Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 372726, 11 pages
http://dx.doi.org/10.1155/2013/372726
Research Article
Hyperbolic Relaxation of a Fourth Order Evolution Equation
Renato Colucci and Gerardo R. Chacón
Departamento de Matemáticas, Pontificia Universidad Javeriana, Cra. 7 No. 43-82, Bogotá, Colombia
Correspondence should be addressed to Renato Colucci; renatocolucci@hotmail.com
Received 26 November 2012; Revised 30 January 2013; Accepted 3 February 2013
Academic Editor: Juan J. Nieto
Copyright © 2013 R. Colucci and G. R. Chacón. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We propose a hyperbolic relaxation of a fourth order evolution equation, with an inertial term 𝜂𝑢𝑡𝑡 , 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)}.
1. Introduction
Leting 𝐼 ⊂ R be an open interval, with |𝐼| ≤ 1, we consider the
following initial-boundary value problem for 𝑢 : 𝐼 × R+ →
R:
𝜂𝑢𝑡𝑡 + 𝑢𝑡 = − 𝜀2 𝑢𝑥𝑥𝑥𝑥
1
+ 𝑊󸀠󸀠 (𝑢𝑥 ) 𝑢𝑥𝑥 ,
2
(𝑥, 𝑡) ∈ 𝐼 × R+ ,
𝑢 (𝑥, 0) = 𝑢0 (𝑥) ,
in 𝐼,
𝑢𝑡 (𝑥, 0) = 𝑢1 (𝑥) ,
in 𝐼,
𝑢 = 𝑢𝑥𝑥 = 0,
(1)
in 𝜕𝐼, 𝑡 ≥ 0,
where the function 𝑊(𝑝) = (𝑝2 − 1)2 is the so-called doublewell 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 𝑛dimensional 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
2
Abstract and Applied Analysis
relaxation of the fourth order evolution equation proposed
in [1]. In particular if V is the solution of the Cahn-Hilliard
equation
V𝑡 + Δ [𝜀2 ΔV − 𝑊󸀠 (V)] = 0,
(2)
𝑥 ∈ {0, 1} ,
(3)
then
𝑥
𝑢 (𝑥) = ∫ V (𝑠) 𝑑𝑠,
(4)
0
is the solution of (1), with 𝜂 = 0, with the corresponding
boundary conditions:
𝑢 = 𝑢𝑥𝑥 = 0,
̃ (𝑢, V) = 𝜂‖V‖2 + 1 𝜀2 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩2 + 𝛽𝜂⟨V, 𝑢⟩,
𝐸
2 󵄩 󵄩
(12)
where 𝛽 ∈ [0, 𝜀2 ].
with Neumann boundary conditions:
𝜕
𝜕
V=
ΔV = 0,
𝜕𝑛
𝜕𝑛
Given functions 𝑢 ∈ 𝐻2 (𝐼) ∩ 𝐻01 (𝐼) and V ∈ 𝐿2 (𝐼) we define
̃ on H𝜂,𝜀 by
the function 𝐸
𝑥 ∈ {0, 1} .
(5)
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
̃ ⋅) induces a
Proposition 1. For all 𝛽 ∈ [0, 𝜀2 ] the function 𝐸(⋅,
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
1
1 󵄩 󵄩2
𝛽𝜂 ⟨V, 𝑢⟩ ≤ 𝛽𝜂 ‖V‖‖𝑢‖ ≤ 𝜂2 ‖V‖2 + 𝜀4 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
2
2
𝜂
𝜀 2 󵄩 󵄩2
≤ ‖V‖2 + 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 = ‖(𝑢, V)‖2H𝜀,𝜂 .
2
2
(13)
̃ V) we
From the previous inequality and by definition of 𝐸(𝑢,
get
̃ (𝑢, V) ≤ 3‖(𝑢, V)‖2 .
𝐸
H𝜀,𝜂
(14)
By a different application of Schwartz inequality and from (11)
we get
𝛽𝜂 ⟨V, 𝑢⟩ ≤ 𝛽𝜂 ‖V‖ ‖𝑢‖ ≤ 𝜂2 ‖V‖2 +
𝛽2
‖𝑢‖2
4
𝜀4
𝜀 2 󵄩 󵄩2
≤ 𝜂‖V‖ + ‖𝑢‖2 ≤ 𝜂‖V‖2 + 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 .
4
4
(15)
2
We begin this section by defining the following Hilbert spaces
that will be helpful for our analysis:
2
H𝜂,𝜀 = [𝐻 (𝐼) ∩
𝐻01
2
(𝐼)] × 𝐿 (𝐼) ,
2
(6)
V𝜂,𝜀 = 𝐷 (𝐴) × 𝐻 (𝐼) ,
(7)
𝐷 (𝐴) = {𝑢 ∈ 𝐻4 (𝐼) : 𝑢 = 𝑢𝑥𝑥 = 0 in 𝜕𝐼}
(8)
where
is the domain of the differential operator 𝐴 = 𝜕4 /𝜕𝑥4 . The
above spaces are equipped with the norms
‖(𝑢, V)‖2H𝜂,𝜀 =
‖(𝑢, V)‖2V𝜂,𝜀 =
𝜀2 󵄩󵄩 󵄩󵄩2 𝜂 2
󵄩𝑢 󵄩 + ‖V‖ ,
2 󵄩 𝑥𝑥 󵄩
2
𝜀2 󵄩󵄩
󵄩󵄩2
󵄩𝑢
󵄩 +
2 󵄩 x𝑥𝑥𝑥 󵄩
𝜂 󵄩󵄩 󵄩󵄩2
󵄩V 󵄩 ,
2 󵄩 𝑥𝑥 󵄩
(9)
(10)
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
󵄩 󵄩 󵄩 󵄩
‖𝑢‖ ≤ 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 .
(11)
Throughout the paper we will use two norms that are equivalent to (9) and (10) in order to simplify the computation.
Combining (13) and (15) we get
̃ (𝑢, V) ≥ max { 𝜂 ‖V‖2 , 1 𝜀2 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩2 } ,
𝐸
2
4 󵄩 󵄩
(16)
and as a consequence
̃ (𝑢, V) ≥ 1 ‖(𝑢, V)‖2 .
𝐸
H𝜀,𝜂
3
(17)
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 2. For every (𝑢0 , 𝑢1 ) ∈ H𝜀,𝜂 there exists a unique
solution 𝑢(𝑡) for the initial value problem (1) such that
𝑢 ∈ 𝐶𝑏 (R+ ; 𝐻2 (𝐼) ∩ 𝐻01 (𝐼)) ∩ 𝐶𝑏1 (R+ ; 𝐿2 (𝐼)) .
(18)
If, moreover, (𝑢0 , 𝑢1 ) ∈ V𝜀,𝜂 , then:
𝑢 ∈ 𝐶𝑏 (R+ ; 𝐷 (𝐴)) ∩ 𝐶𝑏1 (R+ ; 𝐻2 (𝐼) ∩ 𝐻01 (𝐼))
∩ 𝐶𝑏2 (R+ ; 𝐿2 (𝐼)) .
(19)
Abstract and Applied Analysis
3
Proposition 3. For any constant 𝑅 ≥ 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
󵄩󵄩
󵄩
(𝐾2 /𝜀2 )𝑡 󵄩
󵄩󵄩𝑢1 (0) − 𝑢2 (0)󵄩󵄩󵄩 ,
󵄩󵄩𝑆𝜀,𝜂 (𝑡)𝑢1 (0) − 𝑆𝜀,𝜂 (𝑡)𝑢2 (0)󵄩󵄩󵄩
󵄩𝐻𝜂,𝜀
󵄩
󵄩
󵄩𝐻𝜂,𝜀 ≤ 𝑒
(20)
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:
󵄩 󵄩2 1 󵄩 󵄩2 1
𝐸 (𝑢, 𝑢𝑡 ) = 𝜂󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩 + 𝜀2 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + ∫ 𝑊 (𝑢𝑥 ) 𝑑𝑥,
2
2 𝐼
that is not increasing along the solutions; in fact if we multiply
(1) by 𝑢𝑡 and integrate over 𝐼 we obtain
where 𝑆𝜀,𝜂 (𝑡) is the solution semigroup of the problem (1).
Proof. Let 𝑢1 , 𝑢2 , two solutions of (1) with initial data 𝑢1 (0)
and 𝑢2 (0). Let, 𝑤 = 𝑢1 − 𝑢2 then we write
𝜂𝑤𝑡𝑡 + 𝑤𝑡 + 𝜀2 𝑤𝑥𝑥𝑥𝑥 =
1 𝑑
{𝑊󸀠 (𝑢1𝑥 ) − 𝑊󸀠 (𝑢2𝑥 )} .
2 𝑑𝑥
(21)
𝑑
󵄩 󵄩2
𝐸 (𝑢, 𝑢𝑡 ) = −󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩 .
𝑑𝑡
𝑡
𝜂 𝑑 󵄩󵄩 󵄩󵄩2 𝜀2 𝑑 󵄩󵄩
󵄩2 󵄩 󵄩2
󵄩𝑤 󵄩 +
󵄩𝑤 󵄩󵄩 + 󵄩󵄩𝑤 󵄩󵄩
2 𝑑𝑡 󵄩 𝑡 󵄩
2 𝑑𝑡 󵄩 𝑥𝑥 󵄩 󵄩 𝑡 󵄩
(28)
Then for any fixed initial data (𝑢0 , 𝑢1 ) ∈ H𝜀,𝜂 we have that the
corresponding solution satisfies
𝜂 󵄩 󵄩2 𝜀2 󵄩 󵄩2
󵄩
󵄩2
sup { 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 } = sup󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩H𝜀,𝜂
2
2
𝑡≥0
𝑡≥0
= −2 ∫ 𝑤𝑥𝑥 𝑤𝑡 𝑑𝑥
𝐼
2
+ 6 ∫ {𝑢1𝑥
𝑤𝑥𝑥 + 𝑢2𝑥x 𝑤𝑥 [𝑢1𝑥 + 𝑢2𝑥 ]} 𝑤𝑡 𝑑𝑥
(29)
≤ 𝐸 (𝑢0 , 𝑢1 ) .
𝐼
󵄩 󵄩2 󵄩
󵄩
󵄩󵄩 󵄩
󵄩󵄩 󵄩
≤ 2 󵄩󵄩󵄩𝑤𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑤𝑡 󵄩󵄩󵄩 + 6󵄩󵄩󵄩𝑢1𝑥 󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑤𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑤𝑡 󵄩󵄩󵄩
Moreover if we integrate over R+ we get the integral control
1/2
𝐼
∞
𝐼
󵄩
󵄩󵄩 󵄩
󵄩
󵄩2 󵄩 󵄩2
≤ 2𝐾 󵄩󵄩󵄩𝑤𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑤𝑡 󵄩󵄩󵄩 ≤ 𝐾2 󵄩󵄩󵄩𝑤𝑥𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑤𝑡 󵄩󵄩󵄩 ,
(31)
By summing (27) and (31) we get
where we have used the following inequality (see [13] and
inequality (11)):
(23)
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)
and then using Gronwall’s lemma we get
󵄩󵄩
󵄩
(𝐾2 /𝜀2 )𝑡 󵄩
󵄩󵄩(𝑤(0), 𝑤𝑡 (0))󵄩󵄩󵄩 .
󵄩󵄩(𝑤, 𝑤𝑡 )󵄩󵄩󵄩H𝜀,𝜂 ≤ 𝑒
󵄩
󵄩H𝜀,𝜂
𝛽 𝑑
󵄩 󵄩2
‖𝑢‖2 + 𝛽𝜀2 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
2 𝑑𝑡
󵄩 󵄩4
󵄩 󵄩2
+ 2𝛽󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩4 = 2𝛽󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 .
𝛽𝜂 ∫ 𝑢𝑡𝑡 𝑢𝑑𝑥 +
𝑅2 󵄩󵄩
󵄩󵄩 󵄩
} 󵄩𝑤 󵄩󵄩 󵄩󵄩𝑤 󵄩󵄩
𝜀2 󵄩 𝑥𝑥 󵄩 󵄩 𝑡 󵄩
𝑑 󵄩󵄩
2𝐾2 󵄩
󵄩2
󵄩2
󵄩󵄩(𝑤, 𝑤𝑡 )󵄩󵄩󵄩H𝜀,𝜂 ≤ 2 󵄩󵄩󵄩(𝑤, 𝑤𝑡 )󵄩󵄩󵄩H𝜀,𝜂 ,
𝑑𝑡
𝜀
(30)
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
󵄩
󵄩2 󵄩
󵄩2 1/2 󵄩
󵄩󵄩 󵄩
×(󵄩󵄩󵄩𝑢1𝑥𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑢2𝑥𝑥 󵄩󵄩󵄩 ) } 󵄩󵄩󵄩𝑤𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑤𝑡 󵄩󵄩󵄩
󵄩󵄩 󵄩󵄩2
󵄩2
2 󵄩 󵄩2
2󵄩
󵄩󵄩𝑢𝑥 󵄩󵄩∞ ≤ 𝐶 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩𝐻1 (𝐼) ≤ 2𝐶 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 .
󵄩 󵄩2
∫ 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩 𝑑𝑡 ≤ 𝐸 (𝑢0 , 𝑢1 ) .
0
(22)
󵄩
󵄩2
󵄩
󵄩
≤ 2 {1 + 18𝐶2 󵄩󵄩󵄩𝑢1𝑥𝑥 󵄩󵄩󵄩 + 12 󵄩󵄩󵄩𝑢2𝑥𝑥 󵄩󵄩󵄩
≤ 2 {1 + (36𝐶2 + 24√2)
(27)
Moreover, integrating the previous inequality on (0, 𝑡) with
respect to time we get
󵄩 󵄩2
∫ 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩 𝑑𝑠 + 𝐸 (𝑢, 𝑢𝑡 ) = 𝐸 (𝑢0 , 𝑢1 ) .
0
We multiply the above equation by 𝑤𝑡 in 𝐿2 (𝐼); then
󵄩
󵄩
2
2
+ 12 󵄩󵄩󵄩𝑢2𝑥𝑥 󵄩󵄩󵄩 (∫ 𝑤𝑥2 𝑤𝑡2 [𝑢1𝑥
+ 𝑢2𝑥
] 𝑑𝑥)
(26)
(25)
𝑑
𝑑
󵄩 󵄩2
𝐸 + 𝛽𝜂 ⟨𝑢𝑡 , 𝑢⟩ + (1 − 𝛽𝜂) 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩
𝑑𝑡
𝑑𝑡
𝛽 𝑑
󵄩 󵄩2
󵄩 󵄩4
‖𝑢‖2 + 𝛽𝜀2 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + 2𝛽󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩4
2 𝑑𝑡
󵄩 󵄩2
= 2𝛽󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 .
+
(32)
Using the expression of the energy we can rewrite the
previous in the following way:
𝑑
𝑑
󵄩 󵄩2
𝐸 + 𝛽𝜂 ⟨𝑢𝑡 , 𝑢⟩ + (1 − 3𝛽𝜂) 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩
𝑑𝑡
𝑑𝑡
󵄩󵄩 󵄩󵄩4
󵄩𝑢𝑥 󵄩
|𝐼|
+ 𝛽⟨𝑢𝑡 , 𝑢⟩ + 2𝛽 (𝐸 + 󵄩 󵄩4 − ) = 0.
2
2
(33)
We will estimate some of the terms of the previous inequality
in the following lemma.
4
Abstract and Applied Analysis
Lemma 4. Fix 𝜂 ≤ 1 then for all 𝛽 ∈ (0, 1/3) one has
4. Absorbing Sets
󵄩 󵄩2
󵄩 󵄩4
2𝛽2 𝜂⟨𝑢𝑡 , 𝑢⟩ ≤ (1 − 3𝛽𝜂) 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩 + 𝛽󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩4 + 𝛽 |𝐼| + 𝛽⟨u𝑡 , 𝑢⟩.
(34)
Proof. By Holder’s inequality we get
󵄩 󵄩2
󵄩 󵄩2
󵄩 󵄩4
2󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 ≤ 2󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩4 |𝐼|1/2 ≤ 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩4 + |𝐼| ,
(35)
then we use Poincaré’s inequality and the fact that |𝐼| ≤ 1 to
conclude that
󵄩󵄩 󵄩󵄩4
󵄩𝑢𝑥 󵄩
|𝐼|
(36)
‖𝑢‖2 ≤ 󵄩 󵄩4 + .
2
2
Therefore, for a positive constant 𝑐1 to be determined later, we
have that
𝑐 󵄩 󵄩2
1
󵄩 󵄩
⟨𝑢𝑡 , 𝑢⟩ ≤ 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩 ‖𝑢‖ ≤ 1 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩 +
‖𝑢‖2
2
2𝑐1
󵄩 󵄩4
𝑐1 󵄩󵄩 󵄩󵄩2
1 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩4 |𝐼|
≤ 󵄩󵄩𝑢𝑡 󵄩󵄩 +
(
+ ).
2
2𝑐1
2
2
(37)
1 󵄩 󵄩2 󵄩 󵄩2
1
󵄩 󵄩2 |𝐼|
∫ 𝑊 (𝑢𝑥 ) 𝑑𝑥 ≤ 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩𝐿∞ (𝐼) − 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 +
2 𝐼
2
2
≤
1 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 |𝐼|
󵄩𝑢 󵄩 󵄩𝑢 󵄩 +
2 󵄩 𝑥 󵄩 󵄩 𝑥𝑥 󵄩
2
(44)
1 󵄩󵄩 󵄩󵄩4 |𝐼|
󵄩𝑢 󵄩 + .
2 󵄩 𝑥𝑥 󵄩
2
We will use this inequality several times in what follows.
≤
4.1. Absorbing Set in H𝜀,𝜂
Proposition 5. For all 𝑅0 > 1 the set
Consequently, by choosing the parameters 𝑐1 = 1/12 and 𝛽 <
1/3 we have that
(2𝛽2 𝜂 − 𝛽) ⟨𝑢𝑡 , 𝑢⟩
󵄨
󵄨 𝑐 󵄩 󵄩2
≤ 󵄨󵄨󵄨󵄨2𝛽2 𝜂 − 𝛽󵄨󵄨󵄨󵄨 1 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩
2
󵄨󵄨 2
󵄨󵄨
󵄨󵄨󵄨2𝛽2 𝜂 − 𝛽󵄨󵄨󵄨
󵄨
󵄨󵄨 󵄩󵄩 󵄩󵄩4 󵄨󵄨󵄨2𝛽 𝜂 − 𝛽󵄨󵄨󵄨
+󵄨
|𝐼|
󵄩󵄩𝑢𝑥 󵄩󵄩4 +
4𝑐1
4𝑐1
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
𝐵Φ := {(𝑢, 𝑢𝑡 ) ∈ H𝜀,𝜂 : Φ (𝑢, 𝑢𝑡 ) ≤ 𝑅0 }
(45)
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
(38)
󵄩 󵄩4
󵄩 󵄩2
≤ (1 − 3𝛽𝜂) 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩 + 𝛽󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩4 + 𝛽 |𝐼| .
Φ (𝑢, 𝑢𝑡 ) ≤ [𝑅0 − 1] 𝑒−2𝛽𝑡 + 1 ≤ 𝑅0 ,
∀𝑡 > 0.
(46)
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
Φ (𝑢, 𝑢𝑡 ) ≤ 𝑅0 ,
∀𝑡 ≥ 𝜏Φ ,
(47)
where
Now, using the previous lemma and (33) we conclude that
𝑑
𝑑
𝐸 + 𝛽𝜂 ⟨𝑢𝑡 , 𝑢⟩ + 2𝛽2 𝜂 ⟨𝑢𝑡 , 𝑢⟩ + 2𝛽𝐸 ≤ 2𝛽 |𝐼| .
𝑑𝑡
𝑑𝑡
(39)
𝜏Φ :=
1
𝑅−1
log {
}.
2𝛽
𝑅0 − 1
(48)
Then 𝐵Φ is absorbing for the semigroup 𝑆𝜀,𝜂 (𝑡).
Proposition 6. For all 𝑅1 > 1 one has that the set
If we set
Φ (𝑢, 𝑢𝑡 ) = Φ (𝑡) = 𝐸 (𝑢, 𝑢𝑡 ) + 𝛽𝜂⟨𝑢𝑡 , 𝑢⟩,
(40)
̃ (𝑢, 𝑢𝑡 ) ≤ 𝑅1 }
𝐵𝐸̃ := {(𝑢, 𝑢𝑡 ) ∈ H𝜀,𝜂 : 𝐸
(49)
is absorbing for the semigroup 𝑆𝜀,𝜂 (𝑡).
then from (39) we get
𝑑
Φ (𝑡) + 2𝛽Φ (𝑡) ≤ 2𝛽.
𝑑𝑡
(41)
Now, integrating we have that
−2𝛽𝑡
Φ (𝑡) ≤ [Φ (𝑢0 , 𝑢1 ) − 1] 𝑒
+ 1.
(42)
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
̃ and Φ:
functions 𝐸, 𝐸,
̃ (𝑡) + 1 ∫ 𝑊 (𝑢𝑥 ) 𝑑𝑥 ≥ 𝐸
̃ ≥ 0.
Φ (𝑡) = 𝐸
2 𝐼
(43)
Proof. We have from (42) and (43) that
̃ (𝑢, 𝑢𝑡 ) ≤ [Φ (𝑢0 , 𝑢1 ) − 1] 𝑒−2𝛽𝑡 + 1.
𝐸
(50)
̃ 0 , 𝑢1 ) ≤ 𝑅, with 𝑅 > 𝑅1 . Then,
Now, take (𝑢0 , 𝑢1 ) such that 𝐸(𝑢
by (42) and (44) we get that
̃ (𝑡) ≤ Φ (𝑡) ≤ 𝐸
̃ (𝑡) + 1 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩4 + |𝐼|
𝐸
2 󵄩 󵄩4 2
̃ (𝑡) + 1 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩4 + |𝐼|
≤𝐸
2󵄩 󵄩
2
̃ 2 + |𝐼| .
̃ (𝑡) + 4 𝐸(𝑡)
≤𝐸
2
𝜀
2
(51)
Abstract and Applied Analysis
5
Proposition 8. There exists 𝑅3 > 0 such that the closed ball
Therefore,
̃ (𝑢, 𝑢𝑡 ) ≤ {𝐸
̃ 0 , 𝑢1 )2 + |𝐼| − 1} 𝑒−2𝛽𝑡 + 1
̃ (𝑢0 , 𝑢1 ) + 4 𝐸(𝑢
𝐸
𝜀2
2
4
|𝐼|
≤ {𝑅 + 2 𝑅2 +
− 1} 𝑒−2𝛽𝑡 + 1.
𝜀
2
(52)
Consequently, we have that
̃ (𝑢, 𝑢𝑡 ) ≤ 𝑅1 ,
𝐸
∀𝑡 ≥ 𝜏𝐸̃ ,
(53)
𝜏𝐸̃ =
2
𝑅 + (4/𝜀 ) 𝑅 + (|𝐼| /2) − 1
1
}.
log {
2𝛽
𝑅1 − 1
(54)
Now using the equivalence of the norm on H𝜀,𝜂 and (52)
we get
󵄩󵄩
󵄩2
󵄩
󵄩2
󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩H𝜀,𝜂 ≤ 3 {3󵄩󵄩󵄩(𝑢0 , 𝑢1 )󵄩󵄩󵄩H𝜀,𝜂
+
Proof. If we multiply (1) by 𝑢𝑥𝑥𝑡 + 𝑢𝑥𝑥 and integrate over 𝐼 we
obtain
1 󵄩󵄩 󵄩󵄩2
󵄩𝑢 󵄩 + 𝜂⟨𝑢𝑡𝑥 , 𝑢𝑥 ⟩}
2 󵄩 𝑥󵄩
󵄩 󵄩2
󵄩
󵄩2
+ (1 − 𝜂) 󵄩󵄩󵄩𝑢𝑡𝑥 󵄩󵄩󵄩 + 𝜀2 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩
1
1
= − ⟨[𝑊󸀠 (𝑢𝑥 )]𝑥 , 𝑢𝑥𝑥𝑡 ⟩ − ⟨[𝑊󸀠 (𝑢𝑥 )]𝑥 , 𝑢𝑥𝑥 ⟩ .
2
2
(63)
Let us denote the differential term of the previous inequality
as
36 󵄩󵄩
|𝐼|
󵄩4
+
− 1} 𝑒−2𝛽𝑡 + 3.
󵄩(𝑢 , 𝑢 )󵄩󵄩
𝜀2 󵄩 0 1 󵄩H𝜀,𝜂 2
(55)
Ψ0 (𝑡) =
Let 𝑅2 > √3. If (𝑢0 , 𝑢1 ) ∈ H𝜀,𝜂 is such that
󵄩
󵄩󵄩
󵄩󵄩(𝑢0 , 𝑢1 )󵄩󵄩󵄩H𝜀,𝜂 ≤ 𝑅,
then we have that
󵄩󵄩
󵄩
󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩H𝜀,𝜂 ≤ 𝑅2 ,
∀𝑡 ≥ 𝜏𝐻,
(57)
3 {3𝑅2 + (36/𝜀2 ) 𝑅4 + |𝐼| /2 − 1}
1
𝜏𝐻 =
.
log
2𝛽
𝑅22 − 3
(58)
𝜂 󵄩󵄩 󵄩󵄩2 𝜀2 󵄩󵄩
󵄩2
󵄩𝑢 󵄩 + 󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩 + 𝜂 ⟨𝑢𝑡𝑥 , 𝑢𝑥 ⟩
2 󵄩 𝑡𝑥 󵄩
2
1 󵄩 󵄩2
+ 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 .
2
(56)
where
1
1󵄩
󵄩 󵄩 󵄩2
− ⟨[𝑊󸀠 (𝑢𝑥 )]𝑥 , 𝑢𝑥𝑥 ⟩ ≤ 󵄩󵄩󵄩󵄩𝑊󸀠󸀠 (𝑢𝑥 )󵄩󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
2
2
󵄩 󵄩2
󵄩 󵄩2
≤ 2 (3󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩∞ + 1) 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
󵄩 󵄩2
󵄩 󵄩2
≤ 2 (3󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + 1) 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
Proposition 7. For all 𝑅2 > √3 one has that the ball
≤
󵄩
󵄩
𝐵H𝜀,𝜂 := {(𝑢, 𝑢𝑡 ) ∈ H𝜀,𝜂 : 󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩H𝜀,𝜂 ≤ 𝑅2 }
(59)
is an absorbing set for the semigroup 𝑆𝜀,𝜂 (𝑡) in H𝜀,𝜂 .
2
4 2 6𝑅2
𝑅 (
+ 1) := 𝐶1 .
𝜀2 2 𝜀2
Ψ1 =
1
2
𝑑𝑥;
∫ 𝑊󸀠󸀠 (𝑢𝑥 ) 𝑢𝑥𝑥
4 𝐼
(66)
then we can rewrite the the second term of r.h.s of (63)
(60)
equipped with the norm
𝜂 󵄩 󵄩2 𝜀 2 󵄩
󵄩󵄩
󵄩2
󵄩2
󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩U𝜀,𝜂 = 󵄩󵄩󵄩𝑢𝑡𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩 .
2
2
(65)
Let us define
4.2. Absorbing Set in U𝜀,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:
U𝜀,𝜂 = {𝐻3 (𝐼) ∩ 𝐻2 (𝐼) ∩ 𝐻01 (𝐼)} × 𝐻1 (𝐼) ,
(64)
We will estimate the right hand side of (63). The first term can
be estimated as follows:
Thus, we have proved the following.
Then we have the following.
(62)
is a bounded absorbing set for 𝑆𝜀,𝜂 (𝑡) in U𝜀,𝜂 .
𝑑 𝜂 󵄩󵄩 󵄩󵄩2 𝜀2 󵄩󵄩
󵄩2
{ 󵄩𝑢 󵄩 + 󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩 +
𝑑𝑡 2 󵄩 𝑡𝑥 󵄩
2
where
4
󵄩
󵄩
B3 = {(𝑢, 𝑢𝑡 ) ∈ U𝜀,𝜂 : 󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩U𝜀,𝜂 ≤ 𝑅3 } ,
(61)
1
𝑑
1
2
− ⟨[𝑊󸀠 (𝑢𝑥 )]𝑥 , 𝑢𝑥𝑥𝑡 ⟩ = − Ψ1 + ∫ 𝑊󸀠󸀠󸀠 (𝑢𝑥 ) 𝑢𝑥𝑡 𝑢𝑥𝑥
,
2
𝑑𝑡
4 𝐼
(67)
and estimate
6
Abstract and Applied Analysis
1
2
∫ 𝑊󸀠󸀠󸀠 (𝑢𝑥 ) 𝑢𝑥𝑡 𝑢𝑥𝑥
4 𝐼
Then from (72) we get
1
𝑑
̃
Ψ + Ψ ≤ 𝐶 + 𝑐2 := 𝐶,
𝑑𝑡
2
󵄩 󵄩
≤ 6󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩∞ ∫ 𝑢𝑥 𝑢𝑥𝑡 𝑢𝑥𝑥 𝑑𝑥
𝐼
󵄩 󵄩 󵄩 󵄩󵄩
󵄩
≤ 6󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑢𝑥𝑡 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥 𝑢𝑥𝑥 󵄩󵄩󵄩
and by Gronwall’s lemma we obtain
󵄩 󵄩 󵄩 󵄩 󵄩 󵄩2
≤ 6󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑢𝑥𝑡 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
󵄩1/2 󵄩 󵄩
󵄩 󵄩5/2 󵄩
≤ 6󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑡 󵄩󵄩󵄩
5/2
√2
≤ 6( 𝑅2 )
𝜀
(73)
̃ 𝑒−(1/2)𝑡 + 2𝐶,
̃
Ψ (𝑡) ≤ (Ψ (0) − 2𝐶)
(68)
󵄩󵄩
󵄩1/2 󵄩 󵄩
󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑡 󵄩󵄩󵄩
󵄩󵄩
󵄩2
󵄩
󵄩2
̃ 𝑒−(1/2)𝑡
󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩U𝜀,𝜂 ≤ 2 (󵄩󵄩󵄩(𝑢0 , 𝑢1 )󵄩󵄩󵄩U𝜀,𝜂 + 𝑐2 − 2𝐶)
(74)
̃ + 2𝑐1 .
+ 4𝐶
5
≤
𝜂 󵄩󵄩 󵄩󵄩2 18 √2
󵄩
󵄩
󵄩𝑢 󵄩 + ( 𝑅2 ) 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩
2 󵄩 𝑥𝑡 󵄩
𝜂
𝜀
Then if (𝑢0 , 𝑢1 ) ∈ U𝜀,𝜂 such that
5 2
󵄩󵄩
󵄩
󵄩󵄩(𝑢0 , 𝑢1 )󵄩󵄩󵄩U𝜀,𝜂 ≤ 𝑅,
𝜂 󵄩 󵄩2 1 18 √2
𝜀2 󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑢𝑥𝑡 󵄩󵄩󵄩 + [ ( 𝑅2 ) ] + 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩
2
2 𝜂𝜀 𝜀
2
:=
we have that
𝜂 󵄩󵄩 󵄩󵄩2 𝜀2 󵄩󵄩
󵄩2
󵄩󵄩𝑢𝑥𝑡 󵄩󵄩 + 󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩 + 𝐶2 .
2
2
Then if 𝜂 ∈ (0, 1/2) and setting Ψ := Ψ0 + Ψ1 , 𝐶 := 𝐶1 + 𝐶2
we have
𝑑
Ψ+
𝑑𝑡
𝜂 󵄩󵄩 󵄩󵄩2 𝜀2 󵄩󵄩
󵄩2
󵄩𝑢 󵄩 + 󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩 ≤ 𝐶.
2 󵄩 𝑥𝑡 󵄩
2
󵄩󵄩
󵄩
󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩U𝜀,𝜂 ≤ 𝑅3 ,
𝜏𝑈 := 2 log (
1 󵄩 󵄩2
󵄩
󵄩2
Ψ ≥ 󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩U𝜀,𝜂 + 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
2
󵄩 󵄩 󵄩 󵄩 󵄩 󵄩2
− 𝜂 󵄩󵄩󵄩𝑢𝑡𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
∀𝑡 ≥ 𝜏𝑈,
(76)
where
(69)
To conclude the proof we note that
𝜂 󵄩 󵄩2
󵄩
󵄩2
≥ 󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩U𝜀,𝜂 − 󵄩󵄩󵄩𝑢𝑡𝑥 󵄩󵄩󵄩 −
4
(75)
̃
2𝑅2 + 2𝑐2 − 4𝐶
),
2
̃
𝑅3 − 4𝐶 − 2𝑐1
(77)
and this concludes the proof.
3 󵄩󵄩 󵄩󵄩2
󵄩𝑢 󵄩
2 󵄩 𝑥𝑥 󵄩
(70)
1󵄩
3
󵄩2
≥ 󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩U𝜀,𝜂 − 2 𝑅22
2
𝜀
Proposition 9. There exists 𝑅4 > 0 such that the closed ball
1󵄩
󵄩2
:= 󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩U𝜀,𝜂 − 𝑐1 ,
2
󵄩
󵄩
B4 = {(𝑢, 𝑢𝑡 ) ∈ V𝜀,𝜂 : 󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩V𝜀,𝜂 ≤ 𝑅4 }
and that
𝜂 󵄩 󵄩2 󵄩 󵄩2
󵄩
󵄩2
Ψ ≤ 󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩U𝜀,𝜂 + 󵄩󵄩󵄩𝑢𝑡𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
2
󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2
+ 󵄩󵄩𝑢𝑥𝑥 󵄩󵄩 (3󵄩󵄩𝑢𝑥 󵄩󵄩∞ + 1)
2
6
󵄩
󵄩2
≤ 2󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩U𝜀,𝜂 + 2 𝑅22 ( 2 𝑅22 + 2)
𝜀
𝜀
󵄩2
󵄩
:= 2󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩U𝜀,𝜂 + 𝑐2 .
4.3. Absorbing Set in V𝜀,𝜂 . We consider the space V𝜀,𝜂 as
defined in (7), equipped with the norm defined in (10). By
considering more regular initial conditions we can prove the
following result.
is a bounded absorbing set for the semigroup 𝑆𝜀,𝜂 (𝑡) in V𝜀,𝜂 .
Proof. Multiply (1) by 𝑢𝑥𝑥𝑥𝑥𝑡 + 𝑢𝑥𝑥𝑥𝑥 ; then we obtain
(71)
2
𝑑 𝜂 󵄩󵄩
󵄩2 𝜀 󵄩
󵄩2
{ 󵄩󵄩𝑢𝑡𝑥𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑢𝑥𝑥𝑥𝑥 󵄩󵄩󵄩
𝑑𝑡 2
2
1󵄩
󵄩2
+𝜂 ⟨𝑢𝑡𝑥𝑥 , 𝑢𝑥𝑥 ⟩ + 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩 }
2
󵄩
󵄩
󵄩
󵄩2
+ (1 − 𝜂) 󵄩󵄩󵄩𝑢𝑥𝑥𝑡 󵄩󵄩󵄩 + 𝜀2 󵄩󵄩󵄩𝑢𝑥𝑥𝑥𝑥 󵄩󵄩󵄩
From the previous inequalities we get
1 󵄩󵄩
󵄩2
󵄩
󵄩2
󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩U𝜀,𝜂 − 𝑐1 ≤ Ψ ≤ 2󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩U𝜀,𝜂 + 𝑐2 .
2
(78)
(72)
=
1
⟨[𝑊󸀠 (𝑢𝑥 )]𝑥 , 𝑢𝑥𝑥𝑥𝑥𝑡 + 𝑢𝑥𝑥𝑥𝑥 ⟩ .
2
(79)
Abstract and Applied Analysis
7
We call 𝜃0 (𝑡) the differential term of the previous inequality
and we estimate the r.h.s.:
1
⟨[𝑊󸀠 (𝑢𝑥 )]𝑥 , 𝑢𝑥𝑥𝑥𝑥 ⟩
2
4
1
1
2
2
= − ⟨𝑊󸀠󸀠󸀠 (𝑢𝑥 ) 𝑢𝑥𝑥
, 𝑢𝑥𝑥𝑥 ⟩ − ⟨[𝑊󸀠󸀠 (𝑢𝑥 )]𝑥𝑥 , 𝑢𝑥𝑥𝑥
⟩
2
2
2
2
= −12 ∫ 𝑢𝑥 𝑢𝑥𝑥
𝑢𝑥𝑥𝑥 𝑑𝑥 − 2 ∫ (3𝑢𝑥2 − 1) 𝑢𝑥𝑥𝑥
𝑑𝑥
𝐼
12 2
𝑅 + 1) := 𝐴 1 .
𝜀2 2
:=
𝜀2 󵄩󵄩
󵄩2
󵄩󵄩𝑢𝑥𝑥𝑥𝑥 󵄩󵄩󵄩 + 𝐴 2 ,
2
1
2
]𝑥 , 𝑢𝑥𝑥𝑡 ⟩
⟨[𝑊󸀠󸀠󸀠 (𝑢𝑥 ) 𝑢𝑥𝑥
2
=
󵄩
󵄩2
󵄩
󵄩 󵄩2 󵄩 󵄩 󵄩
≤ 2󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩 + 12󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩
󵄩
󵄩2 󵄩 󵄩2
≤ 2󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩 (6󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + 1)
≤ 𝜀2 𝑅32 (
󵄩󵄩
󵄩 󵄩 󵄩2
󵄩󵄩𝑢𝑥𝑥𝑥𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑢𝑥𝑡 󵄩󵄩󵄩
𝑅2 𝑅3/2
1
2
𝜀2 󵄩
󵄩2
≤ 2 (12 ⋅ 23/4 2 7/23 ) + 𝑅32 + 󵄩󵄩󵄩𝑢𝑥𝑥𝑥𝑥 󵄩󵄩󵄩
2𝜀
𝜂
2
𝜀
1
= − ⟨[𝑊󸀠 (𝑢𝑥 )]𝑥𝑥 , 𝑢𝑥𝑥𝑥 ⟩
2
𝐼
≤ (12 ⋅
2 3/2 2
3/4 𝑅2 𝑅3
)
2
𝜀7/2
1
3
+ 𝑊󸀠󸀠󸀠 (𝑢𝑥 ) 2𝑢𝑥𝑥 𝑢𝑥𝑥𝑥 , 𝑢𝑥𝑥𝑡 ⟩
⟨𝑊𝑖V (𝑢𝑥 ) 𝑢𝑥𝑥
2
3
= 12 ∫ 𝑢𝑥𝑥
𝑢𝑥𝑥𝑡 𝑑𝑥 + 24 ∫ 𝑢𝑥 𝑢𝑥𝑥 𝑢𝑥𝑥𝑥 𝑢𝑥𝑥𝑡 𝑑𝑥
𝐼
𝐼
󵄩 󵄩2 󵄩 󵄩
≤ 12 (󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
󵄩 󵄩 󵄩 󵄩 󵄩
󵄩 󵄩
󵄩
+2󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩) 󵄩󵄩󵄩𝑢𝑥𝑥𝑡 󵄩󵄩󵄩
(80)
󵄩 󵄩2 󵄩
󵄩
≤ 12 (󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩
Let us define
1
2
𝜃1 (𝑡) := ∫ 𝑊󸀠󸀠 (𝑢𝑥 ) 𝑢𝑥𝑥𝑥
𝑑𝑥;
4 𝐼
󵄩 󵄩3/2 󵄩
󵄩3/2 󵄩
󵄩
+2󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩 ) 󵄩󵄩󵄩𝑢𝑥𝑥𝑡 󵄩󵄩󵄩
(81)
≤
then
󵄩 󵄩3/2 󵄩
󵄩3/2 2
+2󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩 )
1
⟨[𝑊󸀠 (𝑢𝑥 )]𝑥 , 𝑢𝑥𝑥𝑥𝑥𝑡 ⟩
2
2√2𝑅22 𝑅3
𝜂󵄩
2𝑅2 𝑅3 3/2
󵄩2
≤ 󵄩󵄩󵄩𝑢𝑥𝑥𝑡 󵄩󵄩󵄩 + 72[
+
2(
) ]
2
𝜀3
𝜀2
1
= − ⟨[𝑊󸀠 (𝑢𝑥 )]𝑥𝑥 , 𝑢𝑥𝑥𝑥𝑡 ⟩
2
1
2
= − ⟨𝑊󸀠󸀠󸀠 (𝑢𝑥 ) 𝑢𝑥𝑥
, 𝑢𝑥𝑥𝑥𝑡 ⟩
2
1
− ⟨𝑊󸀠󸀠 (𝑢𝑥 ) 𝑢𝑥𝑥𝑥 , 𝑢𝑥𝑥𝑥𝑡 ⟩
2
=−
+
1
𝑑
2
]𝑥 , 𝑢𝑥𝑥𝑡 ⟩
𝜃1 + ⟨[𝑊󸀠󸀠󸀠 (𝑢𝑥 ) 𝑢𝑥𝑥
𝑑𝑡
2
1
2
𝑑𝑥.
∫ 𝑊󸀠󸀠󸀠 (𝑢𝑥 ) 𝑢𝑥𝑡 𝑢𝑥𝑥𝑥
4 𝐼
We estimate the last two terms of the previous equality:
1
2
𝑑𝑥
∫ 𝑊󸀠󸀠󸀠 (𝑢𝑥 ) 𝑢𝑥𝑡 𝑢𝑥𝑥𝑥
4 𝐼
𝜂 󵄩󵄩
󵄩2
󵄩
󵄩 󵄩2 󵄩
󵄩𝑢 󵄩󵄩 + 72 (󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩
2 󵄩 𝑥𝑥𝑡 󵄩
:=
𝜂 󵄩󵄩
󵄩2
󵄩𝑢 󵄩󵄩 + 𝐴 3 .
2 󵄩 𝑥𝑥𝑡 󵄩
(83)
(82)
Then if we set 𝜃(𝑡) = 𝜃0 (𝑡) + 𝜃1 (𝑡) we get
𝑑
󵄩2
󵄩
𝜃 (𝑡) + 󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩V𝜀,𝜂 ≤ 𝐴 1 + 𝐴 2 + 𝐴 3 := 𝐴.
𝑑𝑡
Moreover we have
2
= 6 ∫ 𝑢𝑥 𝑢𝑥𝑡 𝑢𝑥𝑥𝑥
𝑑𝑥
𝐼
󵄩
󵄩 󵄩 󵄩 󵄩 󵄩󵄩
󵄩
≤ 6󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑢𝑥𝑡 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩
󵄩
󵄩3/2 󵄩
󵄩1/2 󵄩 󵄩 󵄩 󵄩2
≤ 6󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑥𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑡 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
≤ (12 ⋅ 23/4
𝑅22 𝑅33/2 󵄩󵄩
󵄩1/2 󵄩 󵄩
) 󵄩󵄩𝑢𝑥𝑥𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑡 󵄩󵄩󵄩
7/2
𝜀
2
𝜂󵄩
󵄩
󵄩2
󵄩2
𝜃 (𝑡) ≥ 󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩V𝜀,𝜂 − 󵄩󵄩󵄩𝑢𝑥𝑥𝑡 󵄩󵄩󵄩 −
4
1󵄩
1
󵄩2
≥ 󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩V𝜀,𝜂 − 𝑅22
2
2
1󵄩
󵄩2
:= 󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩V𝜀,𝜂 − 𝑎1 ,
2
1 󵄩󵄩 󵄩󵄩2
󵄩𝑢 󵄩
2 󵄩 𝑥𝑥 󵄩
(84)
8
Abstract and Applied Analysis
󵄩󵄩 󵄩
󵄩
󵄩2
󵄩
𝜃 (𝑡) ≤ 󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩V𝜀,𝜂 + 𝜂 󵄩󵄩󵄩𝑢𝑥𝑥𝑡 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
such that
(𝐶1)
1 󵄩 󵄩2 1 󵄩
󵄩 󵄩
󵄩3
+ 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑊󸀠󸀠 (𝑢𝑥 )󵄩󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩
2
4
󵄩󵄩
󵄩󵄩2
󵄩 󵄩2
≤ 2󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩V𝜀,𝜂 + 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
(94)
󵄩
󵄩
lim { sup 󵄩󵄩󵄩󵄩𝑁𝜀,𝜂 (𝑡) 𝑧󵄩󵄩󵄩󵄩H } = 0.
𝑡 → ∞ 𝑧∈H
𝜀,𝜂
(95)
𝜀,𝜂
(𝐶2)
𝜀,𝜂
󵄩2
󵄩 󵄩2
󵄩
+ (3󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩∞ + 1) 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩
󵄩
󵄩2
≤ 2󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩V𝜀,𝜂
󵄩
󵄩
sup sup 󵄩󵄩󵄩󵄩𝐿 𝜀,𝜂 (𝑡)𝑧󵄩󵄩󵄩󵄩V < ∞,
𝜀,𝜂
𝑡≥0 𝑧∈H
Let B ⊂ H𝜀,𝜂 be a fixed bounded set and let (𝑢0 , 𝑢1 ) ∈
B ⊂ H𝜀,𝜂 . We will define the decomposition of 𝑆𝜀,𝜂 as follows:
2
6
2
+ 2 𝑅22 + ( 2 𝑅22 + 1) 2 𝑅32
𝜀
𝜀
𝜀
𝑆𝜀,𝜂 (𝑡) (𝑢0 , 𝑢1 ) = (𝑢, 𝑢𝑡 ) ,
󵄩
󵄩2
:= 2󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩V𝜀,𝜂 + 𝑎2 .
𝐿 𝜀,𝜂 (𝑡) (𝑢0 , 𝑢1 ) = (ℎ, ℎ𝑡 ) ,
(85)
𝑁𝜀,𝜂 (𝑡) (𝑢0 , 𝑢1 ) = (V, V𝑡 ) ,
Then putting all together we get
1 󵄩󵄩
󵄩2
󵄩
󵄩2
󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩V𝜀,𝜂 − 𝑎1 ≤ 𝜃 (𝑡) ≤ 2󵄩󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩V𝜀,𝜂 + 𝑎2 .
2󵄩
where ℎ and V are solutions of the following problems:
(86)
𝜂ℎ𝑡𝑡 + ℎ𝑡 + 𝜀2 ℎ𝑥𝑥𝑥𝑥 − 𝛼ℎ𝑥𝑥
From the previous inequality we get
𝑎
𝑑
1
̃
𝜃 (𝑡) + 𝜃 (𝑡) ≤ 𝐴 + 2 := 𝐴
𝑑𝑡
2
2
1
= 𝑊󸀠󸀠 (ℎ𝑥 ) ℎ𝑥𝑥 − 𝛼𝑢𝑥𝑥 ,
2
(87)
̃
̃ 𝑒−(1/2)𝑡 + 2𝐴.
𝜃 (𝑡) ≤ [𝜃 (0) − 2𝐴]
ℎ𝑡 (𝑥, 0) = 0,
(88)
ℎ = ℎ𝑥𝑥 = 0,
̃ + 2𝑎1 .
+ 4𝐴
in 𝜕𝐼,
𝜂V𝑡𝑡 + V𝑡 + 𝜀2 V𝑥𝑥𝑥𝑥 − 𝛼V𝑥𝑥
Using again (86) we get
󵄩2
󵄩
󵄩2
󵄩󵄩
̃ 𝑒−(1/2)𝑡
󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩V𝜀,𝜂 ≤ 2 [2󵄩󵄩󵄩(𝑢0 , 𝑢1 )󵄩󵄩󵄩V𝜀,𝜂 + 𝑎2 − 2𝐴]
(97)
ℎ (𝑥, 0) = 0,
and by Gronwall’s lemma
1
1
= 𝑊󸀠󸀠 (𝑢𝑥 ) 𝑢𝑥𝑥 − 𝑊󸀠󸀠 (ℎ𝑥 ) ℎ𝑥𝑥 ,
2
2
(89)
(98)
V (0) = 𝑢0 ,
Then if (𝑢0 , 𝑢1 ) ∈ V𝜀,𝜂 such that
󵄩
󵄩󵄩
󵄩󵄩(𝑢0 , 𝑢1 )󵄩󵄩󵄩V𝜀,𝜂 ≤ 𝑅,
then there exists 𝑅4 > 0:
󵄩󵄩
󵄩
󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩V𝜀,𝜂 ≤ 𝑅4 ,
(96)
V𝑡 (0) = 𝑢1 ,
V = V𝑥𝑥 = 0,
(90)
in 𝜕𝐼,
where
∀𝑡 ≥ 𝜏𝑉,
(91)
where
̃
2 (2𝑅2 + 𝑎2 − 2𝐴)
],
𝜏𝑉 := 2 log [
̃ − 2𝑎1
𝑅2 − 4𝐴
(92)
4
6√2
𝑅 > 6√6.
𝜀 2
(99)
Before showing that the semigroups 𝐿 𝜀,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 10. Let 𝜓 : R+ → R be an absolutely continuous
function which fulfills, for some ] > 0 and almost every 𝑡 ≥ 0,
the differential inequality
and this concludes the proof.
5. Global Attractor
In this section we will show the existence of a global attractor
for the semigroup 𝑆𝜀,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:
𝑆𝜀,𝜂 (𝑡) = 𝐿 𝜀,𝜂 (𝑡) + 𝑁𝜀,𝜂 (𝑡) ,
𝛼≥
(93)
𝑑
𝜓 (𝑡) + 2]𝜓 (𝑡) ≤ 𝑓 (𝑡) 𝜓 (𝑡) ,
𝑑𝑡
(100)
where 𝑓 is a positive function satisfying
𝑡
∫ 𝑓 (𝑦) 𝑑𝑦 ≤ ]𝑡 + 𝑐,
0
∀𝑡 ≥ 0.
(101)
𝑡 ≥ 0.
(102)
Then
𝜓 (𝑡) ≤ 𝑐𝜓 (0) 𝑒−]𝑡 ,
Abstract and Applied Analysis
9
5.1. The Semigroup 𝐿 𝜀,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
where 𝑘𝑥 (𝑥) is between ℎ𝑥 (𝑥) and 𝑢𝑥 (𝑥) and satisfies ‖𝑘𝑥 ‖∞ ≤
‖V𝑥 ‖∞ . Moreover we have
⟨𝑊 (𝑢𝑥 ) − 𝑊 (ℎ𝑥 ) − 𝑊󸀠 (ℎ𝑥 ) V𝑥 , 1⟩
= ⟨𝑊󸀠 (ℎ𝑥 + 𝑘̃𝑥 ) − 𝑊󸀠 (ℎ𝑥 ) , V𝑥 ⟩
= ⟨𝑊󸀠󸀠 (𝑠𝑥 ) 𝑘̃𝑥 , V𝑥 ⟩
𝑑 𝜂 󵄩󵄩 󵄩󵄩2 𝜀2 󵄩󵄩 󵄩󵄩2
{ 󵄩ℎ 󵄩 + 󵄩󵄩ℎ𝑥𝑥 󵄩󵄩
𝑑𝑡 2 󵄩 𝑡 󵄩
2
1
𝛼 󵄩 󵄩2
+ ∫ 𝑊 (ℎ𝑥 ) 𝑑𝑥 + 󵄩󵄩󵄩ℎ𝑥 󵄩󵄩󵄩 + 𝛼⟨𝑢𝑥𝑥 , ℎ⟩}
2 𝐼
2
󵄩 󵄩2
󵄩 󵄩
+ 󵄩󵄩󵄩ℎ𝑡 󵄩󵄩󵄩 = 𝛼⟨𝑢𝑡 , ℎ𝑥𝑥 ⟩ ≤ 𝑐 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩 .
= 12 ∫ 𝑠𝑥2 𝑘̃𝑥 V𝑥 𝑑𝑥 − 4 ∫ 𝑘𝑥 V𝑥 𝑑𝑥
𝑡
𝑡
1/2
≤ 𝐴 + 𝐵𝑡
(104)
.
Consequently, for any 𝛾 > 0 there exists a constant 𝐶𝛾 ≥ 0
such that
𝑡
󵄩 󵄩2
∫ 󵄩󵄩󵄩ℎ𝑡 󵄩󵄩󵄩 𝑑𝑠 ≤ 𝛾 𝑡 + 𝐶𝛾 .
0
𝐼
(103)
Then integrating the above equation on (0, 𝑡) and using (30)
we get that there exist constants 𝐴 > 0 and 𝐵 > 0 such that
󵄩 󵄩2
󵄩 󵄩
∫ 󵄩󵄩󵄩ℎ𝑡 󵄩󵄩󵄩 𝑑𝑠 ≤ 𝐴 + 𝑐 ∫ 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩 𝑑𝑠
0
0
(105)
5.2. The Semigroup 𝑁𝜀,𝜂 (𝑡). Let 𝛽 > 0 be a parameter to be
determined later and let us multiply, in 𝐿2 (𝐼), (98) by 2V𝑡 +𝛽V.
Then we get
where 𝑘̃𝑥 (𝑥) is between 0 and V𝑥 (𝑥), 𝑠𝑥 (𝑥) is between 0 and
𝑘̃𝑥 (𝑥) and ‖𝑠𝑥 ‖∞ ≤ ‖𝑘̃𝑥 ‖∞ ≤ ‖V𝑥 ‖∞ . Moreover we have used
that V𝑥2 ≥ 𝑘̃𝑥 V𝑥 ≥ 0.
We denote as F 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
󵄩 󵄩2
F (𝑡) ≥ 𝜂󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩 −
󵄩 󵄩2
󵄩 󵄩2
+ (2 − 𝜂𝛽) 󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩 + 𝜀2 𝛽󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩
−
3 󵄩 󵄩2
󵄩 󵄩2
󵄩 󵄩2
≤ 𝜂󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩 + 𝛽󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩 + 𝜀2 󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩
2
󵄩 󵄩2
󵄩 󵄩2
󵄩 󵄩2
+ 𝛼󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩 + 4 (3󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩∞ + 1) 󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩
(106)
12√2𝑅2
3 󵄩 󵄩2
󵄩 󵄩2
≤ 𝜂󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩 + (𝛽 + 𝜀2 + 𝛼 +
+ 4) 󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩
2
𝜀
= ⟨ℎ𝑡𝑥 , 𝑊󸀠 (𝑢𝑥 ) − 𝑊󸀠 (ℎ𝑥 ) − 𝑊󸀠󸀠 (ℎ𝑥 ) V𝑥 ⟩ .
3 󵄩 󵄩2
𝜀 2 󵄩 󵄩2
:= 𝜂󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩 + 𝐴 󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩
2
2
2
󵄩
󵄩
≤ 𝑐󵄩󵄩󵄩(V, V𝑡 )󵄩󵄩󵄩H𝜀,𝜂 ,
We estimate the term involving 𝑊 and its derivatives:
𝛽
⟨𝑊󸀠 (𝑢𝑥 ) − 𝑊󸀠 (ℎ𝑥 ) , V𝑥 ⟩
2
(110)
𝛽
= − ⟨𝑊󸀠󸀠 (𝑘𝑥 ) V𝑥 , V𝑥 ⟩
2
= 2𝛽 ∫ (1 − 3𝑘𝑥2 ) V𝑥2 𝑑𝑥
𝐼
󵄩 󵄩 󵄩 󵄩2
≥ −6𝛽󵄩󵄩󵄩V𝑥 󵄩󵄩󵄩∞ 󵄩󵄩󵄩V𝑥 󵄩󵄩󵄩
6𝛽√2 󵄩󵄩 󵄩󵄩2
≥−
𝑅2 󵄩󵄩V𝑥 󵄩󵄩 ,
𝜀
(109)
󵄩 󵄩2 𝛽
+ 𝜀2 󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩 + ‖V‖2
2
󵄩 󵄩 󵄩2
󵄩󵄩 󵄩󵄩2 󵄩󵄩󵄩 󸀠󸀠
+ 𝛼󵄩󵄩V𝑥 󵄩󵄩 + 󵄩󵄩𝑊 (𝑘𝑥 )󵄩󵄩󵄩󵄩∞ 󵄩󵄩󵄩V𝑥 󵄩󵄩󵄩
𝛽
⟨𝑊󸀠 (𝑢𝑥 ) − 𝑊󸀠 (ℎ𝑥 ) , V𝑥 ⟩
2
−
𝜂 󵄩󵄩 󵄩󵄩2 𝛽 2
󵄩V 󵄩 − ‖V‖
2 󵄩 𝑡󵄩
2
󵄩 󵄩2
󵄩 󵄩2 6𝛽√2 󵄩󵄩 󵄩󵄩2
+ 𝜀2 󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩 + 𝛼󵄩󵄩󵄩V𝑥 󵄩󵄩󵄩 −
𝑅2 󵄩󵄩V𝑥 󵄩󵄩
𝜀
𝜂 󵄩 󵄩2
󵄩 󵄩2 󵄩
󵄩2
≥ 󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩 + 𝜀2 󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩 ≥ 󵄩󵄩󵄩(V, V𝑡 )󵄩󵄩󵄩H𝜀,𝜂 .
2
Moreover we have
󵄩 󵄩2 𝜂 󵄩 󵄩2 𝛽
F (𝑡) ≤ 𝜂󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩 + 󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩 + ‖V‖2
2
2
𝛽 2
󵄩 󵄩2
‖V‖ + 𝛼󵄩󵄩󵄩V𝑥 󵄩󵄩󵄩
2
+⟨𝑊 (𝑢𝑥 ) − 𝑊 (ℎ𝑥 ) − 𝑊󸀠 (ℎ𝑥 ) V𝑥 , 1⟩}
𝐼
󵄩 󵄩2
≥ −4󵄩󵄩󵄩V𝑥 󵄩󵄩󵄩 ,
𝑑 󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
{𝜂󵄩V 󵄩 + 𝜂𝛽⟨V𝑡 , V⟩ + 𝜀2 󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩
𝑑𝑡 󵄩 𝑡 󵄩
+
(108)
where we have used the same argument as (107) and (108) and
where
(107)
𝐴=
12√2𝑅2
2
(𝛽 + 𝜀2 + 𝛼 +
+ 4) ,
𝜀2
𝜀
𝑐 = max {3, 𝐴} .
(111)
Then we have
󵄩󵄩
󵄩2
󵄩
󵄩2
󵄩󵄩(V, V𝑡 )󵄩󵄩󵄩H𝜀,𝜂 ≤ F (𝑡) ≤ 𝑐󵄩󵄩󵄩(V, V𝑡 )󵄩󵄩󵄩H𝜀,𝜂 .
(112)
10
Abstract and Applied Analysis
We estimate the right hand side of (106):
Then by Lemma 10 and from (105) we conclude that there
exists a positive constant 𝐾 such that
⟨ℎ𝑡𝑥 , 𝑊󸀠 (𝑢𝑥 ) − 𝑊󸀠 (ℎ𝑥 ) − 𝑊󸀠󸀠 (ℎ𝑥 ) V𝑥 ⟩
󸀠󸀠󸀠
󸀠󸀠
= ⟨ℎ𝑡 , 𝑊 (ℎ𝑥 ) ℎ𝑥𝑥 V𝑥 + (𝑊 (ℎ𝑥 ) − 𝑊 (𝑢𝑥 )) 𝑢𝑥𝑥 ⟩
= ⟨ℎ𝑡 , 24ℎ𝑥 ℎ𝑥𝑥 V𝑥 +
12 (ℎ𝑥2
−
𝑢𝑥2 ) 𝑢𝑥𝑥 ⟩
= ⟨ℎ𝑡 , 12ℎ𝑥 V𝑥 (2ℎ𝑥𝑥 − 𝑢𝑥𝑥 ) − 12V𝑥 𝑢𝑥 𝑢𝑥𝑥 ⟩
1/2
Acknowledgment
𝐼
󵄩 󵄩
2 4
+ 12 󵄩󵄩󵄩ℎ𝑡 󵄩󵄩󵄩 (∫ 𝑢𝑥𝑥
V𝑥 )
1/2
The authors would like to thank the editor and the referees for
their helpful suggestions and remarks. This work was partially
supported by projects PPTA 4476 and PPTA 5571 of Pontificia
Universidad Javeriana, Bogotá.
𝐼
󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩
≤ 24 󵄩󵄩󵄩ℎ𝑡 󵄩󵄩󵄩 󵄩󵄩󵄩V𝑥 󵄩󵄩󵄩∞ 󵄩󵄩󵄩ℎ𝑥 󵄩󵄩󵄩∞ 󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩
󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩
+ 12 󵄩󵄩󵄩ℎ𝑡 󵄩󵄩󵄩 󵄩󵄩󵄩V𝑥 󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩∞ 󵄩󵄩󵄩V𝑥 󵄩󵄩󵄩
󵄩 󵄩 󵄩 󵄩2 󵄩 󵄩 󵄩
󵄩
≤ 12 󵄩󵄩󵄩ℎ𝑡 󵄩󵄩󵄩 󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩 (2 󵄩󵄩󵄩ℎ𝑥𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑢𝑥𝑥𝑥 󵄩󵄩󵄩)
󵄩 󵄩 󵄩 󵄩2
≤ 𝜎 󵄩󵄩󵄩ℎ𝑡 󵄩󵄩󵄩 󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩
References
2𝜎2 󵄩󵄩 󵄩󵄩2 𝛽𝜀2 󵄩󵄩 󵄩󵄩2
} 󵄩V 󵄩 ,
󵄩ℎ 󵄩 +
𝛽𝜀2 󵄩 𝑡 󵄩
2 󵄩 𝑥𝑥 󵄩
(113)
where
𝜎=
(118)
Theorem 11. The semigroup 𝑆𝜀,𝜂 (𝑡) possesses a connected
global attractor A𝜀,𝜂 ⊂ H𝜀,𝜂 that is bounded in V𝜀,𝜂 .
= ⟨ℎ𝑡 , 12V𝑥 [−2V𝑥𝑥 ℎ𝑥 − V𝑥 𝑢𝑥𝑥 ]⟩
≤{
and consequently from (112) we conclude
󵄩2
󵄩
󵄩2 −(𝛽/2𝑐)𝑡
󵄩󵄩
.
󵄩󵄩(𝑢, 𝑢𝑡 )󵄩󵄩󵄩H𝜂,𝜀 ≤ 𝑐𝐾󵄩󵄩󵄩(𝑢0 , 𝑢1 )󵄩󵄩󵄩H𝜂,𝜀 𝑒
(117)
This shows that the semigroup 𝑁𝜀,𝜂 (𝑡) satisfies condition (95)
and hence we have proven the following.
= ⟨ℎ𝑡 , 12V𝑥 [ℎ𝑥 (ℎ𝑥𝑥 − V𝑥𝑥 ) − 𝑢𝑥 𝑢𝑥𝑥 ]⟩
󵄩 󵄩
2 2
ℎ𝑥 )
≤ 24 󵄩󵄩󵄩ℎ𝑡 󵄩󵄩󵄩 (∫ V𝑥2 V𝑥𝑥
F (𝑡) ≤ 𝐾F (0) 𝑒−(𝛽/2𝑐)𝑡 ,
󸀠󸀠
12√2
(2𝑅2 + 𝑅3 ) .
𝜀
(114)
Then it remains to estimate the following term:
󵄩 󵄩2
󵄩 󵄩2
(2 − 𝜂𝛽) 󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩 + 𝛽𝜀2 󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩
󵄩 󵄩2 𝛽
+ 𝛼𝛽󵄩󵄩󵄩V𝑥 󵄩󵄩󵄩 − ⟨𝑊󸀠 (𝑢𝑥 ) − 𝑊󸀠 (ℎ𝑥 ) , V𝑥 ⟩
2
𝜂 󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
≥ 𝛽 󵄩󵄩V𝑡 󵄩󵄩 + 𝛽𝜀2 󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩
2
(115)
√ 2 󵄩 󵄩2
󵄩 󵄩2
+ 𝛼𝛽󵄩󵄩󵄩V𝑥 󵄩󵄩󵄩 − 6𝛽 𝑅2 󵄩󵄩󵄩V𝑥 󵄩󵄩󵄩
𝜀
𝜂 󵄩 󵄩2
󵄩 󵄩2
≥ 𝛽 󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩 + 𝛽𝜀2 󵄩󵄩󵄩V𝑥𝑥 󵄩󵄩󵄩 .
2
Then, using the above estimates, we can rewrite (106) in the
following way:
𝛽
𝜎2 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2
𝑑
F (𝑡) + F (𝑡) ≤
󵄩ℎ 󵄩 󵄩V 󵄩
𝑑𝑡
𝑐
2𝛽𝜀2 󵄩 𝑡 󵄩 󵄩 𝑥𝑥 󵄩
≤
𝜎2 󵄩󵄩 󵄩󵄩2 󵄩󵄩
󵄩2
󵄩󵄩ℎ𝑡 󵄩󵄩 󵄩󵄩(V, V𝑡 )󵄩󵄩󵄩H𝜀,𝜂
4
𝛽𝜀
≤
𝜎2 󵄩󵄩 󵄩󵄩2
󵄩ℎ 󵄩 F (𝑡) .
𝛽𝜀4 󵄩 𝑡 󵄩
(116)
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