# Research Article Asymptotic Behavior of Solutions to the Damped Nonlinear Hyperbolic Equation ```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; [email protected]
Received 17 December 2012; Revised 15 May 2013; Accepted 30 May 2013
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 &gt; 0, 𝑘1 &gt; 0 and 𝑘2 &gt; 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 . In , 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 . The local well-posedness for the
Cauchy problem for (1), (2) in three-dimensional space was
obtained by Chen and Da . 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 
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
󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
&times; (󵄩󵄩󵄩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
󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
&times; (󵄩󵄩󵄩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 , 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, .
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 &plusmn; (𝜉) be the corresponding eigenvalues of (10), and
we obtain
𝜆 &plusmn; (𝜉) =
󵄨 󵄨4 󵄨 󵄨2
󵄨 󵄨4
−𝑘2 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 &plusmn; 󵄨󵄨󵄨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)
󵄩
󵄩
󵄩 󵄩
&times; 󵄩󵄩󵄩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)
󵄩 󵄩
󵄩 󵄩
&times; 󵄩󵄩󵄩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)
󵄩 󵄩
󵄩 󵄩
&times; 󵄩󵄩󵄩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)
󵄩
󵄩
󵄩 󵄩
&times; 󵄩󵄩󵄩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)
󵄩
󵄩
󵄩 󵄩
&times; 󵄩󵄩󵄩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)
󵄩
󵄩
󵄩 󵄩
&times; 󵄩󵄩󵄩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
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
&times; (󵄩󵄩󵄩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󵄨󵄨
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
&times; (󵄩󵄩󵄩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 .
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
&times; (󵄩󵄩󵄩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)
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
&times; (󵄩󵄩󵄩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 
Lemma 5. Let 𝑠 and 𝜃 be positive integers, 𝛿 &gt; 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 &lt; 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 &gt; 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
󵄩 󵄩
&times; (󵄩󵄩󵄩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 &lt; ∞} ,
(52)
󵄩
󵄩
+∑(1 + 𝑡)𝑛/8+𝑙/4+1/2 󵄩󵄩󵄩󵄩𝜕𝑥𝑙 𝑢𝑡 (𝑡)󵄩󵄩󵄩󵄩𝐿2 } .
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
&times; (󵄩󵄩󵄩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 &gt; 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∫
󵄩
󵄩
&times; 󵄩󵄩󵄩󵄩𝜕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
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
&times; (󵄩󵄩󵄩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
&times; (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
&times; (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)
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
&times; (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 )
1
&times; (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
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
&times; {(󵄩󵄩󵄩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)
&times; (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
&times; (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
&times; (1 + 𝜏)−(𝑛/4+1/2)𝜃 𝑑𝜏
&times; (‖Δ̃
𝑢‖𝐿∞ + ‖Δ𝑢‖𝐿∞ )
𝑑𝜏
𝑡
≤ 𝐶(1 + 𝑡)−(𝑛/8+𝑙/4+1/2)
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
&times; {(󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿1 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻̇ −2 + 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐻𝑠+4 + 󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩𝐻𝑠 ) + 𝑅𝜃+1 } .
1
(62)
+ 𝐶 ∫ 𝑒−𝑐(𝑡−𝜏)
0
󵄩
󵄩
󵄩
󵄩
&times; {(󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−2)+ Δ̃
𝑢󵄩󵄩󵄩󵄩𝐿2 + 󵄩󵄩󵄩󵄩𝜕𝑥(𝑘−2)+ Δ̃
𝑢󵄩󵄩󵄩󵄩𝐿2 )
&times; ‖Δ (̃
𝑢 − 𝑢)‖𝐿∞ + (‖Δ̃
𝑢‖𝐿∞ + ‖Δ𝑢‖𝐿∞ )
󵄩
󵄩
&times; 󵄩󵄩󵄩󵄩𝜕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 .
&times; (‖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
&times; (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))
&times; (1 + 𝜏)
𝑑𝜏
𝑡
+ 𝐶𝑅𝜃 ‖̃
𝑢 − 𝑢‖𝑋 ∫ 𝑒−𝑐(𝑡−𝜏)
For 𝑢̃, 𝑢 ∈ 𝑋𝑅 , by using (58), we have
0
&times; (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∞ )
&times; ‖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
󵄩
󵄩
󵄩
󵄩
&times; {(󵄩󵄩󵄩󵄩𝜕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.
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