is between hx(x) and ux(x) and satisfies ∥kx∥∞≤∥vx∥∞. Moreover we have
(108)〈W(ux)-W(hx)-W′(hx)vx,1〉=〈W′(hx+k~x)-W′(hx),vx〉=〈W′′(sx)k~x,vx〉=12∫I‍sx2k~xvxdx-4∫I‍kxvxdx≥-4∥vx∥2,
where k~x(x) is between 0 and vx(x), sx(x) is between 0 and k~x(x) and ∥sx∥∞≤∥k~x∥∞≤∥vx∥∞. Moreover we have used that vx2≥k~xvx≥0.We denote asℱ the differential term of (106) and prove that it induces a norm that is equivalent to that of ℋε,η. Then from (99) and (107) we have that
(109)ℱ(t)≥η∥vt∥2-η2∥vt∥2-β2∥v∥2+ε2∥vxx∥2+α∥vx∥2-6β2εR2∥vx∥2≥η2∥vt∥2+ε2∥vxx∥2≥∥(v,vt)∥ℋε,η2.
Moreover we have
(110)ℱ(t)≤η∥vt∥2+η2∥vt∥2+β2∥v∥2+ε2∥vxx∥2+β2∥v∥2+α∥vx∥2+∥W′′(kx)∥∞∥vx∥2≤32η∥vt∥2+β∥vxx∥2+ε2∥vxx∥2+α∥vxx∥2+4(3∥ux∥∞2+1)∥vxx∥2≤32η∥vt∥2+(β+ε2+α+122R2ε+4)∥vxx∥2∶=32η∥vt∥2+Aε22∥vxx∥2≤c∥(v,vt)∥ℋε,η2,
where we have used the same argument as (107) and (108) and where
(111)A-=2ε2(β+ε2+α+122R2ε+4),c=max{3,A-}.
Then we have
(112)∥(v,vt)∥ℋε,η2≤ℱ(t)≤c∥(v,vt)∥ℋε,η2.
We estimate the right hand side of (106):
(113)〈htx,W′(ux)-W′(hx)-W′′(hx)vx〉=〈ht,W′′′(hx)hxxvx+(W′′(hx)-W′′(ux))uxx〉=〈ht,24hxhxxvx+12(hx2-ux2)uxx〉=〈ht,12hxvx(2hxx-uxx)-12vxuxuxx〉=〈ht,12vx[hx(hxx-vxx)-uxuxx]〉=〈ht,12vx[-2vxxhx-vxuxx]〉≤24∥ht∥(∫I‍vx2vxx2hx2)1/2+12∥ht∥(∫I‍uxx2vx4)1/2≤24∥ht∥∥vx∥∞∥hx∥∞∥vxx∥+12∥ht∥∥vx∥∞∥uxx∥∞∥vx∥≤12∥ht∥∥vxx∥2(2∥hxx∥+∥uxxx∥)≤σ∥ht∥∥vxx∥2≤{2σ2βε2∥ht∥2+βε22}∥vxx∥2,
where
(114)σ=122ε(2R2+R3).
Then it remains to estimate the following term:
(115)(2-ηβ)∥vt∥2+βε2∥vxx∥2+αβ∥vx∥2-β2〈W′(ux)-W′(hx),vx〉≥βη2∥vt∥2+βε2∥vxx∥2+αβ∥vx∥2-6β2εR2∥vx∥2≥βη2∥vt∥2+βε2∥vxx∥2.
Then, using the above estimates, we can rewrite (106) in the following way:
(116)ddtℱ(t)+βcℱ(t)≤σ22βε2∥ht∥2∥vxx∥2≤σ2βε4∥ht∥2∥(v,vt)∥ℋε,η2≤σ2βε4∥ht∥2ℱ(t).
Then by Lemma 10 and from (105) we conclude that there exists a positive constant K such that
(117)ℱ(t)≤Kℱ(0)e-(β/2c)t,
and consequently from (112) we conclude
(118)∥(u,ut)∥ℋη,ε2≤cK∥(u0,u1)∥ℋη,ε2e-(β/2c)t.
This shows that the semigroup Nε,η(t) satisfies condition (95) and hence we have proven the following.Theorem 1111.
The semigroupSε,η(t) possesses a connected global attractor 𝒜ε,η⊂ℋε,η that is bounded in 𝒱ε,η.


## 5.1. The SemigroupLε,η(t)

Due to the similarity of problem (97) to (1), Proposition 9 still holds in this setting. We will omit the proof. Consequently we get that condition (94) holds by noting that (h(0),ht(0))=(0,0).Moreover by (97), multiplying the equation by ht and integrating over I, we get that there exists a constant c>0 such that
(103)ddt{η2∥ht∥2+ε22∥hxx∥2+12∫I‍W(hx)dx+α2∥hx∥2+α〈uxx,h〉}+∥ht∥2=α〈ut,hxx〉≤c∥ut∥.
Then integrating the above equation on (0,t) and using (30) we get that there exist constants A>0 and B>0 such that
(104)∫0t‍∥ht∥2ds≤A+c∫0t‍∥ut∥ds≤A+Bt1/2.
Consequently, for any γ>0 there exists a constant Cγ≥0 such that
(105)∫0t‍∥ht∥2ds≤γt+Cγ.


## 5.2. The SemigroupNε,η(t)

Letβ>0 be a parameter to be determined later and let us multiply, in L2(I), (98) by 2vt+βv. Then we get
(106)ddt{{W′(hx)vx,}η∥vt∥2+ηβ〈vt,v〉+ε2∥vxx∥2+β2∥v∥2+α∥vx∥2+〈W(ux)-W(hx)-W′(hx)vx,1〉}+(2-ηβ)∥vt∥2+ε2β∥vxx∥2-β2〈W′(ux)-W′(hx),vx〉=〈htx,W′(ux)-W′(hx)-W′′(hx)vx〉.
We estimate the term involving W and its derivatives:
(107)-β2〈W′(ux)-W′(hx),vx〉=-β2〈W′′(kx)vx,vx〉=2β∫I‍(1-3kx2)vx2dx≥-6β∥vx∥∞∥vx∥2≥-6β2εR2∥vx∥2,
where kx(x) is between hx(x) and ux(x) and satisfies ∥kx∥∞≤∥vx∥∞. Moreover we have
(108)〈W(ux)-W(hx)-W′(hx)vx,1〉=〈W′(hx+k~x)-W′(hx),vx〉=〈W′′(sx)k~x,vx〉=12∫I‍sx2k~xvxdx-4∫I‍kxvxdx≥-4∥vx∥2,
where k~x(x) is between 0 and vx(x), sx(x) is between 0 and k~x(x) and ∥sx∥∞≤∥k~x∥∞≤∥vx∥∞. Moreover we have used that vx2≥k~xvx≥0.We denote asℱ the differential term of (106) and prove that it induces a norm that is equivalent to that of ℋε,η. Then from (99) and (107) we have that
(109)ℱ(t)≥η∥vt∥2-η2∥vt∥2-β2∥v∥2+ε2∥vxx∥2+α∥vx∥2-6β2εR2∥vx∥2≥η2∥vt∥2+ε2∥vxx∥2≥∥(v,vt)∥ℋε,η2.
Moreover we have
(110)ℱ(t)≤η∥vt∥2+η2∥vt∥2+β2∥v∥2+ε2∥vxx∥2+β2∥v∥2+α∥vx∥2+∥W′′(kx)∥∞∥vx∥2≤32η∥vt∥2+β∥vxx∥2+ε2∥vxx∥2+α∥vxx∥2+4(3∥ux∥∞2+1)∥vxx∥2≤32η∥vt∥2+(β+ε2+α+122R2ε+4)∥vxx∥2∶=32η∥vt∥2+Aε22∥vxx∥2≤c∥(v,vt)∥ℋε,η2,
where we have used the same argument as (107) and (108) and where
(111)A-=2ε2(β+ε2+α+122R2ε+4),c=max{3,A-}.
Then we have
(112)∥(v,vt)∥ℋε,η2≤ℱ(t)≤c∥(v,vt)∥ℋε,η2.
We estimate the right hand side of (106):
(113)〈htx,W′(ux)-W′(hx)-W′′(hx)vx〉=〈ht,W′′′(hx)hxxvx+(W′′(hx)-W′′(ux))uxx〉=〈ht,24hxhxxvx+12(hx2-ux2)uxx〉=〈ht,12hxvx(2hxx-uxx)-12vxuxuxx〉=〈ht,12vx[hx(hxx-vxx)-uxuxx]〉=〈ht,12vx[-2vxxhx-vxuxx]〉≤24∥ht∥(∫I‍vx2vxx2hx2)1/2+12∥ht∥(∫I‍uxx2vx4)1/2≤24∥ht∥∥vx∥∞∥hx∥∞∥vxx∥+12∥ht∥∥vx∥∞∥uxx∥∞∥vx∥≤12∥ht∥∥vxx∥2(2∥hxx∥+∥uxxx∥)≤σ∥ht∥∥vxx∥2≤{2σ2βε2∥ht∥2+βε22}∥vxx∥2,
where
(114)σ=122ε(2R2+R3).
Then it remains to estimate the following term:
(115)(2-ηβ)∥vt∥2+βε2∥vxx∥2+αβ∥vx∥2-β2〈W′(ux)-W′(hx),vx〉≥βη2∥vt∥2+βε2∥vxx∥2+αβ∥vx∥2-6β2εR2∥vx∥2≥βη2∥vt∥2+βε2∥vxx∥2.
Then, using the above estimates, we can rewrite (106) in the following way:
(116)ddtℱ(t)+βcℱ(t)≤σ22βε2∥ht∥2∥vxx∥2≤σ2βε4∥ht∥2∥(v,vt)∥ℋε,η2≤σ2βε4∥ht∥2ℱ(t).
Then by Lemma 10 and from (105) we conclude that there exists a positive constant K such that
(117)ℱ(t)≤Kℱ(0)e-(β/2c)t,
and consequently from (112) we conclude
(118)∥(u,ut)∥ℋη,ε2≤cK∥(u0,u1)∥ℋη,ε2e-(β/2c)t.
This shows that the semigroup Nε,η(t) satisfies condition (95) and hence we have proven the following.Theorem 1111.
The semigroupSε,η(t) possesses a connected global attractor 𝒜ε,η⊂ℋε,η that is bounded in 𝒱ε,η.



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*Source: 372726-2013-03-21.xml*