Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 353757, 8 pages
http://dx.doi.org/10.1155/2013/353757
Research Article
Asymptotic Behavior of Solutions to the Damped Nonlinear
Hyperbolic Equation
Yu-Zhu Wang
School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power,
Zhengzhou 450011, China
Correspondence should be addressed to Yu-Zhu Wang; yuzhu108@163.com
Received 17 December 2012; Revised 15 May 2013; Accepted 30 May 2013
Academic Editor: Roberto Natalini
Copyright © 2013 Yu-Zhu Wang. 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 consider the Cauchy problem for the damped nonlinear hyperbolic equation in n-dimensional space. Under small condition
on the initial value, the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces are obtained
by the contraction mapping principle.
1. Introduction
We investigate the Cauchy problem for the following damped
nonlinear hyperbolic equation:
𝑢𝑡𝑡 + 𝑘1 Δ2 𝑢 + 𝑘2 Δ2 𝑢𝑡 = Δ𝑓 (Δ𝑢)
(1)
with the initial value
𝑡 = 0 : 𝑢 = 𝑢0 (𝑥) ,
𝑢𝑡 = 𝑢1 (𝑥) .
(2)
Here 𝑢 = 𝑢(𝑥, 𝑡) is the unknown function of 𝑥 =
(𝑥1 , . . . , 𝑥𝑛 ) ∈ R𝑛 and 𝑡 > 0, 𝑘1 > 0 and 𝑘2 > 0 are constants.
The nonlinear term 𝑓(𝑢) = 𝑂(𝑢1+𝜃 ) and 𝜃 is a positive integer.
Equation (1) is a model in variational form for the neoHookean elastomer rod and describes the motion of a neoHookean elastomer rod with internal damping; for more
detailed physical background, we refer to [1]. In [1], the
authors have studied a general class of abstract evolution
equations
𝑤𝑡𝑡 + 𝐴 1 𝑤 + 𝐴 2 𝑤𝑡 + 𝑁∗ 𝑔 (𝑁𝑤) = 𝑓,
(3)
where 𝐴 1 , 𝐴 2 , 𝑁, and 𝑓 satisfy certain assumptions. For
quite general conditions on the nonlinear term, global existence, uniqueness, regularity, and continuous dependence
on the initial value of a generalized solution to (3) in a
bounded domain of R𝑛 were obtained. Equation (1) fits the
abstract framework of [1]. The local well-posedness for the
Cauchy problem for (1), (2) in three-dimensional space was
obtained by Chen and Da [2]. More precisely, they proved
local existence and uniqueness of weak solutions to (1), (2)
under the assumption that 𝑢0 ∈ 𝐻6 (R3 ), 𝑢1 ∈ 𝐻4 (R3 ).
Local existence and uniqueness of classical solutions to (1),
(2) were also established, provided that 𝑢0 ∈ 𝐻12 (R3 ),
𝑢1 ∈ 𝐻10 (R3 ). Their method is to first establish local-intime well-posedness of a periodic version of (1), (2) and
then construc a solution to (1), (2) as a limit of periodic
solutions with divergent periods. This paper also arrived at
some sufficient conditions for blow-up of the solution in
finite time, and an example was given. Song and Yang [3]
studied the existence and nonexistence of global solutions
to the Cauchy problem for (1) in one-dimensional space.
The boundary value problems for (1) are investigated (see
[4, 5]). Equation (1) is a fifth-order wave equation. For more
higher order wave equations, we refer to [6–8] and references
therein.
The main purpose of this paper is to establish global
existence and asymptotic behavior of solutions to (1), (2) by
using the contraction mapping principle. Firstly, we consider
the decay property of the following linear equation:
𝑢𝑡𝑡 + 𝑘1 Δ2 𝑢 + 𝑘2 Δ2 𝑢𝑡 = 0.
(4)
2
Journal of Applied Mathematics
We obtain the following decay estimate of solutions to (4), (2)
󵄩
󵄩󵄩 𝑘
󵄩󵄩𝜕𝑥 𝑢 (𝑡)󵄩󵄩󵄩 2 ≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑘/4)
󵄩𝐿
󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
× (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 )
1
for (𝑘 ≤ 𝑠 + 4) ,
󵄩
󵄩󵄩 𝑙
󵄩󵄩𝜕𝑥 𝑢𝑡 (𝑡)󵄩󵄩󵄩 2 ≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑙/4+1/2)
󵄩𝐿
󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
× (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 )
1
2. Solution Formula
for (𝑙 ≤ 𝑠) .
(5)
Based on the above estimates, we define a solution space
with time weighted norms, and then global existence and
asymptotic behavior of solutions to (1), (2) are obtained by
using the contraction mapping principle. More precisely, we
prove global existence and the following decay estimate of
solution to (1), (2):
󵄩
󵄩󵄩 𝑘
󵄩󵄩𝜕𝑥 𝑢 (𝑡)󵄩󵄩󵄩 2 ≤ 𝐶𝐸0 (1 + 𝑡)−(𝑛/8+𝑘/4) ,
󵄩𝐿
󵄩
(6)
󵄩󵄩󵄩𝜕𝑙 𝑢 (𝑡)󵄩󵄩󵄩 ≤ 𝐶𝐸 (1 + 𝑡)−(𝑛/8+𝑙/4+1/2)
󵄩󵄩 𝑥 𝑡 󵄩󵄩𝐿2
0
for 𝑘 ≤ 𝑠 + 4, 𝑙 ≤ 𝑠, and 𝑠 ≥ [𝑛/2] + 1. Here 𝑢0 ∈
𝐻𝑠+4 (R𝑛 ) ⋂ 𝐿1 (R𝑛 ), 𝑢1 ∈ 𝐻𝑠 (R𝑛 ) ⋂ 𝐻̇ 1−2 (R𝑛 ), and 𝐸0 =
‖𝑢0 ‖𝐿1 + ‖𝑢1 ‖𝐻̇ 1−2 + ‖𝑢0 ‖𝐻𝑠+4 + ‖𝑢1 ‖𝐻𝑠 is assumed to be suitably
small. When 𝑛 = 3, our result allows for the initial data
𝑢0 ∈ 𝐻6 (R3 ), 𝑢1 ∈ 𝐻2 (R3 ). But in [2], the authors proved
local existence and uniqueness of weak solutions to (1), (2)
under the assumption that 𝑢0 ∈ 𝐻6 (R3 ), 𝑢1 ∈ 𝐻4 (R3 ),
so our result improves the regularity of the initial condition
for the time derivative. This improvement is due to the
strong damping term Δ2 𝑢𝑡 since the strong damping term
Δ2 𝑢𝑡 has stronger dissipative effect than the damping 𝑢𝑡 . The
stronger dissipative effect has been exhibited in the study of
the strongly damped wave equation and related problems; see,
for instance, [9].
The global existence and asymptotic behavior of solutions
to hyperbolic-type equations have been investigated by many
authors. We refer to [10–15] for hyperbolic equations, [16–21]
for damped wave equation, and [22, 23] for various aspects of
dissipation of the plate equation.
We give some notations which are used in this paper. Let
F[𝑢] denote the Fourier transform of 𝑢 defined by
𝑢̂ (𝜉) = F [𝑢] = ∫ 𝑒−𝑖𝜉⋅𝑥 𝑢 (𝑥) 𝑑𝑥,
R𝑛
Finally, in this paper, we denote every positive constant
by the same symbol 𝐶 or 𝑐 without confusion. [⋅] is the Gauss
symbol.
The paper is organized as follows. In Section 2 we derive
the solution formula of our semilinear problem. We study
the decay property of the solution operators appearing in the
solution formula in Section 3. Then, in Section 4, we discuss
the linear problem and show the decay estimates. Finally, we
prove global existence and asymptotic behavior of solutions
for the Cauchy problem (1), (2) in Section 5.
(7)
and we denote its inverse transform by F−1 .
For 1 ≤ 𝑝 ≤ ∞, 𝐿𝑝 = 𝐿𝑝 (R𝑛 ) denotes the usual Lebesgue space with the norm ‖ ⋅ ‖𝐿𝑝 . The usual Sobolev space of
𝑠 is defined by 𝐻𝑝𝑠 = (𝐼 − Δ)−𝑠/2 𝐿𝑝 with the norm ‖𝑓‖𝐻𝑝𝑠 =
‖(𝐼 − Δ)𝑠/2 𝑓‖𝐿𝑝 ; the homogeneous Sobolev space of 𝑠 is
defined by 𝐻̇ 𝑝𝑠 = (−Δ)−𝑠/2 𝐿𝑝 with the norm ‖𝑓‖𝐻𝑝𝑠 =
‖(−Δ)𝑠/2 𝑓‖𝐿𝑝 ; especially 𝐻𝑠 = 𝐻2𝑠 , 𝐻̇ 𝑠 = 𝐻̇ 2𝑠 . Moreover, we
know that 𝐻𝑝𝑠 = 𝐿𝑝 ⋂ 𝐻̇ 𝑝𝑠 for 𝑠 ≥ 0.
The aim of this section is to derive the solution formula for the
problem (1), (2). We first investigate (4). Taking the Fourier
transform, we have
󵄨 󵄨4
󵄨 󵄨4
𝑢̂𝑡𝑡 + 𝑘2 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 𝑢̂𝑡 + 𝑘1 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 𝑢̂ = 0.
(8)
The corresponding initial value is given as
𝑡 = 0 : 𝑢̂ = 𝑢̂0 (𝜉) ,
𝑢̂𝑡 = 𝑢̂1 (𝜉) .
(9)
The characteristic equation of (8) is
󵄨 󵄨4
󵄨 󵄨4
𝜆2 + 𝑘2 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 𝜆 + 𝑘1 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 = 0.
(10)
Let 𝜆 = 𝜆 ± (𝜉) be the corresponding eigenvalues of (10), and
we obtain
𝜆 ± (𝜉) =
󵄨 󵄨4 󵄨 󵄨2
󵄨 󵄨4
−𝑘2 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 ± 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 √𝑘22 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 − 4𝑘1
2
.
(11)
The solution to the problem (8)-(9) is given in the form
̂ (𝜉, 𝑡) 𝑢̂1 (𝜉) + 𝐻
̂ (𝜉, 𝑡) 𝑢̂0 (𝜉) ,
𝑢̂ (𝜉, 𝑡) = 𝐺
(12)
where
̂ (𝜉, 𝑡) =
𝐺
̂ (𝜉, 𝑡) =
𝐻
1
(𝑒𝜆 + (𝜉)𝑡 − 𝑒𝜆 − (𝜉)𝑡 ) ,
𝜆 + (𝜉) − 𝜆 − (𝜉)
(13)
1
(𝜆 (𝜉) 𝑒𝜆 − (𝜉)𝑡 − 𝜆 − (𝜉) 𝑒𝜆 + (𝜉)𝑡 ) .
𝜆 + (𝜉) − 𝜆 − (𝜉) +
(14)
̂ 𝑡)](𝑥)
We define 𝐺(𝑥, 𝑡) and 𝐻(𝑥, 𝑡) by 𝐺(𝑥, 𝑡) = F−1 [𝐺(𝜉,
−1 ̂
and 𝐻(𝑥, 𝑡) = F [𝐻(𝜉, 𝑡)](𝑥), respectively, where F−1
denotes the inverse Fourier transform. Then, applying F−1
to (12), we obtain
𝑢 (𝑡) = 𝐺 (𝑡) ∗ 𝑢1 + 𝐻 (𝑡) ∗ 𝑢0 .
(15)
By the Duhamel principle, we obtain the solution formula to
(1), (2)
𝑢 (𝑡) = 𝐺 (𝑡) ∗ 𝑢1 + 𝐻 (𝑡) ∗ 𝑢0
𝑡
+ ∫ 𝐺 (𝑡 − 𝜏) ∗ Δ𝑓 (Δ𝑢) (𝜏) 𝑑𝜏.
0
(16)
Journal of Applied Mathematics
3
3. Decay Property
The aim of this section is to establish decay estimates of the
solution operators 𝐺(𝑡) and 𝐻(𝑡) appearing in the solution
formula (15).
󵄨2 󵄨 ̂
󵄨󵄨2
󵄨 󵄨4 󵄨 󵄨8 󵄨 ̂
−𝑐𝜔(𝜉)𝑡
,
(󵄨󵄨󵄨𝜉󵄨󵄨󵄨 + 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 ) 󵄨󵄨󵄨󵄨𝐺
(𝜉, 𝑡)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝐺
𝑡 (𝜉, 𝑡)󵄨󵄨󵄨 ≤ 𝐶𝑒
(29)
󵄨2 󵄨 ̂
󵄨󵄨2
󵄨󵄨 󵄨󵄨4 󵄨󵄨 󵄨󵄨8 −𝑐𝜔(𝜉)𝑡
󵄨 󵄨4 󵄨 󵄨8 󵄨 ̂
(󵄨󵄨󵄨𝜉󵄨󵄨󵄨 + 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 ) 󵄨󵄨󵄨󵄨𝐻
(𝜉, 𝑡)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝐻
𝑡 (𝜉, 𝑡)󵄨󵄨󵄨 ≤ 𝐶 (󵄨󵄨𝜉󵄨󵄨 + 󵄨󵄨𝜉󵄨󵄨 ) 𝑒
(30)
Lemma 1. The solution of the problem (8), (9) satisfies
󵄨2 󵄨
󵄨2
󵄨 󵄨4 󵄨 󵄨8 󵄨
(󵄨󵄨󵄨𝜉󵄨󵄨󵄨 + 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 ) 󵄨󵄨󵄨𝑢̂ (𝜉, 𝑡)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑢̂𝑡 (𝜉, 𝑡)󵄨󵄨󵄨
(17)
󵄨2 󵄨
󵄨2
󵄨 󵄨4 󵄨 󵄨8 󵄨
≤ 𝐶𝑒−𝑐𝜔(𝜉)𝑡 ((󵄨󵄨󵄨𝜉󵄨󵄨󵄨 + 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 ) 󵄨󵄨󵄨𝑢̂0 (𝜉)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑢̂1 (𝜉)󵄨󵄨󵄨 )
for 𝜉 ∈ R𝑛 and 𝑡 ≥ 0, where
󵄨󵄨 󵄨󵄨4
󵄨𝜉󵄨 ,
𝜔 (𝜉) = {󵄨 󵄨
1,
for 𝜉 ∈ R𝑛 and 𝑡 ≥ 0, where
󵄨󵄨 󵄨󵄨4
󵄨𝜉󵄨 ,
𝜔 (𝜉) = {󵄨 󵄨
1,
̂ 𝑡) and 𝐻(𝜉,
̂ 𝑡) be the fundamental solution
Lemma 2. Let 𝐺(𝜉,
of (4) in the Fourier space, which are given in (13) and (14),
respectively. Then one has the estimates
󵄨󵄨 󵄨󵄨
󵄨󵄨𝜉󵄨󵄨 ≤ 𝑅0
󵄨󵄨 󵄨󵄨
󵄨󵄨𝜉󵄨󵄨 ≥ 𝑅0 .
(18)
Proof. Multiplying (8) by 𝑢̂𝑡 and taking the real part yield
1 𝑑 󵄨󵄨 󵄨󵄨2
󵄨 󵄨4 2
󵄨 󵄨4 󵄨 󵄨2
{󵄨𝑢̂ 󵄨 + 𝑘2 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 |̂
𝑢| } + 𝑘1 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 󵄨󵄨󵄨𝑢̂𝑡 󵄨󵄨󵄨 = 0.
2 𝑑𝑡 󵄨 𝑡 󵄨
(19)
(31)
Proof. If 𝑢̂0 (𝜉) = 0, from (12), we obtain
̂ (𝜉, 𝑡) 𝑢̂1 (𝜉) ,
𝑢̂ (𝜉, 𝑡) = 𝐺
̂𝑡 (𝜉, 𝑡) 𝑢̂1 (𝜉) .
𝑢̂𝑡 (𝜉, 𝑡) = 𝐺
(32)
Substituting the equalities into (17) with 𝑢̂0 (𝜉) = 0, we get
(29).
In what follows, we consider 𝑢̂1 (𝜉) = 0, and it follows from
(12) that
Multiplying (8) by 𝑢̂ and taking the real part, we obtain
1 𝑑
󵄨 󵄨4 2
󵄨 󵄨4 2 󵄨󵄨 ̂ 󵄨󵄨2
{𝑘 󵄨󵄨𝜉󵄨󵄨 |̂
𝑢𝑡 𝑢̂)} + 𝑘2 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 |̂
𝑢| + 2 Re (̂
𝑢| − 󵄨󵄨𝑢𝑡 󵄨󵄨 = 0.
2 𝑑𝑡 1 󵄨 󵄨
󵄨󵄨 󵄨󵄨
󵄨󵄨𝜉󵄨󵄨 ≤ 𝑅0
󵄨󵄨 󵄨󵄨
󵄨󵄨𝜉󵄨󵄨 ≥ 𝑅0 .
(20)
4
̂ (𝜉, 𝑡) 𝑢̂0 (𝜉) ,
𝑢̂ (𝜉, 𝑡) = 𝐻
(33)
Multiplying both sides of (19) and (20) by 2 and 𝑘1 |𝜉| and
summing up the resulting equation yield
̂𝑡 (𝜉, 𝑡) 𝑢̂0 (𝜉) .
𝑢̂𝑡 (𝜉, 𝑡) = 𝐻
𝑑
𝐸 + 𝐹 = 0,
𝑑𝑡
Substituting the equalities into (17) with 𝑢̂1 (𝜉) = 0, we get the
desired estimate (30). The lemma is proved.
(21)
where
𝑑
𝐸 + 𝑐𝜔 (𝜉) 𝐸 ≤ 0.
𝑑𝑡
(27)
𝐸 (𝜉, 𝑡) ≤ 𝑒−𝑐𝜔(𝜉)𝑡 𝐸 (𝜉, 0) ,
(28)
Lemma 3. Let 𝑘 and 𝑗 be nonnegative integer. Then one has
󵄩󵄩 𝑘
󵄩
󵄩󵄩𝜕𝑥 𝐺 (𝑡) ∗ 𝜙󵄩󵄩󵄩 2 ≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑘/4+𝑗/4−1/2)
󵄩
󵄩𝐿
(34)
󵄩
󵄩
󵄩 󵄩
× 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐻̇ −𝑗 + 𝐶𝑒−𝑐𝑡 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−4)+ 𝜙󵄩󵄩󵄩󵄩𝐿2 ,
1
󵄩󵄩󵄩𝜕𝑘 𝐻 (𝑡) ∗ 𝜙󵄩󵄩󵄩 ≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑘/4+𝑗/4)
󵄩󵄩 𝑥
󵄩󵄩𝐿2
(35)
󵄩 󵄩
󵄩 󵄩
× 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐻̇ −𝑗 + 𝐶𝑒−𝑐𝑡 󵄩󵄩󵄩󵄩𝜕𝑥𝑘 𝜙󵄩󵄩󵄩󵄩𝐿2 ,
1
󵄩󵄩󵄩𝜕𝑘 𝐺 (𝑡) ∗ 𝜙󵄩󵄩󵄩 ≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑘/4+𝑗/4)
󵄩󵄩 𝑥 𝑡
󵄩󵄩𝐿2
(36)
󵄩 󵄩
󵄩 󵄩
× 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐻̇ −𝑗 + 𝐶𝑒−𝑐𝑡 󵄩󵄩󵄩󵄩𝜕𝑥𝑘 𝜙󵄩󵄩󵄩󵄩𝐿2 ,
1
󵄩󵄩󵄩𝜕𝑘 𝐻 (𝑡) ∗ 𝜙󵄩󵄩󵄩 ≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑘/4+𝑗/4+1/2)
󵄩󵄩 𝑥 𝑡
󵄩󵄩𝐿2
(37)
󵄩
󵄩
󵄩 󵄩
× 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐻̇ −𝑗 + 𝐶𝑒−𝑐𝑡 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘+4) 𝜙󵄩󵄩󵄩󵄩𝐿2 ,
1
󵄩󵄩󵄩𝜕𝑘 𝐺 (𝑡) ∗ Δ𝑔󵄩󵄩󵄩 ≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑘/4)
󵄩󵄩 𝑥
󵄩󵄩𝐿2
(38)
󵄩
󵄩
󵄩 󵄩
× 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐿1 + 𝐶𝑒−𝑐𝑡 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−2)+ 𝑔󵄩󵄩󵄩󵄩𝐿2 ,
󵄩󵄩 𝑘
󵄩
󵄩󵄩𝜕𝑥 𝐺𝑡 (𝑡) ∗ Δ𝑔󵄩󵄩󵄩 2 ≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑘/4+1/2)
󵄩
󵄩𝐿
(39)
󵄩
󵄩
󵄩 󵄩
× 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐿1 + 𝐶𝑒−𝑐𝑡 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘+2) 𝑔󵄩󵄩󵄩󵄩𝐿2 .
which together with (23) proves the desired estimates (17).
Then we have completed the proof of the lemma.
Here (𝑘−4)+ = max{0, 𝑘−4} in (34) and (𝑘−2)+ = max{0, 𝑘−2}
in (38).
1 󵄨󵄨 󵄨󵄨8 2
󵄨 󵄨4
𝑢𝑡 𝑢̂) ,
𝑢| + 𝑘1 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 Re (̂
󵄨𝜉󵄨 |̂
2󵄨 󵄨
󵄨 󵄨8 2
󵄨 󵄨4 󵄨 󵄨2
𝑢| + 𝑘1 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 󵄨󵄨󵄨𝑢̂𝑡 󵄨󵄨󵄨 .
𝐹 = 𝑘1 𝑘2 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 |̂
A simple computation implies that
󵄨 󵄨4 2
󵄨 󵄨2
𝐸 = 󵄨󵄨󵄨𝑢̂𝑡 󵄨󵄨󵄨 + 𝑘2 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 |̂
𝑢| +
(22)
𝐶𝐸0 ≤ 𝐸 ≤ 𝐶𝐸0 ,
(23)
󵄨 󵄨4 󵄨 󵄨8
󵄨 󵄨2
𝐸0 = (󵄨󵄨󵄨𝜉󵄨󵄨󵄨 + 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 ) |̂
𝑢|2 + 󵄨󵄨󵄨𝑢̂𝑡 󵄨󵄨󵄨 .
(24)
where
Note that
󵄨 󵄨2
󵄨 󵄨4 󵄨 󵄨4 󵄨 󵄨8
𝑢|2 + 󵄨󵄨󵄨𝑢̂𝑡 󵄨󵄨󵄨 ) ,
{𝑐󵄨󵄨󵄨𝜉󵄨󵄨󵄨 ((󵄨󵄨󵄨𝜉󵄨󵄨󵄨 + 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 ) |̂
𝐹 ≥ { 󵄨 󵄨4 󵄨 󵄨8
󵄨 󵄨 󵄨 󵄨 𝑢|2 + 󵄨󵄨󵄨𝑢̂𝑡 󵄨󵄨󵄨2 ) ,
󵄨 󵄨
{𝑐 ((󵄨󵄨𝜉󵄨󵄨 + 󵄨󵄨𝜉󵄨󵄨 ) |̂
It follows from (23) that
𝐹 ≥ 𝑐𝜔 (𝜉) 𝐸.
󵄨󵄨 󵄨󵄨
󵄨󵄨𝜉󵄨󵄨 ≤ 𝑅0
󵄨󵄨 󵄨󵄨
󵄨󵄨𝜉󵄨󵄨 ≥ 𝑅0 .
(25)
(26)
Using (21) and (26), we get
Thus
4
Journal of Applied Mathematics
Proof. By the Plancherel theorem and (29), Hausdorff-Young
inequality, we obtain
󵄩2
󵄩󵄩 𝑘
󵄩󵄩𝜕𝑥 𝐺 (𝑡) ∗ 𝜙󵄩󵄩󵄩 2
󵄩𝐿
󵄩
=∫
|𝜉|≤𝑅0
|𝜉|≥𝑅0
≤∫
|𝜉|≤𝑅0
󵄨󵄨 󵄨󵄨2|𝑘|−4 −𝑐|𝜉|4 𝑡 󵄨󵄨󵄨 ̂ 󵄨󵄨󵄨2
𝑒
󵄨󵄨𝜉󵄨󵄨
󵄨󵄨𝜙 (𝜉)󵄨󵄨 𝑑𝜉
|𝜉|≥𝑅0
|𝜉|≤𝑅0
+
Theorem 4. Assume that 𝑢0 ∈ 𝐻𝑠+4 (R𝑛 ) ⋂ 𝐿1 (R𝑛 ), 𝑢1 ∈
𝐻𝑠 (R𝑛 ) ⋂ 𝐻̇ 1−2 (R𝑛 )(𝑠 ≥ [𝑛/2] + 1). Then the classical solution
𝑢(𝑥, 𝑡) to (4), (2), which is given by the formula (15), satisfies
the decay estimate
󵄨2 󵄨
󵄨2
󵄨󵄨 󵄨󵄨2|𝑘| 󵄨󵄨󵄨 ̂
󵄨󵄨𝜉󵄨󵄨 󵄨󵄨𝐺 (𝜉, 𝑡)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝜙̂ (𝜉)󵄨󵄨󵄨󵄨 𝑑𝜉
+ 𝐶𝑒−𝑐𝑡 ∫
≤∫
4. Decay Estimate of Solutions to (4), (2)
󵄨2 󵄨
󵄨2
󵄨󵄨 󵄨󵄨2|𝑘| 󵄨󵄨󵄨 ̂
󵄨󵄨𝜉󵄨󵄨 󵄨󵄨𝐺 (𝜉, 𝑡)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝜙̂ (𝜉)󵄨󵄨󵄨󵄨 𝑑𝜉
+∫
󵄩
󵄩󵄩 𝑘
󵄩󵄩𝜕𝑥 𝑢 (𝑡)󵄩󵄩󵄩 2 ≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑘/4)
󵄩𝐿
󵄩
󵄨󵄨 󵄨󵄨2𝑘 󵄨󵄨 󵄨󵄨8 󵄨󵄨 󵄨󵄨4 −1 󵄨󵄨󵄨 ̂ 󵄨󵄨󵄨2
󵄨󵄨𝜉󵄨󵄨 (󵄨󵄨𝜉󵄨󵄨 + 󵄨󵄨𝜉󵄨󵄨 ) 󵄨󵄨𝜙 (𝜉)󵄨󵄨 𝑑𝜉
󵄨󵄨 󵄨󵄨2𝑘−4+2𝑗 −𝑐|𝜉|4 𝑡 󵄨󵄨 󵄨󵄨−2𝑗 󵄨󵄨󵄨 ̂ 󵄨󵄨󵄨2
𝑒
󵄨󵄨𝜉󵄨󵄨
󵄨󵄨𝜉󵄨󵄨 󵄨󵄨𝜙 (𝜉)󵄨󵄨 𝑑𝜉
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
× (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 )
1
(40)
|𝜉|≤𝑅0
for (𝑘 ≤ 𝑠 + 4) ,
(42)
󵄩
󵄩2
𝐶𝑒−𝑐𝑡 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−4)+ 𝜙󵄩󵄩󵄩󵄩𝐿2
󵄩2
󵄩󵄨 󵄨−𝑗
≤ 𝐶󵄩󵄩󵄩󵄩󵄨󵄨󵄨𝜉󵄨󵄨󵄨 𝜙̂ (𝜉)󵄩󵄩󵄩󵄩𝐿∞ ∫
where 𝑅0 is a small positive constant in Lemma 1. Thus (38)
follows. Similarly, we can prove (39). Thus we have completed
the proof of the lemma.
󵄩
󵄩󵄩 𝑙
󵄩󵄩𝜕𝑥 𝑢𝑡 (𝑡)󵄩󵄩󵄩 2 ≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑙/4+1/2)
󵄩𝐿
󵄩
󵄨󵄨 󵄨󵄨2𝑘−4+2𝑗 −𝑐|𝜉|4 𝑡
𝑒
𝑑𝜉
󵄨󵄨𝜉󵄨󵄨
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
× (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 )
1
󵄩
󵄩2
+ 𝐶𝑒−𝑐𝑡 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−4)+ 𝜙󵄩󵄩󵄩󵄩𝐿2
for (𝑙 ≤ 𝑠) ,
(43)
󵄩
󵄩2
≤ 𝐶(1 + 𝑡)−(𝑛/4+𝑘/2+𝑗/2−1) 󵄩󵄩󵄩󵄩(−Δ)−𝑗/2 𝜙󵄩󵄩󵄩󵄩𝐿1
󵄩
󵄩󵄩 ℎ
󵄩󵄩𝜕𝑥 𝑢 (𝑡)󵄩󵄩󵄩 ∞ ≤ 𝐶(1 + 𝑡)−(𝑛/4+ℎ/4)
󵄩𝐿
󵄩
󵄩
󵄩2
+ 𝐶𝑒−𝑐𝑡 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−4)+ 𝜙󵄩󵄩󵄩󵄩𝐿2 .
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
× (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 )
1
Here (𝑘 − 4)+ = max{0, 𝑘 − 4} and 𝑅0 is a small positive constant in Lemma 1. Thus (34) follows.
Similarly, using (29) and (30), respectively, we can prove
(35)–(37).
In what follows, we prove (38). By the Plancherel theorem,
(29), and Hausdorff-Young inequality, we have
󵄩2
󵄩󵄩 𝑘
󵄩󵄩𝜕𝑥 𝐺 (𝑡) ∗ Δ𝑔󵄩󵄩󵄩 2
󵄩𝐿
󵄩
=∫
|𝜉|≤𝑅0
+∫
≤∫
|𝜉|≤𝑅0
󵄩 󵄩
󵄩 󵄩
≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑘/4) (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 )
1
󵄨2 󵄨 󵄨4 󵄨
󵄨󵄨 󵄨󵄨2𝑘 󵄨󵄨󵄨 ̂
󵄨2
󵄨󵄨𝜉󵄨󵄨 󵄨󵄨𝐺 (𝜉, 𝑡)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 󵄨󵄨󵄨𝑔̂ (𝜉)󵄨󵄨󵄨 𝑑𝜉
󵄨󵄨 󵄨󵄨2𝑘 −𝑐|𝜉|4 𝑡 󵄨󵄨 ̂ 󵄨󵄨2
󵄨󵄨𝜉󵄨󵄨 𝑒
󵄨󵄨𝑔 (𝜉)󵄨󵄨 𝑑𝜉
+ 𝐶𝑒−𝑐𝑡 ∫
|𝜉|≥𝑅0
󵄩2
󵄩
≤ 𝐶󵄩󵄩󵄩𝑔̂ (𝜉)󵄩󵄩󵄩𝐿∞ ∫
󵄨2
󵄨󵄨 󵄨󵄨2𝑘
󵄨 󵄨4 −1 󵄨
󵄨󵄨𝜉󵄨󵄨 (1 + 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 ) 󵄨󵄨󵄨𝑔̂ (𝜉)󵄨󵄨󵄨 𝑑𝜉
|𝜉|≤𝑅0
󵄨󵄨 󵄨󵄨2𝑘 −𝑐|𝜉|4 𝑡
𝑑𝜉
󵄨󵄨𝜉󵄨󵄨 𝑒
󵄩
󵄩2
+ 𝐶𝑒−𝑐𝑡 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−2)+ 𝑔󵄩󵄩󵄩󵄩𝐿2
󵄩
󵄩2
󵄩 󵄩2
≤ 𝐶(1 + 𝑡)−(𝑛/4+𝑘/2) 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐿1 + 𝐶𝑒−𝑐𝑡 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−2)+ 𝑔󵄩󵄩󵄩󵄩𝐿2 ,
Proof. Firstly, we prove (42). Using (34) and (35), for 𝑘 ≤ 𝑠+4,
we obtain
󵄩󵄩 𝑘
󵄩
󵄩󵄩𝜕𝑥 𝑢 (𝑡)󵄩󵄩󵄩 2
󵄩
󵄩𝐿
󵄩
󵄩
󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩𝜕𝑥𝑘 𝐺 (𝑡) ∗ 𝑢1 󵄩󵄩󵄩󵄩𝐿2 + 𝐶󵄩󵄩󵄩󵄩𝜕𝑥𝑘 𝐻 (𝑡) ∗ 𝑢0 󵄩󵄩󵄩󵄩𝐿2
󵄨2 󵄨 󵄨4 󵄨
󵄨2
󵄨󵄨 󵄨󵄨2|𝑘| 󵄨󵄨󵄨 ̂
󵄨󵄨𝜉󵄨󵄨 󵄨󵄨𝐺 (𝜉, 𝑡)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 󵄨󵄨󵄨𝑔̂ (𝜉)󵄨󵄨󵄨 𝑑𝜉
|𝜉|≥𝑅0
𝑛
for (ℎ ≤ 𝑠 − [ ] + 3) .
2
(44)
󵄩 󵄩
󵄩 󵄩
+ 𝐶𝑒−𝑐𝑡 (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 )
(41)
≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑘/4)
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
× (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 ) .
1
For 𝑙 ≤ 𝑠, it follows from (36) and (37) that
󵄩
󵄩󵄩 𝑙
󵄩󵄩𝜕𝑥 𝑢𝑡 (𝑡)󵄩󵄩󵄩 2
󵄩𝐿
󵄩
󵄩
󵄩
󵄩󵄩 𝑙
󵄩
≤ 󵄩󵄩󵄩𝜕𝑥 𝐺𝑡 (𝑡) ∗ 𝑢1 󵄩󵄩󵄩󵄩𝐿2 + 𝐶󵄩󵄩󵄩󵄩𝜕𝑥𝑙 𝐻𝑡 (𝑡) ∗ 𝑢0 󵄩󵄩󵄩󵄩𝐿2
(45)
Journal of Applied Mathematics
5
󵄩 󵄩
󵄩 󵄩
≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑙/4+1/2) (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 )
1
where
󵄩 󵄩
󵄩 󵄩
+ 𝐶𝑒−𝑐𝑡 (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 )
󵄩
󵄩
‖𝑢‖𝑋 = sup { ∑ (1 + 𝑡)𝑛/8+𝑘/4 󵄩󵄩󵄩󵄩𝜕𝑥𝑘 𝑢 (𝑡)󵄩󵄩󵄩󵄩𝐿2
𝑡≥0
≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑙/4+1/2)
(46)
Equation (44) follows from (42) and Gagliardo-Nirenberg
inequality. The lemma is proved.
5. Global Existence and Asymptotic Behavior
The purpose of this section is to prove global existence and
asymptotic behavior of solutions to the Cauchy problem (1),
(2). We need the following lemma, which comes from [24]
(see also [25]).
Lemma 5. Let 𝑠 and 𝜃 be positive integers, 𝛿 > 0, 𝑝, 𝑞, 𝑟 ∈
[1, ∞] satisfy 1/𝑟 = 1/𝑝 + 1/𝑟, and let 𝑘 ∈ {0, 1, 2, . . . , 𝑠}.
Assume that 𝐹(V) is class of 𝐶𝑠 and satisfies
󵄨󵄨󵄨𝜕𝑙 𝐹 (V)󵄨󵄨󵄨 ≤ 𝐶 |V|𝜃+1−𝑙 , |V| ≤ 𝛿, 0 ≤ 𝑙 ≤ 𝑠, 𝑙 < 𝜃 + 1,
󵄨󵄨
󵄨󵄨 V
𝑙,𝛿
󵄨󵄨 𝑙
󵄨
󵄨󵄨𝜕V 𝐹 (V)󵄨󵄨󵄨 ≤ 𝐶𝑙,𝛿 , |V| ≤ 𝛿, 𝑙 ≤ 𝑠, 𝜃 + 1 ≤ 𝑙.
󵄨
󵄨
(47)
If V ∈ 𝐿𝑝 ⋂ 𝑊𝑘,𝑞 ⋂ 𝐿∞ and ‖V‖𝐿∞ ≤ 𝛿, then for |𝛼| ≤ 𝑘 one has
󵄩
󵄩 𝛼 󵄩
󵄩󵄩 𝛼
𝜃−1
󵄩󵄩𝜕𝑥 𝐹 (V)󵄩󵄩󵄩𝐿𝑟 ≤ 𝐶𝑘,𝛿 󵄩󵄩󵄩𝜕𝑥 V󵄩󵄩󵄩𝐿𝑞 ‖V‖𝐿𝑝 ‖V‖𝐿∞ .
(48)
Lemma 6. Let 𝑠 and 𝜃 be positive integers, let 𝛿 > 0, 𝑝, 𝑞, 𝑟 ∈
[1, ∞] satisfy 1/𝑟 = 1/𝑝 + 1/𝑟, and let 𝑘 ∈ {0, 1, 2, . . . , 𝑠}. Let
𝐹(V) be a function that satisfies the assumptions of Lemma 5.
Moreover, assume that
󵄨
󵄨󵄨 𝑠
𝑠
󵄨󵄨𝜕V 𝐹 (V1 ) − 𝜕V 𝐹 (V2 )󵄨󵄨󵄨
󵄨
󵄨 󵄨 󵄨 󵄨 max{𝜃−𝑠,𝜃} 󵄨󵄨
(49)
≤ 𝐶𝛿 (󵄨󵄨󵄨V1 󵄨󵄨󵄨 + 󵄨󵄨󵄨V2 󵄨󵄨󵄨)
󵄨󵄨V1 − V2 󵄨󵄨󵄨 ,
󵄨󵄨 󵄨󵄨
󵄨 󵄨
󵄨󵄨V1 󵄨󵄨 ≤ 𝛿, 󵄨󵄨󵄨V2 󵄨󵄨󵄨 ≤ 𝛿.
If V1 , V2 ∈ 𝐿𝑝 ⋂ 𝑊𝑘,𝑞 ⋂ 𝐿∞ and ‖V1 ‖𝐿∞ ≤ 𝛿, ‖V2 ‖𝐿∞ ≤ 𝛿, then
for |𝛼| ≤ 𝑘, one has
󵄩󵄩 𝛼
󵄩
󵄩󵄩𝜕𝑥 (𝐹 (V1 ) − 𝐹 (V2 ))󵄩󵄩󵄩𝐿𝑟
󵄩
󵄩 󵄩
󵄩
󵄩
󵄩
≤ 𝐶𝑘,𝛿 {(󵄩󵄩󵄩𝜕𝑥𝛼 V1 󵄩󵄩󵄩𝐿𝑞 + 󵄩󵄩󵄩𝜕𝑥𝛼 V2 󵄩󵄩󵄩𝐿𝑞 ) 󵄩󵄩󵄩V1 − V2 󵄩󵄩󵄩𝐿𝑝
(50)
󵄩 󵄩
󵄩 󵄩 󵄩
󵄩
+ (󵄩󵄩󵄩V1 󵄩󵄩󵄩𝐿𝑝 + 󵄩󵄩󵄩V2 󵄩󵄩󵄩𝐿𝑝 ) 󵄩󵄩󵄩𝜕𝑥𝛼 (V1 − V2 )󵄩󵄩󵄩𝐿𝑞 }
󵄩 󵄩 𝜃−1
󵄩 󵄩
× (󵄩󵄩󵄩V1 󵄩󵄩󵄩𝐿∞ + 󵄩󵄩󵄩V2 󵄩󵄩󵄩𝐿∞ ) .
Based on the estimates (42)–(44) of solutions to the linear
problem (4), (2), one defines the following solution space:
𝑋 = {𝑢 ∈ 𝐶 ([0, ∞) ; 𝐻𝑠+4 (R𝑛 ))
1
𝑠
𝑛
⋂ 𝐶 ([0, ∞) ; 𝐻 (R )) : ‖𝑢‖𝑋 < ∞} ,
(52)
󵄩
󵄩
+∑(1 + 𝑡)𝑛/8+𝑙/4+1/2 󵄩󵄩󵄩󵄩𝜕𝑥𝑙 𝑢𝑡 (𝑡)󵄩󵄩󵄩󵄩𝐿2 } .
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
× (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 ) .
1
‖𝐹 (V)‖𝑊𝑘,𝑟 ≤ 𝐶𝑘,𝛿 ‖V‖𝑊𝑘,𝑞 ‖V‖𝐿𝑝 ‖V‖𝜃−1
𝐿∞ ,
𝑘≤𝑠+4
(51)
𝑙≤𝑠
For 𝑅 > 0, one defines
𝑋𝑅 = {𝑢 ∈ 𝑋 : ‖𝑢‖𝑋 ≤ 𝑅} ,
(53)
where 𝑅 depends on the initial value, which is chosen in the
proof of main result.
For ℎ ≤ 𝑠−[𝑛/2]+3, using Gagliardo-Nirenberg inequality,
one obtains
󵄩
󵄩󵄩 ℎ
󵄩󵄩𝜕𝑥 𝑢 (𝑡)󵄩󵄩󵄩 ∞ ≤ 𝐶(1 + 𝑡)−(𝑛/4+ℎ/4) ‖𝑢‖𝑋 .
󵄩𝐿
󵄩
(54)
Theorem 7. Assume that 𝑢0 ∈ 𝐻𝑠+4 (R𝑛 ) ⋂ 𝐿1 (R𝑛 ), 𝑢1 ∈
𝑛
𝐻𝑠 (R𝑛 ) ⋂ 𝐻̇ 1−2 (R𝑛 ) (𝑠 ≥ [ ] + 1), and integer 𝜃 ≥ 1. 𝑓(𝑢)
2
satisfies the assumptions of Lemmas 5 and 6. Put
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
𝐸0 = 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 .
1
(55)
If 𝐸0 is suitably small, the Cauchy problem (1)-(2) has a unique
global classical solution 𝑢(𝑥, 𝑡) satisfying
𝑢 ∈ 𝐶 ([0, ∞) ; 𝐻𝑠+4 (R𝑛 )) ,
𝑢𝑡 ∈ 𝐶 ([0, ∞) ; 𝐻𝑠 (R𝑛 )) .
(56)
Moreover, the solution satisfies the decay estimate
󵄩
󵄩󵄩 𝑘
󵄩󵄩𝜕𝑥 𝑢 (𝑡)󵄩󵄩󵄩 2 ≤ 𝐶𝐸0 (1 + 𝑡)−(𝑛/8+𝑘/4) ,
󵄩𝐿
󵄩
󵄩󵄩 𝑙
󵄩󵄩
󵄩󵄩𝜕𝑥 𝑢𝑡 (𝑡)󵄩󵄩 2 ≤ 𝐶𝐸0 (1 + 𝑡)−(𝑛/8+𝑙/4+1/2)
󵄩
󵄩𝐿
(57)
for 𝑘 ≤ 𝑠 + 4 and 𝑙 ≤ 𝑠.
Proof. Define the mapping
T (𝑢) = 𝐺 (𝑡) ∗ 𝑢1 + 𝐻 (𝑡) ∗ 𝑢0
𝑡
+ ∫ 𝐺 (𝑡 − 𝜏) ∗ Δ𝑓 (Δ𝑢 (𝜏)) 𝑑𝜏.
(58)
0
Using (34)-(35), (38), Lemma 5, and (54), for 𝑘 ≤ 𝑠 + 4, we
obtain
󵄩󵄩 𝑘
󵄩
󵄩󵄩𝜕𝑥 T (𝑢)󵄩󵄩󵄩 2
󵄩
󵄩𝐿
󵄩
󵄩
󵄩
󵄩
≤ 𝐶󵄩󵄩󵄩󵄩𝜕𝑥𝑘 𝐺 (𝑡) ∗ 𝑢1 󵄩󵄩󵄩󵄩𝐿2 + 𝐶󵄩󵄩󵄩󵄩𝜕𝑥𝑘 𝐻 (𝑡) ∗ 𝑢0 󵄩󵄩󵄩󵄩𝐿2
𝑡
󵄩
󵄩
+ 𝐶 ∫ 󵄩󵄩󵄩󵄩𝜕𝑥𝑘 𝐺 (𝑡 − 𝜏) ∗ Δ𝑓 (Δ𝑢 (𝜏))󵄩󵄩󵄩󵄩𝐿2 𝑑𝜏
0
6
Journal of Applied Mathematics
󵄩
󵄩
󵄩 󵄩
≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑘/4) 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 𝐶𝑒−𝑐𝑡 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−4)+ 𝑢1 󵄩󵄩󵄩󵄩𝐿2
It follows from (58) that
1
󵄩
󵄩
󵄩 󵄩
+ 𝐶(1 + 𝑡)−(𝑛/8+𝑘/4) 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 𝐶𝑒−𝑐𝑡 󵄩󵄩󵄩󵄩𝜕𝑥𝑘 𝑢0 󵄩󵄩󵄩󵄩𝐿2
+ 𝐶∫
𝑡/2
0
T(𝑢)𝑡 = 𝐺𝑡 (𝑡) ∗ 𝑢1 + 𝐻𝑡 (𝑡) ∗ 𝑢0
𝑡
󵄩
󵄩
(1 + 𝑡 − 𝜏)−(𝑛/8+𝑘/4) 󵄩󵄩󵄩𝑓 (Δ𝑢)󵄩󵄩󵄩𝐿1 𝑑𝜏
+ ∫ 𝐺𝑡 (𝑡 − 𝜏) ∗ Δ𝑓 (Δ𝑢 (𝜏)) 𝑑𝜏.
0
𝑡
󵄩
󵄩
+ 𝐶 ∫ (1 + 𝑡 − 𝜏)−𝑛/8 󵄩󵄩󵄩󵄩𝜕𝑥𝑘 𝑓 (Δ𝑢)󵄩󵄩󵄩󵄩𝐿1 𝑑𝜏
Using (36)-(37), (39), Lemma 5, and (54), for 𝑙 ≤ 𝑠, we have
𝑡/2
𝑡
󵄩
󵄩
+ 𝐶 ∫ 𝑒−𝑐(𝑡−𝜏) 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−2)+ 𝑓 (Δ𝑢)󵄩󵄩󵄩󵄩𝐿2 𝑑𝜏
󵄩󵄩 𝑙
󵄩
󵄩󵄩𝜕𝑥 T(𝑢)𝑡 󵄩󵄩󵄩 2
󵄩
󵄩𝐿
󵄩
󵄩
󵄩
󵄩
≤ 𝐶󵄩󵄩󵄩󵄩𝜕𝑥𝑙 𝐺𝑡 (𝑡) ∗ 𝑢1 󵄩󵄩󵄩󵄩𝐿2 + 𝐶󵄩󵄩󵄩󵄩𝜕𝑥𝑙 𝐻𝑡 (𝑡) ∗ 𝑢0 󵄩󵄩󵄩󵄩𝐿2
0
󵄩 󵄩
󵄩 󵄩
≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑘/4) (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 )
1
𝑡
󵄩
󵄩
+ 𝐶 ∫ 󵄩󵄩󵄩󵄩𝜕𝑥𝑙 𝐺𝑡 (𝑡 − 𝜏) ∗ Δ𝑓 (Δ𝑢 (𝜏))󵄩󵄩󵄩󵄩𝐿2 𝑑𝜏
0
󵄩 󵄩
󵄩 󵄩
+ 𝐶𝑒−𝑐𝑡 (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 )
+ 𝐶∫
𝑡/2
0
󵄩
󵄩
󵄩 󵄩
≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑙/4+1/2) 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 𝐶𝑒−𝑐𝑡 󵄩󵄩󵄩󵄩𝜕𝑥𝑙 𝑢1 󵄩󵄩󵄩󵄩𝐿2
1
(1 + 𝑡 − 𝜏)−(𝑛/8+𝑘/4) ‖Δ𝑢‖2𝐿2 ‖Δ𝑢‖𝜃−1
𝐿∞ 𝑑𝜏
󵄩
󵄩
󵄩 󵄩
+ 𝐶(1 + 𝑡)−(𝑛/8+𝑙/4+1/2) 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 𝐶𝑒−𝑐𝑡 󵄩󵄩󵄩󵄩𝜕𝑙+4 𝑢0 󵄩󵄩󵄩󵄩𝐿2
𝑡
+ 𝐶 ∫ (1 + 𝑡 − 𝜏)−𝑛/8−1/2 ‖Δ𝑢‖𝐿2
𝑡/2
𝑡/2
+ 𝐶∫
󵄩
󵄩
× 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−2)+ Δ𝑢󵄩󵄩󵄩󵄩𝐿2 ‖Δ𝑢‖𝜃−1
𝐿∞ 𝑑𝜏
0
󵄩
󵄩
+ 𝐶 ∫ 𝑒−𝑐(𝑡−𝜏) 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−2)+ Δ𝑢󵄩󵄩󵄩󵄩𝐿2 ‖Δ𝑢‖𝜃𝐿∞ 𝑑𝜏
𝑡/2
𝑡
󵄩
󵄩
+ 𝐶 ∫ 𝑒−𝑐(𝑡−𝜏) 󵄩󵄩󵄩󵄩𝜕𝑥𝑙+2 𝑓 (Δ𝑢)󵄩󵄩󵄩󵄩𝐿2 𝑑𝜏
0
󵄩 󵄩
󵄩 󵄩
≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑘/4) (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 )
1
+ 𝐶𝑒
0
≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑙/4+1/2)
󵄩 󵄩
󵄩 󵄩
(󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 )
+ 𝐶𝑅𝜃+1 ∫
𝑡/2
0
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
× (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 )
1
(1 + 𝑡 − 𝜏)−(𝑛/8+𝑘/4) (1 + 𝜏)−(𝑛/4+1)
𝑡/2
+ 𝐶∫
0
× (1 + 𝜏)−(𝑛/4+1/2)(𝜃−1) 𝑑𝜏
𝑡/2
+ 𝐶𝑅𝜃+1 ∫ (1 + 𝑡 − 𝜏)−𝑛/8−1/2 (1 + 𝜏)−(𝑛/4+𝑘/4)
𝑡/2
𝑡
󵄩
󵄩
+ 𝐶 ∫ 𝑒−𝑐(𝑡−𝜏) 󵄩󵄩󵄩󵄩𝜕𝑥𝑙+4 𝑢󵄩󵄩󵄩󵄩𝐿2 ‖Δ𝑢‖𝜃𝐿∞ 𝑑𝜏
× (1 + 𝜏)−(𝑛/4+1/2)(𝜃−1) 𝑑𝜏
+ 𝐶𝑅
𝑡
−𝑐(𝑡−𝜏)
∫ 𝑒
0
0
≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑙/4+1/2)
−(𝑛/8+((𝑘−2)+ +2)/4)
(1 + 𝜏)
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
× (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 )
1
× (1 + 𝜏)−(𝑛/4+1/2)𝜃 𝑑𝜏
≤ 𝐶(1 + 𝑡)
(1 + 𝑡 − 𝜏)−(𝑛/8+𝑙/4+1/2) ‖Δ𝑢‖2𝐿2 ‖Δ𝑢‖𝜃−1
𝐿∞ 𝑑𝜏
𝑡
󵄩
󵄩2
+ 𝐶 ∫ (1 + 𝑡 − 𝜏)−(𝑛/8+1/2) 󵄩󵄩󵄩󵄩𝜕𝑥𝑙 Δ𝑢󵄩󵄩󵄩󵄩𝐿2 ‖Δ𝑢‖𝜃−1
𝐿∞ 𝑑𝜏
𝑡
𝜃+1
󵄩
󵄩
(1 + 𝑡 − 𝜏)−(𝑛/8+𝑙/4+1/2) 󵄩󵄩󵄩𝑓 (Δ𝑢)󵄩󵄩󵄩𝐿1 𝑑𝜏
𝑡
󵄩
󵄩
+ 𝐶 ∫ (1 + 𝑡 − 𝜏)−(𝑛/8+1/2) 󵄩󵄩󵄩󵄩𝜕𝑥𝑙 𝑓 (Δ𝑢)󵄩󵄩󵄩󵄩𝐿1 𝑑𝜏
𝑡
−𝑐𝑡
(61)
+ 𝐶𝑅𝜃+1 ∫
−(𝑛/8+𝑘/4)
𝑡/2
0
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
× {(󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 ) + 𝑅𝜃+1 } .
1
(59)
(1 + 𝑡 − 𝜏)−(𝑛/8+𝑙/4+1/2) (1 + 𝜏)−(𝑛/4+1)
× (1 + 𝜏)−(𝑛/4+1/2)(𝜃−1) 𝑑𝜏
𝑡
+ 𝐶𝑅𝜃+1 ∫ (1 + 𝑡 − 𝜏)−(𝑛/8+1/2) (1 + 𝜏)−(𝑛/4+𝑙/2+1)
𝑡/2
× (1 + 𝜏)−(𝑛/4+1/2)(𝜃−1) 𝑑𝜏
Thus
󵄩
󵄩
(1 + 𝑡)𝑛/8+𝑘/4 󵄩󵄩󵄩󵄩𝜕𝑥𝑘 T (𝑢)󵄩󵄩󵄩󵄩𝐿2 ≤ 𝐶𝐸0 + 𝐶𝑅𝜃+1 .
𝑡
(60)
+ 𝐶𝑅𝜃+1 ∫ 𝑒−𝑐(𝑡−𝜏) (1 + 𝜏)−(𝑛/8+(𝑙+4)/4)
0
Journal of Applied Mathematics
7
𝜃−1
× (1 + 𝜏)−(𝑛/4+1/2)𝜃 𝑑𝜏
× (‖Δ̃
𝑢‖𝐿∞ + ‖Δ𝑢‖𝐿∞ )
𝑑𝜏
𝑡
≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑙/4+1/2)
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
× {(󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 ) + 𝑅𝜃+1 } .
1
(62)
+ 𝐶 ∫ 𝑒−𝑐(𝑡−𝜏)
0
󵄩
󵄩
󵄩
󵄩
× {(󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−2)+ Δ̃
𝑢󵄩󵄩󵄩󵄩𝐿2 + 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−2)+ Δ̃
𝑢󵄩󵄩󵄩󵄩𝐿2 )
× ‖Δ (̃
𝑢 − 𝑢)‖𝐿∞ + (‖Δ̃
𝑢‖𝐿∞ + ‖Δ𝑢‖𝐿∞ )
󵄩
󵄩
× 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−2)+ Δ (̃
𝑢 − 𝑢)󵄩󵄩󵄩󵄩𝐿2 }
Thus
𝜃−1
󵄩
󵄩
(1 + 𝑡)𝑛/8+𝑙/4+1/2 󵄩󵄩󵄩󵄩𝜕𝑥𝑙 T(𝑢)𝑡 󵄩󵄩󵄩󵄩𝐿2 ≤ 𝐶𝐸0 + 𝐶𝑅𝜃+1 .
× (‖Δ̃
𝑢‖𝐿∞ + ‖Δ𝑢‖𝐿∞ )
(63)
≤ 𝐶𝑅𝜃 ‖̃
𝑢 − 𝑢‖𝑋 ∫
𝑡/2
(1 + 𝑡 − 𝜏)−(𝑛/8+𝑘/4)
0
Combining (60), (63) and taking 𝑅 = 2𝐶𝐸0 and 𝐸0 suitably
small yield
𝑑𝜏
× (1 + 𝜏)−(𝑛/4+1+(𝑛/4+1/2)(𝜃−1)) 𝑑𝜏
𝑡
+ 𝐶𝑅𝜃 ‖̃
𝑢 − 𝑢‖𝑋 ∫ (1 + 𝑡 − 𝜏)−𝑛/8−1/2
𝑡/2
‖T (𝑢)‖𝑋 ≤ 2𝐶𝐸0 .
(64)
−(𝑛/4+𝑘/4+1/2+(𝑛/4+1/2)(𝜃−1))
× (1 + 𝜏)
𝑑𝜏
𝑡
+ 𝐶𝑅𝜃 ‖̃
𝑢 − 𝑢‖𝑋 ∫ 𝑒−𝑐(𝑡−𝜏)
For 𝑢̃, 𝑢 ∈ 𝑋𝑅 , by using (58), we have
0
× (1 + 𝜏)−(𝑛/8+((𝑘−2)+ +2)/4+(𝑛/4+1/2)𝜃) 𝑑𝜏
𝑡
T (̃
𝑢) − 𝑓 (Δ𝑢)] 𝑑𝜏. (65)
𝑢) − T (𝑢) = ∫ 𝐺 (𝑡 − 𝜏) ∗ Δ [𝑓 (Δ̃
0
(66)
Exploiting (65), (38) Lemma 6, and (54), for 𝑘 ≤ 𝑠 + 4, we
obtain
󵄩
󵄩󵄩 𝑘
󵄩󵄩𝜕𝑥 (T (̃
𝑢) − T (𝑢))󵄩󵄩󵄩󵄩𝐿2
󵄩
󵄩
󵄩
≤ ∫ 󵄩󵄩󵄩󵄩𝜕𝑥𝑘 𝐺 (𝑡 − 𝜏) ∗ Δ [𝑓 (Δ̃
𝑢) − 𝑓 (Δ𝑢)]󵄩󵄩󵄩󵄩𝐿2 𝑑𝜏
0
𝑡/2
󵄩
󵄩
𝑢) − 𝑓 (Δ𝑢))󵄩󵄩󵄩𝐿1 𝑑𝜏
(1 + 𝑡 − 𝜏)−(𝑛/8+𝑘/4) 󵄩󵄩󵄩(𝑓 (Δ̃
0
𝑡
󵄩
󵄩
+ 𝐶 ∫ (1 + 𝑡 − 𝜏)−𝑛/8 󵄩󵄩󵄩󵄩𝜕𝑥𝑘 (𝑓 (Δ̃
𝑢) − 𝑓 (Δ𝑢))󵄩󵄩󵄩󵄩𝐿1 𝑑𝜏
𝑡/2
𝑡
󵄩
󵄩
𝑢) − 𝑓 (Δ𝑢))󵄩󵄩󵄩󵄩𝐿2 𝑑𝜏
+ 𝐶 ∫ 𝑒−𝑐(𝑡−𝜏) 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−2)+ (𝑓 (Δ̃
0
≤ 𝐶∫
𝑡/2
0
𝑢‖𝐿2 + ‖Δ𝑢‖𝐿2 )
(1 + 𝑡 − 𝜏)−(𝑛/8+𝑘/4) (‖Δ̃
𝜃−1
𝑢‖𝐿∞ + ‖Δ𝑢‖𝐿∞ )
× ‖Δ (̃
𝑢 − 𝑢)‖𝐿2 (‖Δ̃
𝑡
which implies
󵄩
󵄩
𝑢) − T (𝑢))󵄩󵄩󵄩󵄩𝐿2 ≤ 𝐶𝑅𝜃 ‖̃
𝑢 − 𝑢‖𝑋 .
(1 + 𝑡)𝑛/8+𝑘/4 󵄩󵄩󵄩󵄩𝜕𝑥𝑘 (T (̃
(67)
Similarly for 𝑙 ≤ 𝑠, from (61), (39), and (54), we have
𝑡
≤ 𝐶∫
≤ 𝐶𝑅𝜃 (1 + 𝑡)−(𝑛/8+𝑘/4) ‖̃
𝑢 − 𝑢‖𝑋 ,
𝑑𝜏
󵄩󵄩 𝑙
󵄩
󵄩󵄩𝜕𝑥 (T (̃
𝑢) − T (𝑢))𝑡 󵄩󵄩󵄩󵄩𝐿2
󵄩
𝑡
󵄩
󵄩
𝑢) − 𝑓 (Δ𝑢)]󵄩󵄩󵄩󵄩𝐿2 𝑑𝜏
≤ ∫ 󵄩󵄩󵄩󵄩𝜕𝑥𝑙 𝐺𝑡 (𝑡 − 𝜏) ∗ Δ [𝑓 (Δ̃
0
𝑡/2
󵄩
󵄩
𝑢)−𝑓 (Δ𝑢))󵄩󵄩󵄩𝐿1 𝑑𝜏
≤ 𝐶 ∫ (1+𝑡 − 𝜏)−(𝑛/8+𝑙/4+1/2) 󵄩󵄩󵄩(𝑓 (Δ̃
0
𝑡
󵄩
󵄩
+ 𝐶 ∫ (1+𝑡 − 𝜏)−(𝑛/8+1/2) 󵄩󵄩󵄩󵄩𝜕𝑥𝑙 (𝑓 (Δ̃
𝑢)−𝑓 (Δ𝑢))󵄩󵄩󵄩󵄩𝐿1 𝑑𝜏
𝑡/2
𝑡
󵄩
󵄩
+ 𝐶 ∫ 𝑒−𝑐(𝑡−𝜏) 󵄩󵄩󵄩󵄩𝜕𝑥𝑙+2 (𝑓 (Δ̃
𝑢) − 𝑓 (Δ𝑢))󵄩󵄩󵄩󵄩𝐿2 𝑑𝜏
0
𝜃
≤ 𝐶𝑅 (1 + 𝑡)−(𝑛/8+𝑙/4+1/2) ‖̃
𝑢 − 𝑢‖𝑋 ,
(68)
−𝑛/8−1/2
+ 𝐶 ∫ (1 + 𝑡 − 𝜏)
𝑡/2
󵄩
󵄩
󵄩
󵄩
× {(󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−2)+ Δ̃
𝑢󵄩󵄩󵄩󵄩𝐿2 + 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−2)+ Δ̃
𝑢󵄩󵄩󵄩󵄩𝐿2 ) ‖Δ (̃
𝑢 − 𝑢)‖𝐿2
󵄩
󵄩
+ (‖Δ̃
𝑢‖𝐿2 + ‖Δ𝑢‖𝐿2 ) 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−2)+ Δ (̃
𝑢 − 𝑢)󵄩󵄩󵄩󵄩𝐿2 }
which implies
󵄩
󵄩
𝑢) − T (𝑢))𝑡 󵄩󵄩󵄩󵄩𝐿2 ≤ 𝐶𝑅𝜃 ‖̃
𝑢 − 𝑢‖𝑋 .
(1 + 𝑡)𝑛/8+𝑙/4+1/2 󵄩󵄩󵄩󵄩𝜕𝑥𝑙 (T (̃
(69)
8
Journal of Applied Mathematics
Noting 𝑅 = 2𝐶𝐸0 , by using (67), (69) and taking 𝐸0 suitably
small, yields
1
𝑢) − T (𝑢)‖𝑋 ≤ ‖̃
𝑢 − 𝑢‖𝑋 .
‖T (̃
2
(70)
From (64) and (70), we know that T is strictly contracting
mapping. Consequently, we conclude that there exists a fixed
point 𝑢 ∈ 𝑋𝑅 of the mapping T, which is a classical solution
to (1), (2). We have completed the proof of Theorem 7.
Acknowledgments
This work was supported in part by the NNSF of China (Grant
no. 11101144) and Innovation Scientists and Technicians
Troop Construction Projects of Henan Province.
References
[1] H. T. Banks, D. S. Gilliam, and V. I. Shubov, “Global solvability
for damped abstract nonlinear hyperbolic systems,” Differential
and Integral Equations, vol. 10, no. 2, pp. 309–332, 1997.
[2] G. Chen and F. Da, “Blow-up of solution of Cauchy problem
for three-dimensional damped nonlinear hyperbolic equation,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no.
1-2, pp. 358–372, 2009.
[3] C. Song and Z. Yang, “Existence and nonexistence of global
solutions to the Cauchy problem for a nonlinear beam equation,” Mathematical Methods in the Applied Sciences, vol. 33, no.
5, pp. 563–575, 2010.
[4] G. Chen, “Initial boundary value problem for a damped
nonlinear hyperbolic equation,” Journal of Partial Differential
Equations, vol. 16, no. 1, pp. 49–61, 2003.
[5] G. Chen, R. Song, and S. Wang, “Local existence and global
nonexistence theorems for a damped nonlinear hyperbolic
equation,” Journal of Mathematical Analysis and Applications,
vol. 368, no. 1, pp. 19–31, 2010.
[6] Y.-Z. Wang, “Global existence and asymptotic behaviour of
solutions for the generalized Boussinesq equation,” Nonlinear
Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 465–
482, 2009.
[7] Y.-Z. Wang, F. G. Liu, and Y. Z. Zhang, “Global existence
and asymptotic behavior of solutions for a semi-linear wave
equation,” Journal of Mathematical Analysis and Applications,
vol. 385, no. 2, pp. 836–853, 2012.
[8] S. Wang and H. Xu, “On the asymptotic behavior of solution
for the generalized IBq equation with hydrodynamical damped
term,” Journal of Differential Equations, vol. 252, no. 7, pp. 4243–
4258, 2012.
[9] V. Pata and S. Zelik, “Smooth attractors for strongly damped
wave equations,” Nonlinearity, vol. 19, no. 7, pp. 1495–1506, 2006.
[10] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential
Equations, vol. 26 of Mathématiques and Applications, Springer,
Berlin, Germany, 1997.
[11] C. D. Sogge, Lecures on Nonlinear Wave Equations, vol. 2 of
Monographs in Analysis, International Press, 1995.
[12] W.-R. Dai and D.-X. Kong, “Global existence and asymptotic
behavior of classical solutions of quasilinear hyperbolic systems
with linearly degenerate characteristic fields,” Journal of Differential Equations, vol. 235, no. 1, pp. 127–165, 2007.
[13] D.-X. Kong and T. Yang, “Asymptotic behavior of global classical
solutions of quasilinear hyperbolic systems,” Communications
in Partial Differential Equations, vol. 28, no. 5-6, pp. 1203–1220,
2003.
[14] Y.-Z. Wang, “Global existence of classical solutions to the
minimal surface equation in two space dimensions with slow
decay initial value,” Journal of Mathematical Physics, vol. 50, no.
10, Article ID 103506, 14 pages, 2009.
[15] Y.-Z. Wang and Y.-X. Wang, “Global existence of classical
solutions to the minimal surface equation with slow decay
initial value,” Applied Mathematics and Computation, vol. 216,
no. 2, pp. 576–583, 2010.
[16] Y. Liu and W. Wang, “The pointwise estimates of solutions for
dissipative wave equation in multi-dimensions,” Discrete and
Continuous Dynamical Systems A, vol. 20, no. 4, pp. 1013–1028,
2008.
[17] M. Nakao and K. Ono, “Existence of global solutions to the
Cauchy problem for the semilinear dissipative wave equations,”
Mathematische Zeitschrift, vol. 214, no. 2, pp. 325–342, 1993.
[18] K. Nishihara, “𝐿𝑝 -𝐿𝑞 estimates of solutions to the damped
wave equation in 3-dimensional space and their application,”
Mathematische Zeitschrift, vol. 244, no. 3, pp. 631–649, 2003.
[19] K. Ono, “Global existence and asymptotic behavior of small
solutions for semilinear dissipative wave equations,” Discrete
and Continuous Dynamical Systems A, vol. 9, no. 3, pp. 651–662,
2003.
[20] W. Wang and W. Wang, “The pointwise estimates of solutions
for semilinear dissipative wave equation in multi-dimensions,”
Journal of Mathematical Analysis and Applications, vol. 366, no.
1, pp. 226–241, 2010.
[21] Z. Yang, “Longtime behavior of the Kirchhoff type equation
with strong damping on R𝑛 ,” Journal of Differential Equations,
vol. 242, no. 2, pp. 269–286, 2007.
[22] Y. Liu and S. Kawashima, “Global existence and asymptotic
behavior of solutions for quasi-linear dissipative plate equation,”
Discrete and Continuous Dynamical Systems A, vol. 29, no. 3, pp.
1113–1139, 2011.
[23] Y. Sugitani and S. Kawashima, “Decay estimates of solutions to
a semilinear dissipative plate equation,” Journal of Hyperbolic
Differential Equations, vol. 7, no. 3, pp. 471–501, 2010.
[24] T. T. Li and Y. M. Chen, Nonlinear Evolution Equations,
Scientific Press, 1989 (Chinese).
[25] S. M. Zheng, Nonlinear Evolution Equations, vol. 133 of Monographs and Surveys in Pure and Applied Mathematics, Chapman
& Hall/CRC, 2004.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014