Non uniform decay of the energy of some dissipative evolution systems

0.16 MB·

by  Kaïs Ammari

NON UNIFORM DECAY OF THE ENERGY OF SOME
DISSIPATIVE EVOLUTION SYSTEMS
6
1 KA¨ISAMMARI,AHMEDBCHATNIA,ANDKARIMELMUFTI
0
2

n Abstract. In this paper we consider second order evolution equations with
a bounded damping. We give a characterization of a non uniform decay for
J thedamped problemusingakindofobservabilityestimatefortheassociated
undampedproblem.
3
1

]
S
D Contents
.
h
1. Introduction and main results 1
t
a
2. Proof of Theorem 1.1 3
m
3. Some applications 7
[

3.1. The linear case 8
1
v
3.2. The nonlinear case 9
3
7 References 12
3
3
0
.
1 1. Introduction and main results
0
6
1 LetX beacomplexHilbertspacewithnormandinnerproductdenotedrespectively
: by . and .,. . Let A be a linear unbounded self-adjoint and strictly positive
v ||||X h iX
i operator in X and V = (A12) be the domain of A21, with
X D
r x V = A21x X, x V.
a k k k k ∀ ∈
Denote by ( (A12))′ the dual space of (A12) obtained by means of the inner
D D
product in X. Further, let U be a complex Hilbert space (identified to its dual)
and B (U,X).
∈L
Most of the linear control problems coming from elasticity can be written as
w (t)+Aw(t)+Bu(t)=0,
(1.1) ′′
w(0)=w , w (0)=w ,
0 ′ 1
(cid:26)
where w :[0,T] X is the state of the system, u L2(0,T;U) is the input func-
→ ∈
tion and denote the differentiation with respect to time by “”.
′
2010 Mathematics Subject Classification. 35B30,35B40.
Keywordsandphrases. boundedfeedback,kindofobservabilityestimate,nonuniformdecay.
1
2 KA¨ISAMMARI,AHMEDBCHATNIA,ANDKARIMELMUFTI
We define the energy of w(t) at instant t by
1
E(w(t))= 2 ||w′(t)||2X +||A12w(t)||2X .
n o
Simple formal calculations give
t
(1.2) E(w(0)) E(w(t))= Bu(s),w (s) ds, t 0.
− h ′ iX ∀ ≥
Z0
This is why, in many problems coming in particular from elasticity, the input u is
given in the feedback form u(t) = B w (t), which obviously gives a nonincreasing
∗ ′
energy and which corresponds to collocated actuators and sensors.
The aim of this paper is to give sufficient and necessary conditions on the con-
servative system (1.5) making the corresponding closed loop system
w (t)+Aw(t)+BB w (t)=0,
(1.3) ′′ ∗ ′
w(0)=w ,w(0)=w ,
0 ′ 1
(cid:26)
nonuniformlystable. Thestrategytogetsuchadecayratewillconsisttogeneralize
a kind of observability estimate given in [6].
Any sufficiently smooth solution of (1.3) satisfies the energy estimate
t
(1.4) E(w(0)) E(w(t))= B w (s) 2ds, t 0.
− || ∗ ′ ||U ∀ ≥
Z0
In particular (1.4) implies that
E(w(t)) E(w(0)), t 0.
≤ ∀ ≥
In the natural well-posedness space V X, the existence and uniqueness of finite
×
energy solutions of (1.3) can be obtained by standard semi-group methods.
Denote by φ the solution of the associated undamped problem
φ (t)+Aφ(t)=0,
(1.5) ′′
φ(0)=w , φ(0)=w .
0 ′ 1
(cid:26)
It is well known that (1.5) is well-posed in (A) V and in V X.
D × ×
Our main result is stated as follows:
Let be a continuous positive increasing real function on [0,+ ) and define the
G ∞
function by (x)=x( (x))2.
F F G
Theorem 1.1. (1) Assume that there exists C > 0 such that for all non-
identically zero initial data (w ,w ) V X and for all t>0, the solution
0 1
∈ ×
w of (1.3) satisfies:
1
(1.6) E(w(t)) C (w ,w ) 2 1 ,
≤ || 0 1 ||V×XG− t
(cid:18) (cid:19)
then there exists C >0 such that the solution φ of (1.5) satisfies:
1
(1.7) ||(w0,w1)||2V×X ≤16Z0G(2C1Λ) kB∗φ′(t)k2Udt,
where
(w ,w ) 2
Λ= || 0 1 ||D(A)×V .
(w ,w ) 2
|| 0 1 ||V X
×
NON UNIFORM DECAY OF THE ENERGY 3
(2) Assumethatx x 1(1)isanincreasingfunctionandthereexistsC >0
7→ F− x
such that for all non-identically zero initial data (w ,w ) (A) V, the
0 1
∈D ×
solution φ of (1.5) satisfies:
1
(1.8) ||(w0,w1)||2V×X ≤CZ0G(2C1Λ) kB∗φ′(t)k2Udt.
Then there exists C > 0 such that for all t > 0, the solution w of (1.3)
satisfies:
1
(1.9) E(w(t))≤C||(w0,w1)||2D(A)×VF−1 √t .
(cid:18) (cid:19)
Corollary 1.2. The weak observability i.e. there exist T,C >0 such that for all
∗
(w ,w ) V X the solution φ of (1.5) satisfies
0 1
∈ ×
(w ,w ) 2
(1.10) Z0T ||B∗φ′(t)||2Udt≥C||(w0,w1)||2V×XG|| |0|(w01,w||1X)×||(2VD×(AX12))′,
 
implies in particular (1.7).
The paper is organized as follows: In section 2 we prove our main result and
in the last section we give some applications both in the linear and the nonlinear
case. Decay rates for nonlinear dissipations were obtained under our generalized
observabilityestimate. Here we mentionthatthe literatureis less provide. We cite
essentially [1, 7].
2. Proof of Theorem 1.1
The following lemma will be very useful.
Lemma2.1. Let (resp. )beacontinuouspositivedecreasing(resp. increasing)
H G
real function on [0,+ ). Suppose that is bounded by one and there exists a
∞ H
positive constant c such that
c 1
(2.11) (s) (s) +s , s>0.
H ≤ ( (s)) 2 H −H ( (s)) ∀
G H (cid:18) (cid:18)G H (cid:19)(cid:19)
Suppose that x7→xF(cid:0)−1(x1) is(cid:1)an increasing function, then there exists C >0 such
that for any t>0,
1
(2.12) (t) C −1 .
H ≤ F √t
(cid:18) (cid:19)
The proofis similar to that of Lemma B in [6]. Since our result is more general,
we give it for the reader’s convenience.
Proof. Let t>0. We distinguish two cases:
If
•
1 1
(2.13) ( (s))< , then (s) 1 .
−
G H t H ≤G t
(cid:18) (cid:19)
∗see[2,3,4]formoredetails
4 KA¨ISAMMARI,AHMEDBCHATNIA,ANDKARIMELMUFTI
If
• 1 1
( (s)) , then +s t+s,
G H ≥ t ( (s)) ≤
G H
therefore
1
(2.14) (t+s) +s
H ≤H ( (s))
(cid:18)G H (cid:19)
and we get
(2.15) ( (s)) (s) (t+s).
F H ≤H −H
The inequalities (2.13) and (2.15) give
1
(2.16) (s) 1( (s) (t+s))+ 1 , s,t>0.
− −
H ≤F H −H G t ∀
(cid:18) (cid:19)
We introduce the function Ψ defined on ]0,+ [ by:
t
∞
1
(2.17) Ψ (s)= .
t 1 t + 1 1
F− s G− t
We distinguish two cases: (cid:0) (cid:1) (cid:0) (cid:1)
If (s) (t+s)< t then (s) Ψ (t+s) and we deduce
• H −H t+s H ≤ t
(2.18) Ψ (t+s) (t+s) 1.
t
H ≤
If (s) (t+s)> t . Taking into account that (s) 1 we obtain
• H −H t+s H ≤
t t
(s) ,
t+sH ≤ t+s
so
t t
(2.19) (s) < (s) (t+s),
t+sH ≤ t+s H −H
and we deduce
s
(2.20) (t+s)< (s) .
H H t+s
Consequently,
s
Ψ (t+s) (t+s) < Ψ (t+s) (s)
t H t H t+s
Ψ (t+s) s
= t (s)Ψ (s)
t+s H t Ψ (s)
t
Ψt(t+s)
(2.21) = (s)Ψ (s) t+s .
H t Ψt(s)
s
Using the increasing property of x x 1(1), we obtain
7→ F− x
(2.22) Ψ (t+s) (t+s)<Ψ (s) (s).
t t
H H
We have proved that for all s,t>0, we have either
Ψ (t+s) (t+s) 1orΨ (t+s) (t+s)<Ψ (s) (s).
t t t
H ≤ H H
In particular, we deduce that for any t>0 and n N , either
∗
∈
(2.23) Ψ ((n+1)t) ((n+1)t) 1 or Ψ ((n+1)t) ((n+1)t)<Ψ (t) (nt).
t t nt
H ≤ H H
NON UNIFORM DECAY OF THE ENERGY 5
Hence, we have
(2.24) Ψ ((n+1)t) ((n+1)t) max(1,Ψ (t) (t))=1.
t t
H ≤ H
Therefore, for all t>0 and n N ,
∗
∈
1 1
(2.25) ((n+1)t) −1 + −1 .
H ≤F n+1 G t
(cid:18) (cid:19) (cid:18) (cid:19)
Choosensuchthatn+1 t<n+2andmakeuseagainoftheincreasingproperty
of x x 1(1), we get f≤or all t 2 :
7→ F− x ≥
1 1
(2.26) (t2) −1 + −1 .
H ≤F t G t
(cid:18) (cid:19) (cid:18) (cid:19)
Since 1(x) 1(x) close to zero,the desiredresultfollows immediately. (cid:3)
− −
|G |≤|F |
After, we give the proof of the main result.
Proof of (1). We combine (1.6) and the following formula:
t
(2.27) E(w(t))=E(φ(0)) 2 B φ(t) 2ds t>0,
− k ∗ ′ kU ∀
Z0
to get:
t 1
(2.28) E(φ(0)) 2 B φ(t) 2 ds C 1 (w ,w ) 2 .
− Z0 k ∗ ′ kU ≤ G− (cid:18)t(cid:19)|| 0 1 ||D(A)×V
Take t= 1 , we obtain
( 1 )
G 2CΛ
1
(2.29) E(φ(0))−2Z0G(2C1Λ) kB∗φ′(t)k2Uds≤ 21||(w0,w1)||2V×X.
We deduce that
1
(2.30) (w ,w ) 2 4 G(2C1Λ) (B φ)(t) 2ds.
|| 0 1 ||V×X ≤ Z0 k ∗ ′ kU
Now, let us consider v =φ w, then v satisfies the following system:
−
v (t)+Av(t)+BB v (t)=BB⋆φ(t), t>0,
(2.31) ′′ ∗ ′ ′
(v(0),v (0))=(0,0).
′
(cid:26)
Multiply the first equation of (2.31) by v , and integer by parts to get
′
t t
(2.32) E(v(t))+2 B v (s) 2 ds=2 B φ(s),B v (s) ds.
k ∗ ′ kU h ∗ ′ ∗ ′ iU
Z0 Z0
Make use of Young inequality,
t t
(2.33) E(v(t))+2 B v (s) 2 ds B φ(s) 2 + B v (t) 2 ds.
k ∗ ′ kU ≤ k ∗ ′ kU k ∗ ′ kU
Z0 Z0
Hence, (cid:0) (cid:1)
t t
(2.34) E(v(t))+ B v (s) 2 ds B φ(s) 2ds.
k ∗ ′ kU ≤ k ∗ ′ kU
Z0 Z0
Since
kB∗w′k2U =kB∗φ′−B∗v′k2U ≤2 kB∗φ′k2U +kB∗v′k2U ,
(cid:0) (cid:1)
6 KA¨ISAMMARI,AHMEDBCHATNIA,ANDKARIMELMUFTI
therefore
1 1
(2.35) G(2C1Λ) kB∗w′(t)k2Udt≤2 G(2C1Λ) kB∗φ′k2U +kB∗v′k2U dt.
Z0 Z0
(cid:0) (cid:1)
Thanks to (2.34) and (2.35), we obtain
1 1
(2.36) G(2C1Λ) kB∗w′(t)k2Udt≤4 G(2C1Λ) kB∗φ′(t)k2Udt.
Z0 Z0
Finally, by virtue of (2.30), we conclude the following estimate
1
(2.37) ||(w0,w1)||2V×X ≤16Z0G(2C1Λ) kB∗φ′(t)k2Udt.
(cid:3)
Proof of (2). Using similar arguments as previously, the inequality (1.8) becomes
1
E(w(0))≤2C G(2C1Λ) kB∗w′(t)k2U +kB∗v′(t)k2U dt.
Z0
(cid:0) (cid:1)
Multiply the first equation of (2.31) by v and integer by parts. It follows that
′
t B w (s) 2
(2.38) E(v(t))≤ k ∗ ε′ kU +εkB∗v′(s)k2U ds,∀ε>0.
Z0 (cid:18) (cid:19)
Let T >0 be fixed, for all 0 t T, we have
≤ ≤
T B w (t) 2
sup E(v(t)) ≤ k ∗ ε′ kU +εkB∗v′(t)k2U dt
0≤t≤T Z0 (cid:18) (cid:19)
T B w (t) 2 T
k ∗ ′ kU dt+εC E(v(t))dt
≤ ε
Z0 Z0
T B w (t) 2
k ∗ ′ kU dt+εCT sup E(v(t)).
≤ ε
Z0 0≤t≤T
Choose ε= 1 , so we have
2CT
T
(2.39) sup E(v(t))≤4CT kB∗w′k2Udt.
0≤t≤T Z0
On the other hand, as
kv′(t)k2X ≤E(v(t)),
we integer on [0,T], to get
T T
kv′(t)k2Xdt ≤ E(v(t)) dt
Z0 Z0
T sup E(v(t))
≤
0 t T
≤ ≤
T
4CT2 B w 2dt,
≤ k ∗ ′kU
Z0
so
T T
(2.40) B v 2dt 4C2T2 B w 2dt.
k ∗ ′kU ≤ k ∗ ′kU
Z0 Z0
NON UNIFORM DECAY OF THE ENERGY 7
Hence, for T = 1 we conclude that
( 1 )
G 2CΛ
2 1
(2.41) E(w(0))≤2C1+4C2 G 21C1Λ ! Z0G(2C1Λ) kB∗w′k2Udt.
 (cid:0) (cid:1) 
One can easily verify that
E(w(0))+E(w (0))
Λ ′ :=Λ˜,
≤ E(w(0))
and consequently
2
1
(2.42) E(w(0))≤2C1+4C2G 21C1Λ˜  Z0G(2C1Λ˜) kB∗w′(t)k2Udt.
 
Since Λ˜ is minimized, we get  (cid:16) (cid:17) 
2
1
(2.43) E(w(0))≤C G 21C1Λ˜  Z0G(2C1Λ˜) kB∗w′(t)k2Udt.
 (cid:16) (cid:17)
By translating the time variable and using the formula
t1
E(w(t )) E(w(t ))+ B w (t) 2 dt=0,
1 − 2 k ∗ ′ kU
Zt2
we obtain:
2
E(w(0)) C 1 G(21C1Λ˜)+s kB∗w′(t)k2U dt
E(w(0))+E(w′(0)) ≤ G 2C1Λ˜  Zs E(w(0))+E(w′(0))
 (cid:16) (cid:17)
2 E w 1 +s
1 E(w(s)) ( 1 )
C  (cid:18) (cid:18)G 2CΛ˜ (cid:19)(cid:19).
≤  1  E(w(0))+E(w′(0)) − E(w(0))+E(w′(0))
G 2CΛ˜  
 (cid:16) (cid:17)  
Put (s)= E(w(s)) . 
H 2C(E(w(0))+E(w′(0)))
Make use of the previous inequality and the decay of , it follows that
H
1 2 1
(s) C (s) +s , s>0.
H ≤ ( (s)) H −H ( (s)) ∀
(cid:18)G H (cid:19) (cid:18) (cid:18)G H (cid:19)(cid:19)
Thanks to (2.11) and Lemma 2.1, there exists C such that
E(w(t)) 1
C 1 .
−
E(w(0))+E(∂ w(0)) ≤ F √t
t (cid:18) (cid:19)
We conclude the desired result. (cid:3)
3. Some applications
We give some applications of Theorem 1.1.
8 KA¨ISAMMARI,AHMEDBCHATNIA,ANDKARIMELMUFTI
3.1. The linear case.
3.1.1. Example1. Let begivenby (x)=xp on(0,r ],r >0andp R [ 1,0].
G G 0 0 ∈ \ −2
Then The following two statements are equivalent.
i) ThereexistsC >0suchthatforallnon-identicallyzeroinitialdata(w ,w )
0 1
∈
V X, the solution φ of (1.5) satisfies:
×
CΛp
(w ,w ) 2 16 B φ(t) 2 dt.
|| 0 1 ||V×X ≤ Z0 k ∗ ′ kU
ii) ThereexistsC >0suchthatforallnon-identicallyzeroinitialdata(w ,w )
0 1
∈
(A) V and for all t>0, the solution w of (1.3) satisfies:
D ×
C
E(w(t)) (w ,w ) 2 .
≤ tp|| 0 1 ||D(A)×V
Remark 3.1. In [6], the author construct a geometry with a trapped ray for the
lineardissipativewaveequation(thegeometriccontrolconditionisthennotfulfilled)
and establish a polynomial decay rate when (w ,w ) H2(Ω) H1(Ω) H1(Ω),
0 1 ∈ ∩ 0 × 0
the estimate (1.7) is satisfied for (x)=xδ, δ >0.
G (cid:2) (cid:3)
3.1.2. Example 2. Let G be given by G(x) = exp√(−xx1p) on (0,r0], p ∈ R+. The
following statements hold.
i) TheexistenceofaconstantC >0suchthatthesolutionφof(1.5)satisfies:
1
||(w0,w1)||2H01(Ω)×L2(Ω) ≤16Z0G(2C1Λ) kB∗φ′(t)k2Udt,
implies the existence of a constant C >0 such that for all non-identically
1
zero initial data (w ,w ) (A) V and for all t > 0, the solution w of
0 1
∈ D ×
(1.3) satisfies:
C
E(w(t)) 1 (w ,w ) 2 .
≤ (lnt)p1 || 0 1 ||D(A)×V
ii) TheexistenceofaconstantC >0suchthatforallnon-identically(w ,w )
1 0 1
∈
(A) V and for all t>0, the solution w of (1.3) satisfies:
D ×
C
E(w(t)) 1 (w ,w ) 2 .
≤ (lnt)p1 || 0 1 ||D(A)×V
implies the existence of a constant C >0 such that the solution φ of (1.5)
satisfies:
1
||(w0,w1)||2V×X ≤16 Z0F(2C1Λ) kB∗φ′(t)|2dt.
NON UNIFORM DECAY OF THE ENERGY 9
3.2. The nonlinear case. Let Ω be a bounded connected open set of Rn, n > 1
witha 2boundary∂Ω. LetalsoM = αij Ω¯;Rn n beasymmetric
C 1 i,j n ∈C∞ ×
≤ ≤
anduniformlypositivedefinitematrix.(cid:0)Den(cid:1)oteby∇= Σ(cid:0)nj=1β1j∂x(cid:1)j,...,Σnj=1βnj∂xj
and ∆ = Σn ∂ αij∂ . We deal with the followi(cid:16)ng second order differential(cid:17)
i,j=1 xi xj
equation:
(cid:0) (cid:1)
∂2u ∆u+a(x)g(∂ u)=0, in Ω (0,+ ),
t − t × ∞
(3.44) u=0, on ∂Ω (0,+ ),
 × ∞
(u,∂ u)(.,0)=(u ,u ), in Ω,
 t 0 1

where a=a(x) L (Ω) is a bounded function with a(x) 0 for all x Ω and
∞
g :R R is a cont∈inuousstrictly increasingfunction with g(0≥)=0, sg(s) ∈ 0. We
→ ≥
assume the additional conditions:
(i) r [1, ), c ,c >0, s 1 c sr g(s) c s1/r.
1 2 1 2
∃ ∈ ∞ ∃ | |≤ ⇒ | | ≤| |≤ | |
(ii) k [0,1], p [1, ), c , c >0, s >1 c sk g(s) c sp.
3 4 3 4
∃ ∈ ∃ ∈ ∞ ∃ | | ⇒ | | ≤| |≤ | |
(iii) (n 2)(1 k) 4r and (n 2)(p 1) 1.
− − ≤ − − ≤
For(u ,u ) H1(Ω) L2(Ω),thereexistsauniquesolutionu [0,+ ),H1(Ω)
0 1 ∈ 0 × ∈C ∞ 0 ∩
1 [0,+ ),L2(Ω) . For more regular initial data (u ,u ) H(cid:16)2(Ω) H1(Ω) (cid:17)
C ∞ 0 1 ∈ ∩ 0 ×
H1(cid:16)(Ω),thesolution(cid:17)uhasthefollowingregularityu L 0,+ (cid:2);H2(Ω) H1(Ω)(cid:3)
0 ∈ ∞ ∞ ∩ 0 ∩
W1, 0,+ ;H1(Ω) W2, 0,+ ;L2(Ω) . (cid:16) (cid:17)
∞ ∞ 0 ∩ ∞ ∞
The en(cid:16)ergy of a solut(cid:17)ion is defi(cid:16)ned at instant(cid:17)t 0 by
≥
1
E(u(t))= ∂ u(x,t)2+ u(x,t)2 dx.
t
2 | | |∇ |
ZΩ(cid:16) (cid:17)
E(u(t)) is a non-increasing function of time and satisfies, for all t > t 0 the
2 1
≥
identity
t2
E(u(t )) E(u(t ))= a(x)g(∂ u(x,t))∂ u(x,t)dxdt 0.
2 1 t t
− − ≤
Zt1 ZΩ
Denote by
X(u0,u1)=E(u(0))+E1(u(0))+[E1(u(0))](2p−1)+[E1(u(0))](1+rr−+k1),
where
E (u(0))= (∆u ag(u ),u ) 2 .
1 k 0− 1 1 kL2(Ω)×H01(Ω)
We introduce u the solution of the linear locally damped problem:
l
∂2u ∆u +a(x)∂ u =0, in Ω (0,+ ),
t l− l t l × ∞
(3.45) u =0, on ∂Ω (0,+ ),
l
 × ∞
(u ,∂ u )(.,0)=(u ,u ) H2(Ω) H1(Ω) H1(Ω),
 l t l 0 1 ∈ × 0 ∩ 0
and make the following assumption: (cid:2) (cid:3)
10 KA¨ISAMMARI,AHMEDBCHATNIA,ANDKARIMELMUFTI
(A) Assumethatx x 1(1)isanincreasingfunctionandthereexistsC >0
7→ F− x
suchthatforallnon-identicallyzeroinitialdata(u ,u ) H2(Ω) H1(Ω)
0 1 ∈ × 0 ∩
H1(Ω), the solution φ of
0 (cid:2) (cid:3)
∂2φ(t) ∆φ(t)=0,
(3.46) t −
φ(0)=u , ∂ φ(0)=u ,
0 t 1
(cid:26)
satisfies:
1
(3.47) ||(u0,u1)||2H01(Ω)×L2(Ω) ≤C Z0G(2C1Λr) ZΩa(x)|∂tφ(x,t)|2dxdt,
where
(r 1)+X(u ,u )
Λ = − 0 1 .
r E(u(0))
The following result is deduced from Theorem 1.1 and [7, Proposition 3].
Proposition 3.2. Let (A) holds. There exists c > 0 such that for any (u ,u )
0 1
∈
H2(Ω) H1(Ω) H1(Ω), the solution u of (3.44) satisfies
× 0 ∩ 0
(cid:2)(3.48) E(u(cid:3)(s)) ch((r 1)+X(u0,u1))
≤ −
s+ 1
G(h)
(3.49) +c a(x)g(∂ u(x,t))∂ u(x,t)dxdt,
t t
Zs ZΩ
for any h>0 and any s 0 where
≥
G(h):=Ch(2r+1) (h)4(r+1).
F
We have the following stabilization result for the nonlinear damped wave equa-
tion.
Theorem3.3. Let(A)holdsandsupposethatthereexistsc suchthatthefunction
0
G satisfies G 1(x) c G 1 x(c +1) for all x 0. Then the energy of the
− ≥ c+1 − 0 ≥
solution of (3.44) satisfies the estimate:
(cid:0) (cid:1)
c
(3.50) E(u(t))≤CG−1 t′ (r−1)+X(u0,u1) , for t sufficiently large,
(cid:18) (cid:19)
and all non-identically zero initia(cid:0)l data (u0,u1) H(cid:1)2(Ω) H1(Ω) H1(Ω), the
∈ ∩ 0 × 0
constant C depend on the initial data (u0,u1).
(cid:2) (cid:3)
Proof. Choosing
1
h= ,
2CΛ
r
this implies the existence of a constant c>0 such that
s+ 1
(3.51) E(u(s)) c G(2C1Λr) a(x)g(∂ u(x,t))∂ u(x,t)dxdt.
t t
≤
Zs ZΩ
Denoting by (s)= E(u,s) , we deduce from (3.51) that
H (r−1)+X(u0,u1)
1 1
s+ (s) c (s) s+ ,
H G( (s)) ≤H ≤ H −H G( (s))
(cid:18) H (cid:19) (cid:18) (cid:18) H (cid:19)(cid:19)
which gives
1 c
s+ (s).
H G( (s)) ≤ c+1H
(cid:18) H (cid:19)