global existence and optimal decay rate of solutions for the degenerate quasilinear wave equation...
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Global existence and optimal decayrate of solutions for the degeneratequasilinear wave equation with astrong dissipationSalima Mimouni a , Abbes Benaissa a & Nour-Eddine Amroun aa Laboratory of Mathematics , Djillali Liabes University , P. O. Box89, Sidi Bel Abbes 22000, AlgeriaPublished online: 18 May 2010.
To cite this article: Salima Mimouni , Abbes Benaissa & Nour-Eddine Amroun (2010) Globalexistence and optimal decay rate of solutions for the degenerate quasilinear wave equationwith a strong dissipation, Applicable Analysis: An International Journal, 89:6, 815-831, DOI:10.1080/00036811003649090
To link to this article: http://dx.doi.org/10.1080/00036811003649090
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Applicable AnalysisVol. 89, No. 6, June 2010, 815–831
Global existence and optimal decay rate of solutions for the
degenerate quasilinear wave equation with a strong dissipation
Salima Mimouni, Abbes Benaissa* and Nour-Eddine Amroun
Laboratory of Mathematics, Djillali Liabes University, P. O. Box 89,Sidi Bel Abbes 22000, Algeria
Communicated by I. Lasiecka
(Received 27 March 2009; final version received 25 January 2010)
In this article, we study the initial boundary value problem of thedegenerate quasilinear wave equation with a strong dissipation of the form
u00 � �
Z�
jrxuj2 dx
� �Dxu� �ðtÞDxu
0 þ juj�u ¼ 0 in �� IRþ:
We prove global existence of solutions in Sobolev spaces and generalstability estimates using multiplier method and general weighted integralinequalities. Without imposing any growth condition at the origin �, weshow that the energy of the system is bounded above by a quantity,depending on � and �, which tends to zero (as time goes to infinity). Wealso prove the optimality of decay rate of the energy for �� 1 and � isslowly degenerates. These estimates allows us to consider large class offunctions � and � with general growth at the origin. We give manysignificant examples to illustrate how to derive from our general estimatesthe polynomial, exponential or logarithmic decay.
Keywords: degenerate quasilinear wave equation; global existence; strongdissipative term; multiplier method; integral inequalities
AMS Subject Classifications: 35L70; 35L80
1. Introduction
We consider the initial boundary value problem for the degenerate quasilinear wave
equation of Kirchhoff type with a strong dissipation, that is,
u00 � �
Z�
jrxuj2 dx
� �Dxu� �ðtÞDxu
0 þ juj�u ¼ 0 in �� ½0,þ1½,
u ¼ 0 on �� ½0,þ1½,
uðx, 0Þ ¼ u0ðxÞ, u0ðx, 0Þ ¼ u1ðxÞ on �,
8>>><>>>:
ðPÞ
*Corresponding author. Email: [email protected]
ISSN 0003–6811 print/ISSN 1563–504X online
� 2010 Taylor & Francis
DOI: 10.1080/00036811003649090
http://www.informaworld.com
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where � and � are given functions and � is a bounded domain in IRn (n� 1) with a
smooth boundary �¼ @�. The functions (u0, u1) are the given initial data.In the case n¼ 1, Problem (P) describes a small amplitude vibration of an elastic
string. The original equation is
�h@2u
@t2¼ P0 þ
E h
2L
Z L
0
@u
@xðx, tÞ
��������2 ds
!@2u
@x2þ f,
where 0� x�L and t40, u(x, t) is the lateral displacement at the space coordinate x
and the time t, � the mass density, h the cross-section area, L the length, P0 the initial
axial tension,E the Youngmodulus and f the external force (e.g. the action of gravity).The equation was first introduced by Kirchhoff [1] in 1876, and is called the
Kirchhoff string after his name. The local existence of solutions in Sobolev space was
investigated by many authors [2–7]. Concerning a global existence of solutions
to degenerate Kirchhoff equations, it is natural to add a dissipative term
(e.g. u0,�Dxu0,D2
xu).When we have �� 1 and �(r)¼ r�(�� 1) in the problem (P), Nishihara and Ono
[8–10] proved the existence and uniqueness of a global solution for initial data
ðu0, u1Þ 2H2 \H1
0 � L2ð�Þ and the polynomial decay of the solution.Unfortunately, the method used by Nishihara and Ono which is based on a decay
Lemma of Nakao [11] does not seem to be applicable for more general functions �and �.
In this article, we shall obtain global solutions in time in Sobolev spaces and derive
decay properties of the solution when �� 1 and � have a general form. Moreover,
when �� 1 and � are slowly degenerate, we prove that the decay rate obtained is
optimal by deriving the decay estimate from below for the solution. We extend the
results obtained by Nishihara and Ono [8–10]. We use a method introduced by
Martinez [12] and generalized to the non-dissipative case by Guesmia in [13] and [14]
to study the decay rate of solutions to the wave equation u00 �Dxuþ g(u0)¼ 0 in
�� IRþ where � is a bounded domain of IRn. This method is based on new integral
inequalities (Lemma 2.5) proved in [13] and [14] that generalize the results of Haraux
[15], Komornik [16] and Martinez [12], and improve the ones of Eller et al. [17].The proof is based on new integral inequalities and some properties of convex
functions, in particular, the dual function of convex function to use the general
Young and Jensen’s inequalities (instead of Holder inequality widely used in the
classical case of linear or polynomial growth of g at the origin) in objective to prove
our general decay estimate under a general growth of � at the origin. These
arguments of convexity were introduced and developed by Lasiecka et al. [18–22],
and used by Eller et al. [17] and Alabau-Boussouira [23]. In [24], decay and existence
for non-degenerate quasilinear wave equation with some general nonlinear boundary
dissipation is investigated.Throughout this article, the functions considered are all real valued. We omit the
space and time variables x and t of u(t,x), ut(t, x) and simply denote u(t, x), ut(t, x) by
u, u0, respectively, when no confusion arises. Let l be a number with 2� l�þ1. We
denote by k.kl the Ll-norm over �. In particular, L2-norm is denoted by
k.k2. (�) denotes the usual L2 inner product. We use familiar function spaces H10
and H10.
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2. Preliminaries and main results
First assume the following hypotheses:
(H1) � : IRþ !]0,þ1[ is a non-increasing function of class C1(IRþ) satisfyingZ þ10
�ð�Þ d� ¼ þ1: ð1Þ
(H2) � : IRþ! IRþ is a continuous function on IRþ satisfying the followinghypotheses:
(i) �2C1(]0, a])\C 0([0, a]) for some a40;(ii) degenerate case: �(s)40 on ]0, a] and � is non-decreasing. In this case there(exist m140 such that
s�ðsÞ � m1
Z s
0
�ð�Þd� on IRþ: ð2Þ
or(ii)0 non-degenerate case: there exist m0,m140 such that �(s)�m0 on IRþ and
satisfy (2);(iii) there exists a constant C such that
js �0ðsÞj � C, 05 s � a:
Now we define ~�ðsÞ ¼ 12
R s0 �ð�Þd� and the energy associated with the solution of the
problem (P) by the following formula:
EðtÞ ¼1
2
Z�
ju0j2 dxþ ~�
Z�
jrxuj2 dx
� �þ
1
�þ 2kuk�þ2�þ2: ð3Þ
By a simple computation, we have
E 0ðtÞ ¼ ��ðtÞ
Z�
jrxu0j2 dx: ð4Þ
We first state some lemmas which will be needed later.
LEMMA 2.1 (Sobolev–Poincare’s inequality) Let q be a number with 2� q5þ1(n¼ 1, 2, . . . , p) or 2� q� np/(n� p) (n� pþ 1). Then there is a constant c�¼ c�(�, q)
such that
kukq � c�krukp for u2W1, p0 ð�Þ:
The case p¼ q¼ 2 gives the known Poincare’s inequality.
LEMMA 2.2 (Gagliardo–Nirenberg) Let 1� r5q�þ1 and p� 2. Then, the
inequality
kukp � Ckrmx uk
�2kuk
1��r for u2Dðð�DÞ
m2 Þ \ Lr
holds with some constant C40 and
� ¼1
r�1
p
� �m
nþ1
r�1
2
� ��1
provided that 05�� 1 (we assume 05�51 if m� n2 is a non-negative integer).
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LEMMA 2.3 (Modified Gronwall inequality) Let M and g be non-negative functionson [0,þ1) satisfying
0 �MðtÞ � Kþ
Z t
0
gðsÞMðsÞrþ1 ds
with K40 and r40. Then
MðtÞ � K�r � r
Z t
0
gðsÞds
� ��1=r
as long as the right-hand side exists.
LEMMA 2.4 [25] Let ’ : IRm! IR be a differentiable function. Assume that ’ is a
convex function of class C1 on IRm, then
’ð yÞ � ’ðxÞ þ hr’ðxÞ, y� xi, 8x2 IRm, 8y2 IRm:
hr’ð yÞ � r’ðxÞ, y� xi � 0, 8x2 IRm, 8y2 IRm:
Now we recall the following local existence theorem, which can be established byusing the argument in [7] and [10].
THEOREM 2.1 Let �(r) be a non-negative continuous function for r� 0, and
0 � � � 4=ðn� 4Þ ð0 � �5 þ1 if 0 � n � 4Þ:
If u0 2H2 \H1
0ð�Þ and u12L2(�), then there exists a unique local weak solution u
of (P) satisfying
uðtÞ 2C 0ð½0,T ½;H2 \H10ð�ÞÞ and u0ðtÞ 2C 0ð½0,T ½;L2ð�ÞÞ \ C 0ð½0,T ½;H1
0ð�ÞÞ,
moreover, at least one of the following statements hold true:
(i) T¼þ1,(ii) eðuðtÞÞ � ku0ðtÞk22 þ kDxuðtÞk
22!þ1 as t! T�.
LEMMA 2.5 [11,13] Let E : IRþ! IRþ differentiable function, 2 IRþ and� : IRþ! IRþ convex and increasing function such that �(0)¼ 0. Assume that
R Ts �ðEðtÞÞdt � EðsÞ, 80 � s � T,E 0ðtÞ � EðtÞ, 8t � 0:
�
Then E satisfies the following estimate:
EðtÞ � e�0T0d�1ðeðt�hðtÞÞ�ð �1ðhðtÞ þ ðEð0ÞÞÞÞÞ, 8t � 0
where
ðtÞ ¼
Z 1
t
1
�ðsÞds, 8t4 0,
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dðtÞ ¼�ðtÞ if ¼ 0,R t0
�ðsÞs ds if 4 0,
8t � 0,
(
hðtÞ ¼K�1ðDðtÞÞ, 8t4T0,
0, 8t2 ½0,T0,
�
KðtÞ ¼ DðtÞ þ �1ðtþ ðEð0ÞÞÞ�ð �1ðtþ ðEð0ÞÞÞÞ
et, 8t � 0,
DðtÞ ¼R t0 e
s ds, 8t � 0,
(
T0 ¼ D�1Eð0Þ
�ðEð0ÞÞ
� �, �0 ¼
0, 8t4T0 ,
1, 8t2 ½0,T0.
�
Remark 2.1 If ¼ 0 (that is E is non-increasing), then we have
EðtÞ � �1 hðtÞ þ ðEð0ÞÞð Þ, 80 � t � T, ð5Þ
where ðtÞ ¼R 1t
1�ðsÞ ds for t40, h(t)¼ 0 for 0 � t � Eð0Þ
�ðEð0ÞÞ, and
h�1ðtÞ ¼ tþ �1 tþ ðEð0ÞÞð Þ
� �1 tþ ðEð0ÞÞð Þð Þ, 8t � 0:
This particular result generalizes the one obtained by Komornik [16] and
Martinez [12] in the particular case �(t)¼ dtpþ1 with p� 0 and d40, and improves
the one obtained by Eller et al. [17].
Our main results are the following.
THEOREM 2.2 Assume that (H1)–(H2) hold. Let ~�ðtÞ ¼R t0 �ð�Þd�. Then there exist !,
040 such that the energy E satisfies
A. Degenerate case:
EðtÞ � ~�ð �1ðhð ~�ðtÞÞ þ ð ~��1ðEð0ÞÞÞÞÞ, 80 � t � T, ð6Þ
where ðtÞ ¼R 1t
1! ~�ð�Þ
d� for t40, h(t)¼ 0 for 0 � t � ~��1~��1ðEð0ÞÞ
!Eð0Þ
� �,
h�1ðtÞ ¼ tþ �1 tþ ðEð0ÞÞð Þ
! ~� �1 tþ ðEð0ÞÞð Þð Þ, 8t4 ~��1
~��1ðEð0ÞÞ
!Eð0Þ
!
and T is the life span of a local solution.
B. Non-degenerate case:
EðtÞ � e�! hð ~�ðtÞÞ�1! lnEð0Þð Þ, 80 � t � T, ð7Þ
where h(t)¼ 0 for 0 � t � ~��11
!
� �,
h�1ðtÞ ¼ tþ1
!, 8t4 ~��1
1
!
� �
and T is the life span of a local solution.
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Remark 2.2 We use the energy estimate (6) and (7) to prove global existence of the
solution, indeed we need to estimate theH2-norm (i.e kDxuk2) of the solution because
from (4) E(t) (the energy of first order) is bounded and does not blow up in finite
time, so to prove global existence we will estimate the second energy (the H2-norm of
the solution) by using the energy estimate E(t).In the non-degenerate case, � is not a non-decreasing function and we can take
some examples of � as
�ðsÞ ¼ 2þ s cos ln1
s
� ��, 05 � � 1:
In the degenerate case, we can replace the condition that � is an increasing function
by the following:There exists a function �2C1([0, a[) such that
0 � �ðsÞ � s�ðsÞ, 8s2 ½0, a½,
� is strictly increasing in ½0, a½:
So, we can add some interesting examples as
�ðsÞ ¼ e�2s��
2þ cos1
s
� �, �4 0:
THEOREM 2.3 Let ðu0, u1Þ 2H2 \H1
0 � L2ð�Þ, suppose that �� 1, � degenerate,
0��� 4/(n� 2) (�5þ1 if n¼ 1, 2) andZ 10
ð �1ðc hðtÞÞÞ�2 dt5þ1: ð8Þ
Then there exists a unique global solution of (P) satisfying
uðtÞ 2Cð½0,þ1Þ; H2 \H10ð�ÞÞ \ L
1ðð0,þ1Þ; H2 \H10ð�ÞÞ,
u0ðtÞ 2Cð½0,þ1Þ; L2ð�ÞÞ \ L1ðð0,þ1Þ; L2ð�ÞÞ \ L2ðð0,þ1Þ; H10ð�ÞÞ:
THEOREM 2.4 Let ðu0, u1Þ 2H2 \H1
0 � L2ð�Þ, suppose that �� 1, � degenerate,
4/(n� 2)��� 4/(n� 4)þ (�5þ1 if n¼ 1, 2) and
Z 10
ð �1ðc hðtÞÞÞ�ð1��0 Þ
2 dt5þ1,
where �0� [(n� 2)�� 4]þ/(2�). Then there exists a unique global solution of (P)
satisfying
uðtÞ 2Cð½0,þ1Þ; H2 \H10ð�ÞÞ \ L
1ðð0,þ1Þ; H2 \H10ð�ÞÞ,
u0ðtÞ 2Cð½0,þ1Þ; L2ð�ÞÞ \ L1ðð0,þ1Þ; L2ð�ÞÞ \ L2ðð0,þ1Þ; H2 \H10ð�ÞÞ:
Remark 2.3 In this article, we are not able to establish the global in time solvability
in Sobolev spaces for the degenerate quasilinear wave equation with a strong
dissipation when � 6¼ const (�(t)! 0 as t!1) but we can easily establish the global
existence of analytic solutions in the case when � has a general form such that
�(r)� 0 and without any smallness conditions on initial data and with general
function �(t). The proof of the global existence is basically the same as the one in
[26–28].
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Examples Using the last two estimates, we give several significant examples of
growth at the origin of �, and the corresponding decay estimates (see the appendix
for more details).
(1) Polynomial or logarithmic growth of � and polynomial growth of g: if
�(s)¼ csr(ln(sþ1))q (degeneracy of finite order) for c, c040, r� 0, q��r and
p� 1 (note that c01srþqþ1 � ~�ðsÞ � c02s
rþqþ1 for c01, c02 4 0 when t is near 0), then
(as in the Example 1) there exists 40 such that for all t� 0
EðtÞ � e�! ~�ðtÞ if rþ q ¼ 0,
EðtÞ � ~�ðtÞ þ 1ð Þ�
2ðrþqþ1Þ2ðrþqÞ if rþ q 6¼ 0:
(2) Exponential growth of � (degeneracy of infinite order): if �(s)¼ e�s��
, �40,
(note that c01s�þ1e�s
��� ~�ðsÞ � c02s
�þ1e�s��
for c01, c02 4 0 when s is near 0)
then there exist �, 40 such that
EðtÞ � �ðhð ~�ðtÞÞÞ�1 lnð hð ~�ðtÞÞ þ 2Þð Þ��þ1� , 8t � 0:
(3) Slow than polynomials of � (slow degeneracy): if �(s)¼ jln sj�� near of 0
where �40, (note that c01sð� ln sÞ�� � ~�ðsÞ � c02sð� ln sÞ�� for c01, c02 4 0 when
s is near 0, and then (s)� c(�ln s)�þ1) then there exists 40 such that
EðtÞ � ð ~�ðtÞÞ���þ1e�ð ~�ðtÞÞ
1�þ1
, 8t � 0:
(4) Slow than polynomials of � (slow degeneracy): if �(s)¼ e�jln sj�
near of 0
where 05�51, (note that c01se�j ln sj� � ~�ðsÞ � c02se
�j ln sj� for c01, c02 4 0 when s
is near 0, and then ðsÞ � ðln sÞ1��eðln1sÞ�
) then there exists 40 such that
EðtÞ � ð ~�ðtÞÞ�1e�ðln ~�ðtÞÞ1�, 8t � 0:
Proof of the energy decay From now on, we denote by c various positive constants
which may be different at different occurrences.If E(t0)¼ 0 for some t0� 0, then E(t)¼ 0 for all t� t0, and then we have nothing
to prove in this case. So we assume that E(t)40 for all t� 0 without loss of
generality.We multiply the first equation of (P) by �(t)u and we integrate by parts, we have,
for all 0�S�T,
0 ¼
Z T
S
�ðtÞ
Z�
u u00 � �
Z�
jrxuj2dx
� �Dxu� �ðtÞDxu
0 þ juj�u
� �dx dt
¼ �ðtÞ
Z�
uu0dx
TS
�
Z T
S
Z�
u0 �0ðtÞuþ �ðtÞu0ð Þdx dt
þ
Z T
S
�ðtÞ�
Z�
jrxuj2dx
� �Z�
jruj2dxdt�
Z T
S
�2ðtÞ
Z�
uDxu0dxdt
þ
Z T
S
�ðtÞ
Z�
juj�þ2 dx dt:
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On the other hand, we have s�ðsÞ � c ~�ðsÞ. We deduce thatZ T
S
�ðtÞEðtÞdt � � �ðtÞ
Z�
uu0 dx
TS
þ
Z T
S
Z�
u0 �0ðtÞuþ3
2�ðtÞu0
� �dxdt
þ
Z T
S
�2ðtÞ
Z�
uDxu0 dx dt:
Using Lemma 2.1 for p¼ q¼ 2 (Poincare’s inequality), the fact that E is
non-increasing, � is a bounded non-negative function on IRþ and ~��1
is
non-decreasing, we have (note also that ~� is a bijection from IRþ to IRþ)
�ðtÞ
Z�
uu0 dx
�������� � c
Z�
u2 dx
� �12Z
�
ju0j2dx
� �12
� c
Z�
jrxuj2dx
� �12Z
�
ju0j2dx
� �12
� c
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEðS Þ ~�
�1ðEðS ÞÞ
q8t � S ð9Þ
Z T
S
�0ðtÞ
Z�
u0u dxdt
�������� � c
Z T
S
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEðtÞ ~�
�1ðEðtÞÞ
qð��0ðtÞÞdt
� c
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEðS Þ ~�
�1ðEðS ÞÞ
q Z T
S
ð��0ðtÞÞdt � c
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEðS Þ ~�
�1ðEðS ÞÞ
q: ð10Þ
Using the fact that ~� is convex, increasing and ~�ð0Þ ¼ 0 (then s� s~��1ðsÞ
is
non-decreasing), we haveffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEðS Þ ~�
�1ðEðS ÞÞ
q�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEðS Þ
~��1ðEðS ÞÞ
s~��1ðEðS ÞÞ � c ~�
�1ðEðS ÞÞ:
Then we deduce thatZ T
S
�ðtÞEðtÞdt � c ~��1ðEðS ÞÞ þ c
Z T
S
�ðtÞ
Z�
ðju0j2 þ jrxurxu0ÞjÞdxdt: ð11Þ
Now we estimate the last integral of (11). For all t� 0, we denote by
�þt ¼ fx2� : rxurxu0 � 0g, ��t ¼ fx2� : rxurxu
0 � 0g:
Using (4), we deduceZ T
S
�ðtÞ
Z�
ju0j2 dxdt � c
Z T
S
�ðtÞ
Z�
jrxu0j2 dx dt � cEðS Þ
and (note that �0 � 0)Z T
S
�ðtÞ
Z�
jrxurxu0jdxdt
� �ðtÞ
Z�þt
jrxuj2 dx��ðtÞ
Z��t
jrxuj2 dx
" #T
S
þ
Z T
S
�0ðtÞ �
Z�þt
jrxuj2dxþ
Z��t
jrxuj2 dx
!dt
� c ~��1ðEðS ÞÞþ c ~�
�1ðEðSÞÞ
Z T
S
ð��0ðtÞÞdt� c ~��1ðEðS ÞÞ:
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Thus, we obtain from (11) thatZ þ1S
�ðtÞEðtÞdt � c ~��1ðEðS ÞÞ:
Let ~E ¼ ~��1 E ~��1 (note that ~� is a bijection from IRþ to IRþ). Then, for
some !40, Z þ1S
~�ð ~EðtÞÞdt �1
!~EðS Þ:
Using Lemma 2.5 [13,14] for ~E in the particular case �ðsÞ ¼ ! ~�ðsÞ and ¼ 0,
we deduce that
~EðtÞ � �1ðhðtÞ þ ð ~��1ðEð0ÞÞÞÞ, 8t � 0:
Then, using the definition of ~E, we obtain (6).
B. Non-degenerate case: Using Young and Poincare’s inequalities we have,
for all 40,Z T
S
�ðtÞ
Z�
uDxu0 dx dt
�������� �
Z T
S
�ðtÞ
Z�
jrxuj2 dxdtþ cðÞ
Z T
S
�ðtÞ
Z�
jrxu0j2dxdt
�
Z T
S
�ðtÞ ~��1ðEÞdtþ cðÞ
Z T
S
�ðtÞ
Z�
jrxu0j2dx dt:
Using the fact that ~��1ðsÞ � cs and choosing small enough, we obtain
from (11) thatZ 1S
�ðtÞEðtÞdt � c ~��1ðEðS ÞÞ þ c
Z T
S
�ðtÞ
Z�
ðju0j2 þ jrxu0j2Þdx dt: ð12Þ
Using (H2), (4), we haveZ T
S
�ðtÞ’ðEÞ
E
Z�1
t
ðju0j2 þ jrxu0j2Þdxdt � c
Z T
S
�ðtÞ
Z�
jrxu0j2 dxdt � cEðS Þ:
Since, using (12) and choosing ~’ðsÞ ¼ s, we deduce thatZ þ1S
�ðtÞEðtÞdt � cðEðS Þ þ ~��1ðEðS ÞÞ � c0EðS Þ:
Using Lemma 2.5 [13,14] for ~E in the particular case �(s)¼!s and ¼ 0, we deduce
our estimate.
Proof of the global existence Multiplying the first equation of the problem (P)
by �2Dxu and integrating it over �, we have
d
dtfkDxuk
22 � 2ðu0ðtÞ,DxuðtÞÞg þ 2�ðkrxuk
22ÞkDxuk
22
¼ 2krxu0k22 � 2ðjuj�u,DxuÞ: ð13Þ
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We shall show that the solution u belongs to L1ð0,þ1;H2 \H10ð�ÞÞ. It follows
from (13) that
d
dtkDxuðtÞk
22 þ 2�ðkrxuðtÞk
22ÞkDxuðtÞk
22
¼ 2d
dtðu0ðtÞ,DxuðtÞÞ þ 2krxu
0k22 � 2ðjuj�u,DxuðtÞÞ:
Integrating it over [0, t] and applying Young’s inequality, we have that
1
2kDxuðtÞk
22 þ 2
Z t
0
�ðkrxuðtÞk22ÞkDxuðtÞk
22 ds
� kDxu0k22 � 2ðu1,Dxu0Þ þ 2ku0ðtÞk22 þ 2
Z t
0
krxu0ðtÞk22 ds
þ 2
Z t
0
jðjuðtÞj�uðtÞ,DxuðtÞÞj ds,
and hence, we see that
kDxuðtÞk22 � I20 þ 4
Z t
0
jðjuðsÞj�uðsÞ,DxuðsÞÞjds ð14Þ
with
I 20 � 2fkDxu0k22 � 2ðu1,Dxu0Þ þ 2Eð0Þg:
Thus, it follows from (14) and (6) that
kDxuðtÞk22 � I 20 þ c0
Z t
0
krxuðsÞk�kDxuðsÞk
22 ds
� I 20 þ c0Z t
0
ð �1ðchðsÞÞÞ�2kDxuðsÞk
2 ds
with some constant c0, and hence by Gronwall lemma we obtain
kDxuðtÞk22 � I 20 e
Z t
0
ð �1ðchðsÞÞÞ�2 ds
by (8) we conclude
kDxuðtÞk22 � const5 þ1
for any t� 0. The proof of Theorem 2.3 is now completed.
Proof of the theorem 2.4 If n� 3 and 4/(n� 2)5�52/(n� 4), by applying
Lemma 2.2 we see that
jðjuj�u,DxuÞj � ð�þ 1Þ
Z�
juj�jrxuj2 dx
� ð�þ 1Þkuk�n�=2krxuk22n=ðn�2Þ
� c�þ1� ð�þ 1Þkrxuk�ð1��0Þ2 kDxuk
��0þ22
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with �0� [(n� 2)�� 4]þ/(2�) (we note 0� �051). Thus, it follows from (14) that
kDxuðtÞk22 � I 20 þ c0
Z t
0
kDxuðsÞk�ð1��0Þ2 kDxuðsÞk
��0þ2 ds
and hence, by Lemma 2.3,
kDxuðtÞk2 � I���00 � c0��02
Z t
0
krxuðsÞk�ð1��0Þ2 ds
� ��1=ð��0Þ,
kDxuðtÞk2 � I���00 � c0��02
Z t
0
ð �1ðc hðsÞÞÞ�ð1��0 Þ
2 ds
� ��1=ð��0Þ:
Therefore, if initial data {u0, u1} is small, we deduce that
kDxuðtÞk2 � const:
Next, we shall derive another decay property, which is the estimate from below.
THEOREM 2.5 Under the assumption of Theorem 2.3 or Theorem 2.4, let the initial
data {u0, u1} be small and satisfy
Fðu0, u1Þ � krxu0k22 þ 2ðu0, u1Þ4 0 ð15Þ
and
m2
R s0 �ð�Þd� � s�ðsÞ,
lims!0
s�þ22
~�ðsÞ� const if 0 � � �
4
n� 2,
lims!0
sð�þ2Þð1��Þ
2
~�ðsÞ� const if
4
n� 2� � �
4
n� 4:
Then the solution u of (P) has the following decay estimate for large time t�T �40
such that
C1 �1ðctÞ � krxuðtÞk
22 � C2
�1ðctÞ ð16Þ
for t�T �, where Ci, i¼ 1, 2 are some constants depending on krxu0k22 þ ku1k
22 and
ðtÞ ¼R 1t
1! ~�ð�Þ
d� for t40.
Proof We shall show only that there exists a T � such that krxuðtÞk22 � C1
�1ðctÞ
for t�T �. Multiplying the first equation of the problem (P) by u and integrating it
over �, we have
d
dtFðuðtÞ, u0ðtÞÞ þ 2GðuðtÞ, u0ðtÞÞ ¼ 0, ð17Þ
where we set
FðuðtÞ, u0ðtÞÞ ¼ krxuðtÞk22 þ 2ðu, u0Þ,
GðuðtÞ, u0ðtÞÞ ¼ �ðkrxuðtÞk22ÞkrxuðtÞk
22 þ kuðtÞk
�þ2�þ2 � ku
0ðtÞk22:
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In what follows, we put F(t)¼F(u(t), u0(t)) and G(t)¼G(u(t), u0(t)) for simplicity.
Since F(0)40 near t¼ 0. Let
T3 ¼ supft2 ½0,þ1½;FðsÞ4 0 for 0 � s � tg,
then T340 and F(t)40 on [0,T3[. If T35þ1, then F(T3)¼ 0. As ~� is a convex
function of class C1 on IR, then applying Lemma 2.4, we deduce
~�ðFðtÞÞ � ~�ðkrxuk22Þ þ 2�ðkrxuk
22Þðu
0, uÞ
� ~�ðkrxuk22Þ � �ðkrxuk
22Þkuk
22 �
1
�ðkrxuk
22Þku
0k22
� ð1� m2 c�Þ ~�ðkrxuk22Þ �
1
�ðkrxuk
22Þku
0k22: ð18Þ
From the definition of the energy (see (3)), we have
~�ðkrxuk22Þ � EðtÞ � Eð0Þ,
~� is a non-decreasing function (s� ~��1ðsÞ is non-decreasing), then
krxuk22 �
~��1ðEð0ÞÞ,
� is a non-decreasing function, thus we have
�ðkrxuk22Þ � �ð
~��1ðEð0ÞÞÞ: ð19Þ
By (18), (19) and choosing small enough, we deduce that
~�ðFðtÞÞ � ð1� m2 c�Þ ~�ðkrxuk22Þ �
1
�ð ~��1ðEð0ÞÞÞku0k22
� c1 ~�ðkrxuk22Þ � c2�ð ~�
�1ðEð0ÞÞÞku0ðtÞk22,
where c1 and c2 are two positive constants. While we see easily
GðuðtÞ, u0ðtÞÞ ¼ �ðkrxuk22Þkrxuk
22 þ kuk
�þ2�þ2 � ku
0k22
� c ~�ðkrxuk22Þ þ c0krxuk
ð�þ2Þð1��Þ2 � ku0k22
� c� ~�ðkrxuk22Þ � ku
0k22,
where
� ¼
0 if 0 � � �4
n� 2,
ðn� 2Þ�� 4
2ð�þ 2Þif
4
n� 2� � �
4
n� 4:
8>><>>:
Indeed by applying Lemmas 2.1 and 2.2, we have
kuk�þ2�þ2 � ckruk�þ22 if 0 � � �4
n� 2ð�51 if n ¼ 1, 2Þ,
kuk�þ2�þ2 � ckukð�þ2Þð1��0Þ2nn�2
kDukð�þ2Þ�2
� ckrukð�þ2Þð1��0Þ2 kDukð�þ2Þ�
0
2
� ckrukð�þ2Þð1��0Þ2
if4
n� 2� � �
4
n� 4ð�51 if n ¼ 3, 4Þ:
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where �0 ¼ ((n� 2)�� 4)/2(�þ 2). Thus we obtain
C ~�ðFðtÞÞ � GðtÞ � ð1� c3�ð ~��1ðEð0ÞÞÞÞku0ðtÞk22 � 0: ð20Þ
where we have used the assumption E(0)� 1. Then, we see from (17) and (20) that
C ~�ðFÞ þd
dtFðtÞ � 0: ð21Þ
Observe that
m2
Z s
0
�ð�Þd� � s�ðsÞ �
Z s
0
�ð�Þd� ð0 � s � s0Þ ) C1s �
Z s
0
�ð�Þd� � C2sm2 :
In fact, by the assumption,
m2
s�
�ðsÞR s0 �ð�Þd�
�1
s
and
d
dsln ðsm2 Þ �
d
dsln
Z s
0
�ð�Þd�
� ��
d
dsln ðsÞ ðm1 ¼ 1 and m2 � 1Þ:
Integrating this on (s, s0]
s0s
� �m2
�
R s00 �ð�Þd�R s0 �ð�Þd�
�s0s
� �, 05 s � s0
and R s00 �ð�Þd�
s0s �
Z s
0
�ð�Þd� �
R s00 �ð�Þd�
sm2
0
sm2 ð0 � s � s0Þ ð22Þ
and hence, we see from F(0)40 thatZ FðtÞ
Fð0Þ
1
~�ðsÞds � �Ct: ð23Þ
If F(t)�F(0), then from (22) and (23)
Z FðtÞ
Fð0Þ
1
c2sm2ds � �Ct
then
FðtÞ � ðFð0Þ1�m2 þ c0tÞ� 1
m2�1 4 0:
If F(t)�F(0), then from (22) and (23)
Z Fð0Þ
FðtÞ
1
c1sds � Ct:
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Then
FðtÞ � Fð0Þe�ct 4 0,
which contradicts F(T3)¼ 0. Therefore, we see that T3¼þ1 and the above estimate
holds true for all t� 0. Also from (22), we have
ðFðtÞÞ � Ct� ðFð0ÞÞ, is a decreasing function.
Thus, for large t�T 040, we deduce that
FðtÞ � �1ðCtÞ: ð24Þ
Moreover, by (24), Sobolev Poincare inequality and by the energy decay estimate (6),
it follows that
krxuðtÞk22 �
�1ðCtÞ � 2ku0k2kuk2
� �1ðCtÞ � Cku0k2krxuk2
� �1ðCtÞ � Cð ~�ð �1ðCtÞÞÞ12ð �1ðCtÞÞ
12
� C1 �1ðCtÞ, t � T�
because
limt!þ1
~�ð �1ðCtÞÞ
�1ðCtÞ¼ lim
s!0
~�ðsÞ
s¼ lim
s!0�ðsÞ ¼ �ð0Þ ¼ 0,
which imply the desired estimate (16).
Remark 2.4 We can replace the term juj�u by rF with a positive sign, where
F :D(�D)! IR is a functional such that
(a) F is Gateaux differentiable at any point x2D(D);(b) for any x2D(D) there exists a constant c(x)40 such that
jDFðxÞð yÞj � cðxÞk yk, for any y2DðDÞ, ð25Þ
where DF(x) denotes the Gateaux derivative of F in x; consequently, DF(x) can be
extended to the whole space X (and we will denote by rF(x) The Unique
Vector Representing DF(x) in the Riesz isomorphism, that is, hrF(x), yi¼ DF(x)(y),
for any y2X );(c) for any R40 there exists a constant CR40 such that
krFðxÞ � rFð yÞk � CRjrx� ryj ð26Þ
for all x,y2D(D) satisfying krxk, kryk�R.
We suppose that there exists a strictly increasing continuous function : [0,1[!
[0,1[ such that
jFðxÞj � ðkrxk2Þkrxk2 8x2DðDÞ: ð27Þ
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Acknowledgements
The authors thank the anonymous referee for the helpful comments that improved this article.
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Appendix
Proof of the examples To keep this article from becoming too long, we will prove only thelast two examples given in Section 2, and the proof of the other examples can be done exactlyas the same way.
Example 3 Let � be given by �(s)¼ e�jln sj�
when 05�51, and �� const. We have
~�ðtÞ ¼
Z t
0
�ðsÞds ¼
Z t
0
ej ln sj�
ds ¼
Z þ11t
1
z2e�ðln zÞ
�
dz:
Applying Theorem 10.5 (ii) in [29, p. 95], we obtain
~�ðtÞ � ct e�j ln tj�
:
Also, we have
ðtÞ ¼
Z 1
t
1
! ~�ðsÞds ¼ c
Z 1
t
e�j ln sj�
sds
¼ c
Z 1
t
1
eðln zÞ�
zdz
¼ c
Z ln1t
0
ey�
dy,
where we use the following change of variable ln z¼ y. Applying Theorem 10.7 (i) in [29, p. 96],we obtain
ðtÞ � cðln tÞ1��eðln1tÞ�
as t! 0,
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then using a result of Dieudonne [29, p. 95] for asymptotic development, we have
�1ðtÞ � ce�ðln tÞ1�
as t!1:
Using the fact that h(t)� t as t goes to infinity, we deduce that
EðtÞ � t�1e�ðln tÞ1�, 8t � 0
and
c1e�ðln tÞ
1�� krxuk
22 � c2 e�ðln tÞ
1�:
Example 4 Let � be given by �(s)¼ s�jln sj� when �40, �2 IR and �� const. We have
~�ðtÞ ¼
Z t
0
�ðsÞds ¼
Z t
0
s�j ln sj� ds ¼
Z þ11t
1
z�þ2ðln zÞ� dz:
Applying Theorem 10.5 (ii) in [29, p. 95], we obtain
~�ðtÞ � ct�þ1j ln tj� as t! 0:
Also, we have
ðtÞ ¼
Z 1
t
1
! ~�ðsÞds ¼ c
Z 1
t
1
s�þ1j ln sj�ds
¼ c
Z 1t
1
z��1
ðln zÞ�dz,
where we use the following change of variable ln z¼ y. Applying Theorem 10.5 (i) in [29, p. 95],we obtain
ðtÞ � ct�� ln1
t
� ��as t! 0,
then using a result of Dieudonne [29, p. 95] for asymptotic development, we have
�1ðtÞ � ct�1�ðln tÞ�
�� as t!1:
Using the fact that h(t)� t as t goes to infinity, we deduce that
EðtÞ � � t��þ1� ðln tÞ�
�� , 8t � 0
and
c1t�1�ðln tÞ�
�� � krxuk
22 � c2t
�1�ðln tÞ�
�� :
Applicable Analysis 831
Dow
nloa
ded
by [
Uni
vers
ity o
f T
enne
ssee
At M
artin
] at
20:
48 0
5 O
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