Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 724815, 36 pages
doi:10.1155/2011/724815
Research Article
Dynamic Analysis of a Nonlinear
Timoshenko Equation
Jorge Alfredo Esquivel-Avila
Departamento de Ciencias Básicas, Análisis Matemático y sus Aplicaciones, UAM-Azcapotzalco,
Avenida San Pablo 180, Col. Reynosa Tamaulipas, 02200 México, DF, Mexico
Correspondence should be addressed to Jorge Alfredo Esquivel-Avila, jaea@correo.azc.uam.mx
Received 8 February 2011; Accepted 28 April 2011
Academic Editor: Norimichi Hirano
Copyright q 2011 Jorge Alfredo Esquivel-Avila. 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 characterize the global and nonglobal solutions of the Timoshenko equation in a bounded
domain. We consider nonlinear dissipation and a nonlinear source term. We prove blowup of
solutions as well as convergence to the zero and nonzero equilibria, and we give rates of decay
to the zero equilibrium. In particular, we prove instability of the ground state. We show existence
of global solutions without a uniform bound in time for the equation with nonlinear damping. We
define and use a potential well and positive invariant sets.
1. Introduction
We consider
utt kΔ2 u − M ∇u22 Δu gut fu in Ω,
1.1
with initial conditions
ux, 0 u0 ,
ut x, 0 v0 ,
x ∈ Ω,
1.2
and with one set of the following boundary conditions:
u 0,
Δu 0 on ∂Ω,
1.3
u 0,
∂u
0
∂ν
1.4
or
on ∂Ω,
2
Abstract and Applied Analysis
where Ω ⊂ Rn is a bounded domain with sufficiently smooth boundary, · 2 is the norm in
L2 Ω,
M s2 α βs2γ ,
α ≥ 0, β ≥ 0, α β > 0, γ ≥ 1, k 1,
gut δut |ut |λ−2 ,
fu μu|u|r−2 ,
δ > 0, λ ≥ 2,
μ > 0, r > 0.
1.5
1.6
1.7
When the source term f ≡ 0, there is a considerable set of works studying several properties
of equation 1.1, see for instance, the early papers by Ball 1, 2
, Haraux and Zuazua 3
, and
the books by Hale 4
, Haraux 5
, and references therein. For a destabilizing source term,
sfs > 0, s ∈ R \ {0}, in the works of Payne and Sattinger 6
, Georgiev and Todorova 7
,
and Ikehata 8
, qualitative properties of 1.1 are studied, when k 0 β. To understand
the dynamics of second-order equations in time, similar to 1.1, active research is reported in
Alves and Cavalcanti 9
, Barbu et al. 10
, Cavalcanti et al. 11–16
, Rammaha 17
Rammaha
and Sakuntasathien 18
, and Todorova and Vitillaro 19
, Vitillaro 20
. For the Timoshenko
equation, with g ≡ 0, Bainov and Minchev 21
gave sufficient conditions for the nonexistence
of smooth solutions of 1.1, with negative initial energy, and gave an upper bound of the
maximal time of existence. For positive and sufficiently small initial energy, blowup and
globality properties are characterized in Esquivel-Avila 22
. For the Kirchhoff equation,
that is, 1.1 with k 0, the nonexistence of global solutions is studied in 23
. In 24, 25
,
we characterized properties such as blowup and asymptotic behavior of solutions, for 1.1
with k 0 and β 0. To the knowledge of the author, such problems are still open for
the Timoshenko equation 1.1. Here, we want to give some results about the dynamics of
problem 1.1. To do that we will generalize the concept of the depth of the potential well in
such manner that our results of the dynamics be as sharp as the ones in 24, 25
. Furthermore,
for particular cases, our definition of depth of the potential well will coincide with the one
introduced in 6
.
2. Preliminaries and Framework
We begin this section with an existence, uniqueness, and continuation theorem for 1.1. The
proof is similar to the ones in 7, 8
, where semilinear wave equations are studied.
Theorem 2.1. Assume that r > 2 and r ≤ 2n − 2/n − 4 if n ≥ 5. For every initial data u0 , v0 ∈
H ≡ B × L2 Ω, where B is defined either by B ≡ H 2 Ω ∩ H01 Ω, or B ≡ H02 Ω, there exists a
unique (local) weak solution ut, vt ≡ Stu0 , v0 of problem 1.1, that is,
d
vt, w2 Δut, Δw2 M ∇ut22 ∇ut, ∇w2 gvt, w 2 fut, w 2 ,
dt
2.1
a.e. in 0, T and for every w ∈ B ∩ Lλ Ω, such that
u ∈ C0, T ; B ∩ C1 0, T ; L2 Ω ,
v ≡ ut ∈ Lλ 0, T × Ω.
2.2
Abstract and Applied Analysis
3
Here, St denotes the corresponding semigroup on H, generated by problem 1.1, and ·, ·2
is the inner product in L2 Ω.
The following energy equation holds:
E0 Et t
0
δvτλλ dτ,
2.3
where
Et ≡ Eut, vt ≡
Ju ≡
1
vt22 Jut,
2
2.4
1
1
1
au cu − bu,
2
r
2 γ 1
2.5
with
au ≡ u2B ,
bu ≡ μurr ,
2γ1
cu ≡ β∇u2
.
2.6
Here, E0 ≡ Eu0 , v0 is the initial energy, and · q denotes the norm in the Lq Ω space.
If the maximal time of existence TM < ∞, then Stu0 , v0 → ∞ as t TM , in the norm of
H:
u, v2H ≡ u2B v22 ≡ Δu22 α∇u22 v22 .
2.7
In that case, from 2.3–2.6, utr → ∞ as t TM .
Now, we define, respectively, the stable potential well and unstable sets:
W ≡ Iu > 0
∪ {0} ∩ Ju < d
,
V ≡ Iu < 0
∩ Ju < d
,
2.8
where
Iu ≡ au cu − bu.
2.9
Here, Iu < 0
denotes the set of u ∈ B with that property, and the depth of the
potential well is defined as follows:
d≡
r − 2 r/r−2
S
,
2r
2.10
4
Abstract and Applied Analysis
where
√
S ≡ inf
0/
u∈B
bu>0
κ1 ≡
r−2 γ 1
,
r − 2 γ 1
,
bu
≡ bu κ2 cu.
au ≡ au κ1 cu,
with
au1/2
1/r
bu
κ2 ≡ κ1 − 1 −rγ
.
r − 2 γ 1
2.11
2.12
2.13
We assume that r ≥ 2γ 1, and since γ ≥ 1, then κ1 ∈ 0, 1/2, and κ2 ∈ −1, −1/2.
Also note that if r 2γ 1, then κ1 0, κ2 −1, and we have the following characterization
of the depth of the potential well 2.10-2.11:
d inf supJλu,
0/
u∈B λ≥0
2.14
which is the definition given in 6
, where a nondissipative nonlinear wave equation is
studied.
Consider any u ∈ B, r > 2, and r ≤ 2n/n − 4 if n ≥ 5, then
au ≥ CΩbu2/r κ1 cu ≥ CΩbu2/r ,
2.15
where CΩ > 0, is any constant in the Sobolev-Poincaré’s inequality
Δu22 α∇u22
1/2
≥
CΩμ1/r ur .
2.16
Moreover, if bu
> 0, from 2.15 and since bu ≥ bu,
2/r .
au ≥ CΩbu
Hence, S ≥ CΩ, and d ≥ D ≡ r − 2/2rCΩr/r−2 > 0.
If ue denotes any nonzero equilibria of equation 1.1,
E ≡ 0/
ue ∈ B : Δ2 ue − M u22 Δue fue ,
2.17
2.18
then, by 2.1 in Theorem 2.1 with ut ue w, we get that ue belongs to the Nehari
manifold, N, that is,
E ⊂ N ≡ 0
Iu 0 ,
/ u ∈ B : Iu
where Iu
≡ au − bu.
2.19
Abstract and Applied Analysis
5
e aue > 0. Furthermore, from 2.17 which is an equality when
Consequently, bu
CΩ S, we conclude that the Nehari manifold can be represented by the line: y x, in the
plane with axes x bu
and y au, beginning at the point: y x Sr/r−2 2r/r −2d.
We also note that
Ju 1
1
au − bu.
2
r
2.20
From these facts it follows that the depth of the potential well 2.10 is characterized
by
d inf Ju u∈N
r−2
,
2r
2.21
where
0 < ≡ inf au inf bu.
u∈N
u∈N
2.22
Hence, any equilibrium is such that ue ∈ Ju ≥ d
. Moreover, like in 6
, the set of
extremals of 2.21 is characterized by set of equilibria with least energy, that is the ground
state
e .
N∗ ≡ ue ∈ E : Jue d
ue ∈ E : aue bu
2.23
Observe that Ju d is a tangent line to the curve defined by the equality in 2.17 with
> 0. On the other hand, we notice that
CΩ S, at the point N∗ , which holds if bu
κ1 bu
− κ2 au κ1 bu − κ2 au > 0,
2.24
and is equal to zero if and only if au 0 bu. Hence, if bu
< 0, then
r−2 γ 1
au >
bu.
−rγ
Therefore, next results about the stable and unstable sets follow.
Lemma 2.2. The following properties of V and W hold:
i W is a neighborhood of 0 ∈ B.
/ V.
ii 0 ∈
/ Iu < 0
(closure in B), in particular 0 ∈
2.25
6
Abstract and Applied Analysis
iii W W ∪ W− ∪ {0}, where
2
r−2
2/r
r−2/r , 0 < bu < ,
W ≡ W ∩ bu > 0 bu ≤ au < bu r
r
2.26
r−2 γ 1
2
r
−
2
, −γ < bu
<0 .
W− ≡ W ∩ bu
bu
< au < bu
<0 ⊂
−rγ
r
r
r−2 2/r ≤ au < 2 bu
, bu > .
iv V r−2/r bu
r
r
∗
e .
v N W ∩ V W ∩ V ue ∈ N, aue bu
vi W Iu < 0
c ∩ Ju < d
, V Iu > 0
∪ {0}c ∩ Ju < d
.
The following result follows easily like in 23
.
Lemma 2.3. One has that
r − 2
r−2
au >
bu,
2r
2r
r−2 γ 1
r−2
au Ju >
cu
2r
2r γ 1
r−2 γ 1
γ
> au bu,
2 γ 1
2r γ 1
Ju >
2.27
2.28
for any u ∈ B such that Iu > 0, in particular if 0 / u ∈ W, and
r−2
r − 2
au <
bu,
2r
2r
r−2 γ 1
r−2
au d <
cu
2r
2r γ 1
r−2 γ 1
γ
< au bu,
2 γ 1
2r γ 1
d<
2.29
2.30
for any u ∈ B, such that Iu < 0, in particular if u ∈ V .
A set V ⊂ H is positive invariant, with respect to problem 1.1, if the corresponding
generated semigroup St on H is such that
StV ⊂ V.
2.31
Lemma 2.4. Let u, v denote any solution of 1.1, given by Theorem 2.1. Then, the sets
are positive invariant.
S ≡ Eu, v < d
∩ u, v ∈ H : u ∈ W
,
2.32
U ≡ Eu, v < d
∩ u, v ∈ H : u ∈ V ,
2.33
Abstract and Applied Analysis
7
Proof. First, we show that S is positive invariant. In order to do that, we take u0 , v0 ∈ S.
Then, by 2.4, Jut ≤ Eut, vt ≤ E0 < d, for any t ≥ 0. Now, if S is not positive
invariant, there exists some t > 0, such that Iut 0, with ut /
0. Then, by 2.21, d ≤
Jut. But this is impossible because Jut < d. The proof of the positive invariance of U
is quite similar. Indeed, if this is not true there exists some t > 0, such that Iut 0. From
ii of Lemma 2.2 ut /
0, and this implies the same contradiction as before.
Next result gives an interpretation of sets S and U and follows from Lemma 2.2.
Lemma 2.5. The sets S and U have the properties
r−2/r S ⊂ Eu, v < d
∩ u, v ∈ H : 2/r
bu
2
r−2
, 0 < bu < ,
≤ au < bu r
r
r−2 γ 1
2
r
−
2
, −γ < bu
<0 ,
∪ Eu, v < d
∩ u, v ∈ H :
bu
< au < bu
−rγ
r
r
r−2 2/r ≤ au < 2 bu
, bu > ,
U Eu, v < d
∩ u, v ∈ H : r−2/r bu
r
r
2.34
S ∩ U ue , 0 ∈ H : ue ∈ N∗ e .
ue , 0 ∈ H : ue ∈ N, aue bu
2.35
The following result is a direct consequence of vi in Lemma 2.2 and Lemma 2.4.
Lemma 2.6. For every solution of 1.1, only one of the following holds:
i there exists some t0 ≥ 0 such that ut0 , vt0 ∈ S, and remains there for every t > t0 ,
ii there exists some t0 ≥ 0 such that ut0 , vt0 ∈ U, and remains there for every t > t0 ,
iii ut, vt ∈ Eu, v ≥ d
for every t ≥ 0.
Hence, we notice that the sets S and U play an important role in the dynamics of
1.1. Moreover, we will prove that any solution eventually contained in S converges to the
zero equilibrium. If enters in U, either blowups in a finite time or it is global but without a
uniform bound in H for every t ≥ 0, in the case that λ > 2, in 1.6. Also, we will prove that
any solution with ut, vt ∈ Eu, v ≥ d
, for every t ≥ 0, is bounded and converges to the
set of nonzero equilibria E.
We will need the following inequalities to show blowup and convergence to the zero
equilibrium, respectively, in the dissipative case.
1,1
R be a nonnegative function such that
Lemma 2.7. Let F ∈ Wloc
Ḟt ≥ CFa t
a.e. for t ≥ 0,
with a > 1 and C > 0.
Then, there exists some T ∗ > 0 such that limtT ∗ Ft ∞.
2.36
8
Abstract and Applied Analysis
Proof. Define Gt ≡ F1−a t, then
Ġt ≤ 1 − aC < 0
a.e. for t ≥ 0.
2.37
Hence, 0 < G0 1 − aCt, which is only possible if t < T ∗ ≡ 1/Ca − 1F1−a 0.
1,1
Lemma 2.8. Let F ∈ Wloc
R be a nonnegative function such that
Ḟt ≤ −CFa t a.e. for t ≥ 0,
2.38
−1/a−1
,
Ft ≤ F0 1 tCa − 1F0a−1
2.39
Ft ≤ F0 e−Ct .
2.40
with a ≥ 1 and C > 0.
Then, for t ≥ 0, if a > 1
and, if a 1
˙
Proof. Consider a > 1, and notice that F1−a t
≥ a − 1C. Then, we integrate and obtain
the first inequality. Now, let a → 1, and the second one follows.
3. Timoshenko Equation
Due to our assumptions on r and γ, we restrict our analysis to dimensions n ≤ 5. Indeed,
since γ ≥ 1, 2γ 1 < r and r ≤ 2n − 2/n − 4, if n ≥ 5, then our analysis considers, n 5
whenever γ < 2. We also notice that in any case we do not consider the interval 2 < r ≤ 4.
Moreover, r ≤ 6 whenever n 5. We begin with a characterization of blowup when δ > 0 and
λ ≥ 2.
Theorem 3.1. Let ut, vt Stu0 , v0 be a solution of problem 1.1, and suppose that r >
2γ 1. A necessary and sufficient condition for nonglobality, blowup by Theorem 2.1, is that λ < r
and there exists t0 ≥ 0 such that ut0 , vt0 ∈ U.
Proof. Sufficiency
By Lemma 2.4, ut, vt ∈ U for all t > t0 .
Now, we consider the function defined, along the solution, by
Vt ≡ d − Et,
3.1
and notice that because of energy equation 2.3,
Vt ≥ d − E0 ≡ V0 > 0,
where, now E0 ≡ Eut0 , vt0 .
3.2
Abstract and Applied Analysis
9
Notice that from 2.29 in Lemma 2.3,
Vt ≤ d − Jut
≤d−
−
1
r
d but
r−2
r
3.3
γ
1
2
d but −
cut.
r−2
r
r − 2 γ 1
We will need some estimates. First, we notice that from energy equation in terms of
Vt and 3.3,
δ ut, vt|vt|λ−2 ≤ δutλ vtλ−1
λ
2
≤ CΩδutr vtλ−1
λ
k
λ−1
≤ CΩδut1−k
r utr vtλ
1
λ
kλ
≤ CΩδut1−k
νut
vt
r
r
λ
Cν
1
< CV 1−k/r t νδutkλ
V̇t ,
r Cν
3.4
where k ∈ 1, r/λ, C ≡ CΩr/μ1−k/r , CΩ > 0 is the constant in the continuous
embedding Lr Ω ⊂ Lλ Ω, Cν > 0, and ν > 0 will be chosen later.
Consider a positive number q to be chosen later, from 3.2-3.3, we obtain
q−2 γ 1
q−2
r−q
aut but
−Iut −qJut cut 2
r
2 γ 1
q−2 γ 1
r−q
q−2
≥ qV0 − d aut but.
cut 2
r
2 γ 1
3.5
If V0 ≥ d, we choose q ≡ 2γ 1, and from 3.5 we get
r−2 γ 1
but.
−Iut ≥
r
3.6
If V0 < d, then we notice that from 3.2-3.3,
r − 2V0 − d
V0 − d ≥
r − 2V0 2d
γ
1
but −
cut .
r
r − 2 γ 1
3.7
10
Abstract and Applied Analysis
Hence and from 3.5, we have the estimate
−Iut ≥
q−2
aut
2
q−2 γ 1
q
2γd − V0 cut
q
r − 2V0 2d
2 γ 1
q r − q r − 2d − V0 −
but.
r
q
r − 2V0 2d
3.8
In this case, we choose the number q so that the coefficient of cut in 3.8 be equal
to zero, then
2 γ 1 r − 2V0 2d
q≡ .
r − 2 γ 1 V0 2 γ 1 d
3.9
We note that 2 < q < 2γ 1, and we get
γraut r − 2 γ 1 but
−Iut ≥ V0 .
r − 2 γ 1 V0 2 γ 1 d
3.10
Therefore, from 3.6 and 3.10,
−Iut ≥ Cbut,
3.11
where
≡
C
r−2 γ 1
rV0
min 1, > 0.
r
r − 2 γ 1 V0 2 γ 1 d
3.12
Now, we define the function, along the solution, by
Ft ≡ V 1/a t ut, vt2 ,
where a ≡ 1 1 − k/r−1 ∈ 1, 2 and > 0 will be choosen later.
3.13
Abstract and Applied Analysis
11
We intend to apply Lemma 2.7 to functional 3.13. First, we calculate the derivative,
along solutions, with respect to t. Let us start with the second term of 3.13. From 3.2–3.4
and 3.11, one has
d
ut, vt2 vt22 − Iut − δ ut, vt|vt|λ−2
2
dt
r
1−k/r
≥ vt22 Cμut
t νδutkλ
r − CV
r kλ−r/r Ca 1/a
− νCδ r
V̇ t
Cμ
V0b utrr −
μ
Cν
≥
vt22
1
V̇t
Cν
≥ vt22 3.14
Cμ
Ca 1/a
V̇ t,
utrr −
2
Cν
where b ≡ kλ − 1 − r − 1/r < 0, and ν > 0 is sufficiently small.
Consequently, if > 0 is sufficiently small,
vt22 utrr > 0,
Ḟt ≥ C
3.15
≡ min1, Cμ/2
where C
> 0.
From 3.15 and choosing > 0 small enough, we get
Ft ≥ F0 ≡ V0 ut0 , vt0 2 > 0.
3.16
Utilizing two times 3.3, we get
Fa t ≤ 2a−1 Vt a |ut, vt2 |a
r
a
a
a
a−1 μ
a
≤2
utr CΩ utr vt2
r
r
2a/2−a
2
a
a−1 μ
a
vt2
≤2
utr CΩ utr
r
μ
μr − 2 c
≤ 2a−1 utrr a CΩa
utrr vt22
r
2rd
≤ C utrr vt22 ,
3.17
where CΩ > 0 is the imbedding constant of Lr Ω ⊂ L2 Ω, c ≡ 1 − 2a/r2 − a > 0, and
C > 0.
Hence and from 3.15, we obtain the inequality in order to apply Lemma 2.7.
Therefore, the maximal time of existence is finite: T < ∞.
12
Abstract and Applied Analysis
Necessity
Suppose that λ ≥ r. Define the function, along the solution, by
Wt ≡ Et 2μ
utrr .
r
3.18
Then,
Ẇt Ėt 2μ ut|ut|r−2 , vt
2
≤ −δvtλλ δ
vtrλ Cutrr
2
3.19
≤ CWt
1,
≡ maxδ/2, Cr/μ, C ≡ 2μCΩδ/2r , and CΩ > 0 is the imbedding constant of
where C
λ
L Ω ⊂ Lr Ω.
Hence, by Gronwall inequality, it follows that u, v is bounded in H for any finite
time. A contradiction.
Proceeding again by contradiction suppose that, for all t ≥ 0, ut, vt ∈
/ U. Then, by
Lemma 2.6, we have either ut ∈ S for all t ≥ 0, or Et ≥ d for all t ≥ 0. In the first case, from
2.28 in Lemma 2.3
E0 ≥
1
r−2
aut,
vt22 2
2r
3.20
that is, ut, vt is bounded in H. This is not possible. In the second case,
t
δ
0
vτλλ dτ ≤ E0 − d,
3.21
where E0 ≡ Eu0 , v0 . Hence, by the Hölder inequality,
δt
λ
t
vτdτ ≤ E0 − d,
0
3.22
utλ ≤ CT ,
3.23
−λ−1 λ
and consequently
for t ∈ 0, T , where CT ≡ u0 λ E0 − d/δ1/λ T λ−1/λ .
From Theorem 2.1,
lim utr ∞;
tTMAX
3.24
Abstract and Applied Analysis
13
hence, by Sobolev-Poincaré’s inequality 2.16, for every M > E0 , there exists some t > 0, such
that
M<
r−2
aut,
2r
3.25
for every t ≥ t. This implies the first inequality of 2.29 in Lemma 2.3, replacing d by M.
Now, we consider the function 3.13
Ft ≡ Vt1/a ut, vt2 ,
3.26
defined for t ≥ t, where a ∈ 1, 2, > 0 is sufficiently small, and here
Vt ≡ M − Et > 0,
3.27
and repeat the sufficiency part of the proof. Then, by Lemma 2.7, Ft blowups as t T ∗ ,
T ∗ > t. Moreover, for t ≤ t < T ∗ ,
F t
Ft ≥ 1/a−1 ,
1 − t − t / T ∗ − t
3.28
hence and from 3.26, 3.27, and since Et ≥ d,
⎧ ⎛
⎞ ⎫
⎪
⎨ t
⎬
2 2 ⎪
F t
⎜
⎟
2
1/a
−
V
τ
ut2 ≥ u t ⎝
⎠dτ
1/a−1
⎪
2
⎪
⎩ t
⎭
1 − τ − t / T ∗ − t
2 2 t − t
≥ u t −
M − d1/a
2
⎧
⎫
−2−a/a−1
⎬
t − t
2a − 1 ∗ ⎨
1−
F t T −t
−1 .
⎩
⎭
2 − a
T ∗ − t
3.29
Consequently,
lim ut22 ∞.
tT ∗
3.30
But this contradicts 3.23, since Lλ Ω ⊂ L2 Ω. The proof is complete.
Remark 3.2. From the last result, if λ 2 and r > 2γ 1, any solution of problem 1.1,
ut, vt, is global if and only if either i there exists t0 ≥ 0 such that ut0 , vt0 ∈ S or
ii ut, vt ∈ Eu, v ≥ d
, for every t ≥ 0. On the other hand, if λ > 2 and r > 2γ 1,
then any solution is global if and only if one of the following holds: i, ii, or iii λ ≥ r and
there exists t0 ≥ 0 such that ut0 , vt0 ∈ U.
14
Abstract and Applied Analysis
We next prove a characterization of convergence to the zero equilibrium, and we give
rates of decay.
Theorem 3.3. Let ut, vt Stu0 , v0 be a solution of problem 1.1 with λ ≥ 2. Suppose that
r > 2γ 1 and that λ ≤ 10, if n 5. A necessary and sufficient condition for ut, vt → 0, 0,
strongly in H as t → ∞, is that there exists t0 ≥ 0 such that ut0 , vt0 ∈ S.
In this case, if Ft denotes either the energy
Et ≡ Eut, vt,
3.31
ωt ≡ ut, vt2H ≡ aut vt22 ,
3.32
or the norm of the solution in H
One has the rates of decay, for t ≥ T ,
'
(−2/λ−2
λ−2
λ−2/2
Ft ≤ K0 1 t
,
K1 K0
2
3.33
and, for linear dissipation, λ 2,
Ft ≤ K0 e−K1 t ,
3.34
where T > 0 is sufficiently large, and K0 > 0, K1 > 0 are constants depending only on initial
conditions.
Proof. Necessity
By ii in Lemma 2.2, 0, 0 ∈
/ U, and since the equilibrium 0, 0 ∈
/ Eut, vt ≥ d
, strong
closures in H, then, by Lemma 2.6, the solution must eventually enter in S.
Sufficiency
By energy equation and 2.27 in Lemma 2.3, the solution must be global and uniformly
bounded in the norm of H, that is ωt < 2rd/r − 2, for any t ≥ 0. Hence, there exists a
u, v weakly in
sequence of times, {tn }, such that if n → ∞ then tn → ∞, utn , vtn → u. Also, notice that the
H and, since the embedding B ⊂ Lr Ω is compact, butn → b
energy is such that
0 ≤ E∞ ≡ lim Et inf Et < ∞.
t→∞
L Ω,
t≥0
3.35
Consequently, from the energy equation and the continuous embedding Lλ Ω ⊂
2
t1
lim
t→∞
t
vτλ2 dτ 0,
3.36
Abstract and Applied Analysis
15
in particular, for any sequence of times {sn } such that sn → ∞ as n → ∞,
1
lim
n→∞
hn τdτ 0,
3.37
0
where hn τ ≡ vsn τλ2 , for τ ∈ 0, 1
. By Fatou Lemma,
lim inf vsn τλ2 lim inf hn τ 0,
n→∞
n→∞
3.38
for a.e. τ ∈ 0, 1
, and by the weak convergence to v ,
v2 ≤ lim inf vtn 2 0,
n→∞
3.39
where we choose {sn } such that tn sn τ0 , for some τ0 ∈ 0, 1
.
It can be shown that the semigroup generated by problem 1.1 is continuous in H
with the weak topology, and then that the weak limit set is positive invariant, see Ball 26
.
Consequently u, v ue , 0 must be an equilibrium of 1.1. Furthermore, by the lowersemicontinuity of the norm in H, one has
e aue ≤ lim inf vtn 2 autn bu
2
'
n→∞
2
lim 2Etn but
n n→∞
r
(
3.40
2
2E∞ bu
e .
r
Hence,
r − 2
bue ≤ E∞ < d.
2r
3.41
Then, by 2.19 and 2.21, ue 0, and
lim ut, vt 0, 0
t→∞
weakly in H.
3.42
Strong convergence follows if we get the rates of decay in our statement. Here, we will
adapt the technique used in Haraux and Zuazua 3
, to 1.1. That technique is based on the
construction of suitable Liapunov functions defined along solutions and the application of
Lemma 2.8. One of them is the energy, and we will need one more, defined by
Wt ≡ Et κEtλ/2−1 ut, vt2 ,
3.43
16
Abstract and Applied Analysis
where κ > 0 is a constant to be chosen later. We next prove that Wt is equivalent to both, the
energy Et and the norm ωt of the solution, in the sense of 2.30 and 2.28 below. First
we note that from 2.27 in Lemma 2.3,
K
r−2
ωt ≤ Et ≤ ωt,
2r
2
3.44
where K − 1 ≡ β/αγ 12rd/αr − 2γ .
Also, notice that from 3.43,
|Wt − Et| ≤ κE0λ/2−1 ut2 vt2 ≤ κE0λ/2−1 C1 Ωωt,
3.45
where C1 Ω > 0 is a constant that depends on the continuous embedding B ⊂ L2 Ω. Hence
and from 3.44, if κ is sufficiently small, then
1
3
Et ≤ Wt ≤ Et.
2
2
3.46
We will need the following estimate:
δ ut, vt|vt|λ−1 ≤ δutλ vtλ−1
λ
2
≤ δC2 Ωautλ−2/2λ aut1/λ vtλ−1
λ
2r
E0
≤ δC2 Ω
r−2
≤
λ−2/2λ
λ−1/λ
aut1/λ −Ėt
3.47
1
Ėt,
aut − C
λ
where we applied 3.44 in the third step and Young inequality in last step, and the constants
> 0 depend on the continuous embedding B ⊂ Lλ Ω, and C
also depends on
C2 Ω > 0, C
E0 .
It follows that, by 3.42 and since B ⊂ Lr Ω is compact, for any > 0, there exists
some T > 0 such that for any t > T
but ≤ C3 Ωbutr−2/r aut ≤ Et,
3.48
where C3 Ω > 0 is the corresponding embedding constant and we used 3.44 in the last
step.
Abstract and Applied Analysis
17
Since we will apply Lemma 2.8, we need to calculate the time derivative of 3.43 and
we begin with
d
ut, vt2 vt22 − aut − cut but − δ ut, vt|vt|λ−1
2
dt
1
Ėt Et
≤ vt22 − aut − cut − C
2
≤
3.49
3
Ėt,
vt22 − 1 − Et − C
2
which holds for any t > T , and where we used 3.47, 3.48 and definition of Et.
We notice that for any small η > 0, and by Young inequality and energy equation
3
Eλ/2−1 t vt22 ≤ Eλ/2−1 tC4 Ωvt2λ ≤ ηEλ/2 t − C η Ėt,
2
3.50
where C4 Ω > 0, Cη > 0 depend on the continuous embedding Lλ Ω ⊂ L2 Ω, and Cη
depends on η.
Then, for and η sufficiently small, 3.49 and 3.50, imply
Eλ/2−1 t
d
1
Ėt,
ut, vt2 ≤ − Eλ/2 t − C
dt
2
3.51
≡ Cη CE
λ/2−1 .
for any t > T , where C
0
Consequently, for κ sufficiently small and any t > T
Ẇt ≤ Ėt − κ
1
λ−2
Ėt ≤ − κ Eλ/2 t ≤ −κ0 W λ/2 t,
rC1 ΩE0λ/2−1 Ėt − κ Eλ/2 t C
r−2
2
2
3.52
where κ0 ≡ κ/22/3λ/2 and C1 Ω > 0 is the constant in 3.45; also we used 3.44, the
fact that the energy is decreasing and 3.46. Then, from 3.52 and Lemma 2.8, we obtain the
desired rates of decay for Wt. The result now follows by 3.46 and 3.44, and the proof is
complete.
Remark 3.4. By 2.35, the ground state is: u, 0 ∈ H : u ∈ N∗ S ∩ U. Then, in any Hneighborhood of that subset of nonzero equilibria, one can choose initial conditions either
in U or in S. Hence, by Theorem 3.1 and 3.3, the ground state is unstable in the sense of
Liapunov when the dissipation term gut is either linear or nonlinear.
Next we will study the behavior of solutions such that ut, vt ∈ Eu, v ≥ d
for
all t ≥ 0. First, we prove that those solutions are uniformly bounded in time. To that end we
will study the cases: λ 2 and λ > 2 separately, First, we consider the case λ 2.
Theorem 3.5. Let ut, vt Stu0 , v0 be a solution of problem 1.1. Assume that r > 2γ 1,
and λ 2. Also, assume that r ≤ 26/n 1 if n ≥ 2. If ut, vt ∈ Eu, v ≥ d
for all t ≥ 0,
then the solution is global and uniformly bounded in H, for all t ≥ 0.
18
Abstract and Applied Analysis
Proof. Suppose that ut, vt is not global, then by Theorem 2.1 blowups and by
Theorem 3.1, ut0 , vt0 ∈ U for some t0 ≥ 0. Hence, ut, vt ∈ Eu, v < d
for all
t ≥ t0 . A contradiction.
Next, we will prove that ut2 is uniformly bounded for all t ≥ 0.
Let Ft ≡ 1/2ut22 − C, where C > 0 is the constant given below. Then, we obtain
F̈t δḞt vt22 − aut but
r2
r−2
aut − rEt
vt22 2
2
3.53
≥ CΩr − 2Ft,
where CΩ > 0 is the imbedding constant of B ⊂ L2 Ω, and C ≡ rE0 /r − 2CΩ.
We define Wt ≡ F t ≡ sup{Ft, 0}, the positive part of Ft. We claim that, along
solutions of 1.1, the time derivative satisfies Ẇt ≤ 0. Indeed, if this is no the case, there
exists some t0 > 0 such that
Ft0 > 0,
Ḟt0 > 0.
3.54
By a standard comparison result for ordinary differential equations, 3.53 and 3.54
imply that Ft → ∞ as t → ∞. Consequently, for any constant C > E0 , there exists some
t0 > 0, such that for t ≥ t0
C<
r−2
aut.
2r
3.55
This is 2.29 in Lemma 2.3, replacing d by C. If we now define, for t ≥ t0 , the function
Vt ≡ C − Et,
3.56
we can repeat the sufficiency part of the proof of Theorem 3.1 and show that the solution
blowups in a finite time, consequently is nonglobal. A contradiction. Then, ut2 ≤ C < ∞,
for all t ∈ R , and some constant C > 0.
Next, we will prove that uniform boundedness of ut2 implies uniform boundedness of ut, vt in H, for all t ∈ R . To that end, we consider the functions Ht ≡
Gt − kE0 ≡ Ḟt δFt − kE0 , where now Ft ≡ 1/2ut22 and k > 0 is defined below.
From the second line in 3.53,
Ḣt ≥ KHt,
3.57
where K ≡ min{r 2, αCΩr − 2/1 δ} > 0 and k ≡ r/K.
Hence, for 0 ≤ s ≤ τ,
Hτ ≥ HseKτ−s ,
3.58
Abstract and Applied Analysis
19
and consequently, from definition of Ht, for 0 ≤ s ≤ τ ≤ t,
Ft Fse−δt−s t
Hτ kE0 eδτ−t dτ
s
≥ Fse−δt−s t
HseKτ−s kE0 eδτ−t dτ
3.59
s
≥
kE Hs Kt−s
0
e
1 − eδs−t .
− eδs−t δK
δ
Notice that if Hs > 0, for some s ≥ 0, we obtain from 3.59 that limt → ∞ Ft ∞. A
contradiction. Then, for all t ≥ 0,
Gt ≤ kE0 .
3.60
0 , and like in 3.57
Now, we define Lt ≡ Gt kE
L̇t ≥ −KLt,
3.61
≡ min{r 2, CΩr − 2} > 0 and k ≡ r/K.
Hence,
where K
Lt ≥ L0e−Kt ≥ min{L0, 0},
3.62
0 .
Gt ≥ min G0, −kE
3.63
and consequently,
Hence and from 3.60, Gt is uniformly bounded in time.
We integrate the second line of 3.53 in terms of Gt and, by the energy equation, we
obtain
r − 2 γ 1 t1
1
cuτ dτ − rE0 .
Gt 1 − Gt ≥
vτ22 auτ 2
γ 1
t
3.64
Hence and since Gt is uniformly bounded in time,
t1
ωτdτ ≤ C,
3.65
t
where C > 0 is a constant, and
ωt ≡
1
1
1
1
cut Et but.
vt22 aut 2
2
r
2 γ 1
3.66
20
Abstract and Applied Analysis
Next, we will show that there exists a constant κ > 0, such that
ωt ≤ κωs 1,
3.67
ω̇t vt, fut 2 − δvt22 ≤ μvt2 utr−1
2r−1 .
3.68
for any 0 ≤ s ≤ t ≤ s 1.
To this end, we calculate
If n 1, we integrate and obtain, for 0 ≤ s ≤ t ≤ s 1, that
ωt ≤ ωs μ
t
s
μ
≤ ωs 2
≤ ωs μ
2
vτ2 uτr−1
2r−1 dτ
t
s
2r−1
vτ22 uτ2r−1 dτ
s1
s
2r−1
uτ2r−1 dτ μ
3.69
t
ωτdτ.
s
By Gronwall inequality and 3.65,
μ
2r−1
ωt ≤ ωs uL2r−1 Ω×s,s1 eμ
2
⎧
s1
r−1 ⎫
⎨
⎬
ωs ≤C
uτ22 α∇uτ22 vτ22 dτ
⎩
⎭
s
⎧
⎨
ωs ≤C
⎩
3.70
r−1 ⎫
⎬
s1
ωτdτ
⎭
s
ωs Cr−1 ,
≤C
> 0, C
> 0 depend on the continuous embeddings B ⊂ L2 Ω, and H 1 Ω×s, s1 ⊂
where C
2r−1
Ω × s, s 1.
L
If 2 ≤ n ≤ 5, we use Galiardo-Niremberg’s inequality,
r−1a
ur−1
2r−1 ≤ CΩuB
r−11−a
u2
,
3.71
where CΩ > 0, and a nr − 2/4r − 1. Notice that a < 1 if 2 ≤ n ≤ 4, and a ≤ 1 if n 5,
because r ≤ 2n − 2/n − 4 6. Then, from
ω̇t ≤ μvt2 utr−1
2r−1 ,
3.72
Abstract and Applied Analysis
21
we integrate and apply Gronwall inequality, for 0 ≤ s ≤ t ≤ s 1,
ωt ≤ ωs μ
t
s
vτ2 uτr−1
2r−1 dτ
≤ ωs μCΩ
t
s
≤ ωs μCΩ
r−1a
vτ2 uτB
t
s
)
r−1a−1
ωτuτB
≤ ωs exp μCΩ
t
s
r−11−a
uτ2
r−11−a
uτ2
dτ
dτ
3.73
*
r−1a−1
r−11−a
dτ
uτB
uτ2
) t
*
r−1a−1
uτ
≤ ωs exp C
dτ ,
s
B
r−11−a
≡ μCΩsup ut
where C
.
t≥0
2
Notice that r − 1a − 1 ≤ 2 because by hypothesis r ≤ 26/n 1, then we use the
Hölder inequality, and from 3.65 we get
⎧ {r−1a−1}/2 ⎫
t
⎬
⎨
ωt ≤ ωs exp C
uτ2B dτ
⎭
⎩
s
⎧ {r−1a−1}/2 ⎫
s1
⎬
⎨
ωτdτ
≤ ωs exp C
⎭
⎩
s
3.74
{r−1a−1}/2 .
≤ ωs exp CC
Then 3.67 holds for any n ≥ 1, under our assumptions on r.
Consequently, 3.65 and 3.67 imply that
vt22 ut2B ≤
t
2ωtds
t−1
≤ 2κ
t
ωs 1ds
3.75
t−1
≤ 2κC 1,
and the proof is complete.
Next, we consider the case λ > 2. Due to our assumptions on r, we restrict our analysis
to nγ < 4. Since γ ≥ 1, our analysis considers, at most, dimensions n ≤ 3, whenever γ < 4/3.
Theorem 3.6. Let ut, vt Stu0 , v0 be a solution of problem 1.1. Assume that r > 2γ 1, and λ > 2 with λ ≤ 2n 1/n − 1 if n ≥ 2. Also assume that r < 24/n 1 for n ≥ 1. If
22
Abstract and Applied Analysis
ut, vt ∈ Eu, v ≥ d
for all t ≥ 0, then the solution is global and uniformly bounded in H, for
all t ≥ 0.
Proof. Globality follows like in last Theorem. Suppose that
ut22 ≤ K,
3.76
for some constant K > 0, and every t ≥ 0.
We recall Galiardo-Nirenberg’s inequality
r1−a
urr ≤ CΩauar/2 u2
,
3.77
where CΩ > 0 is a constant, a n/2r − 2/2r, and a ∈ 0, 1
.
Hence and from 3.76 in the energy equation, we obtain, for any time t ≥ 0,
E0 ≥
1
1
ar/2
,
vt22 aut − Caut
2
2
3.78
≡ K r1−a/2 CΩμ/r, and therefore ut, vt, t ≥ 0, is uniformly bounded in H,
where C
since ar < 2 if and only if r < 24/n 1. This implies, since r > 2γ 1, that nγ < 4.
Now, we prove 3.76. First, we notice that from energy equation,
t
δ
0
vτλλ dτ ≤ E0 − d.
3.79
λ
t
vτdτ ≤ E0 − d,
0
3.80
utλ ≤ Ct,
3.81
Hence, by Hölder inequality,
−λ−1 δt
λ
and then
for every t ≥ 0, where Ct ≡ u0 λ E0 − d/δ1/λ tλ−1/λ .
Next, we define
2ωt ≡ ut, vt2H ,
3.82
Abstract and Applied Analysis
23
and obtain the following estimate for t ∈ 0, T , and T > 0 finite and arbitrary:
t t
δ uτ, |vτ|λ−2 vτ dτ ≤ δuτλλ vτλ−1
λ dτ
0
2
0
≤
t
0
uτλλ dτ
1/λ t
λ−1/λ
λ
δ
vτλ dτ
0
≤ C0 uLλ Ω×0,T ≤C
t
uτ22
0
≤C
t
3.83
α∇uτ22
vτ22
1/2
dτ
1/2
ωτdτ
.
0
Here, we used the Hölder inequality, the energy equation and the fact that Et ≥ d,
> 0, C
> 0 depend on the continuous embedding
also C0 ≡ E0 − dλ−1/λ δ1/λ > 0, and C
1
λ
also depends
H Ω × 0, T ⊂ L Ω × 0, T , valid for λ > 2 and λ ≤ 2n 1/n − 1 if n ≥ 2. C
on the embedding B ⊂ L2 Ω.
Now, we define the function
Ft ≡
1
ut22 − κ,
2
3.84
where κ > 0 is the constant given below. Then, the second derivative is
F̈t vt22 − aut − cut − δ ut, |vt|λ−2 vt but
2
r−2 γ 1
r2
r−2
2
aut cut − δ ut, |vt|λ−2 vt − rEt
vt2 2
2
2
2 γ 1
≥ r − 2ωt − δ ut, |vt|λ−2 vt − rE0 .
2
3.85
If we integrate 3.85, and we use 3.83 and that ωt ≥ Et ≥ d, we obtain
Ḟt ≥ Ḟ0 r − 2
t
ωτdτ − C
0
t
t
1/2
− rE0 t
ωτdτ
0
)
C
≥ Ḟ0 r − 2 ωτdτ 1 −
+
r − 2 Tm d
0
*
− rE0 t
24
Abstract and Applied Analysis
Ḟ0 r − 2
2
t
ωτdτ − rE0 t
0
r − 2CΩ
≥ Ḟ0 2
t
Fτdτ,
0
3.86
+
− 2 > 0, CΩ > 0 is the embedding constant of
for t ∈ Tm , T , where Tm d ≡ 2C/r
2
B ⊂ L Ω, and κ ≡ 2rE0 /r − 2CΩ > 0.
Now, we define for every t ≥ 0,
Gt ≡
t
Fτdτ 0
Ḟ0
,
C
3.87
where C ≡ r − 2CΩ/2. Hence, 3.86 has the form
G̈t − CGt ≥ 0,
3.88
for every t ∈ Tm , T .
We define Ht ≡ F t ≡ sup{Ft, 0}, the positive part of Ft. We claim that, for
every t > Tm , the time derivative satisfies Ḣt ≤ 0. Otherwise, there exists some t0 ∈ Tm , T such that
Ft0 > 0,
Ḟt0 > 0,
3.89
Ġt0 > 0,
G̈t0 > 0.
3.90
that is,
By a standard comparison result for ordinary differential equations, 3.88 and 3.90
imply that
Ft Ġt ≥ M0 e
√
Ct
,
3.91
for t ∈ t1 , T and some t1 ∈ t0 , T , where M0 > 0 depends on Ft0 √and Ḟt0 . Furthermore,
for any constant M > 0, there exists some t2 ∈ t1 , T , such that M0 e Ct > M1 t2λ−1/λ , for
t ∈ t2 , T . Then, 3.91 contradicts 3.81, since Lλ Ω ⊂ L2 Ω. Consequently, 3.76 holds
with K ≡ max{max0≤t≤Tm ut22 , 2κ}, and the proof is complete.
From Remark 3.2 and Theorems 3.1, 3.3, and 3.5, any global solution of 1.1, with
λ 2, is necessarily uniformly bounded in H, for all t ≥ 0. However, this is false when λ > 2,
as we show next.
Theorem 3.7. Let ut, vt Stu0 , v0 be a solution of problem 1.1. Assume the conditions,
on r and λ, made in Theorem 3.6. Then, ut, vt is global and not uniformly bounded in H, for
t ≥ 0, if and only if λ ≥ r and there exists t0 ≥ 0 such that ut0 , vt0 ∈ U.
Abstract and Applied Analysis
25
Proof. Sufficiency
By Lemma 2.4, ut, vt ∈ U, for every t ≥ t0 . This solution must be global. Otherwise, by
Theorem 2.1, blowups in a finite time and, by Theorem 3.1, necessarily λ < r. A contradiction.
Now, suppose that, for t ≥ t0 , the solution is uniformly bounded in H. Hence, there
exists a sequence of times {tn }, such that if n → ∞, then tn → ∞, utn , vtn →
u, v weakly in H and, since the imbedding B ⊂ H 1 Ω ∩ Lr Ω is compact, but
n →
u. Moreover, since the energy is nonincreasing and bounded, E∞ ≡ limt → ∞ Et ∈ R.
b
Consequently, from the energy equation and since Lλ Ω ⊂ L2 Ω,
t1
lim
t→∞
t
vτλ2 dτ 0,
3.92
in particular, for any sequence of times {sn } such that sn → ∞ as n → ∞,
1
lim
n→∞
hn τdτ 0,
3.93
0
where hn τ ≡ vsn τλ2 , for τ ∈ 0, 1
. By Fatou Lemma,
lim inf vsn τλ2 lim inf hn τ 0,
n→∞
n→∞
3.94
for a.e. τ ∈ 0, 1
, and by the weak convergence vtn → v , in L2 Ω,
v2 ≤ lim inf vtn 2 0,
n→∞
3.95
where we choose {sn } such that tn sn τ0 , for some τ0 ∈ 0, 1
.
It can be shown that the semigroup generated by problem 1.1 is continuous in H
with the weak topology, and then that the weak limit set is positive invariant; see Ball 26
.
Consequently u, v ue , 0 must be an equilibrium of 1.1. Since utn , vtn ∈ U then,
by definition 2.33 and 2.29,
2rd
e lim but
.
bu
n ≥
n→∞
r−2
3.96
26
Abstract and Applied Analysis
Since the norm in H is weak lower-semicontinuous, from 2.4, 2.19, and 2.20, we
get that
e aue bu
≤ lim inf vtn 22 autn n→∞
'
(
2
lim 2Etn butn n→∞
r
3.97
2
2E∞ bu
e .
r
Hence,
r − 2
bue ≤ E∞ < d.
2r
3.98
This contradicts 3.96. Then, the solution cannot be uniformly bounded in H, for all t ≥ 0.
Necessity
This follows from Lemma 2.6 and Theorems 3.1, 3.3, and 3.6. The proof is complete.
Next, we characterize the convergence to the set of nonzero equilibria of equation 1.1.
Due to our assumptions on r and γ, our result considers, at most, dimensions n ≤ 3 for λ > 2,
and n ≤ 5 for λ 2.
Theorem 3.8. Let ut, vt Stu0 , v0 be a solution of problem 1.1. If λ > 2, assume
the conditions, on r and λ, made in Theorem 3.6. If λ 2, and assume the conditions, on r
made in Theorem 3.5. Then, ut, vt → E∞ , strongly in H as t → ∞, if and only if
ut, vt ∈ Eu, v ≥ d
for all t ≥ 0, where E∞ ≡ ue , 0 ∈ E : Jue E∞ ≥ d
and
E∞ ≡ limt → ∞ Eut, vt.
Proof. Sufficiency
By Theorems 3.5 and 3.6, the solution is global and uniformly bounded in H, that is, ωt ≡
ut, vt2H ≤ K, for all t ≥ 0, and some constant K > 0. Then, like in the sufficiency
part of the proof of Theorem 3.7, there exists a sequence of times such that tn → ∞ and
utn , vtn → ue , 0, weakly in H, as n → ∞, where ue , 0 is an equilibrium. If
{ut, vt}t≥0
is precompact in H,
3.99
then strong convergence to ue , 0 follows. In this situation, utn , vtn converges to the set
E∞ , because 0, 0 ∈
/ Eu, v ≥ d
, strong closure in H.
Abstract and Applied Analysis
27
Haraux 5
, developed a technique to prove precompactness of bounded orbits of
some kind of semilinear wave equations. We will follow that method to prove 3.99. To this
end we define, for every > 0, and t ≥ 0,
u t ≡ ut − ut,
v t ≡ vt − vt,
2ω t ≡ u t, v t2H ,
3.100
and note that from 2.1, we get the energy equation for u t, v t
ω 0 ω t t
τ, v τ 2 dτ,
g τ − f τ − m
0
3.101
where,
g t ≡ gvt − gvt,
f t ≡ fut − fut,
2γ
m ∇ut22 ≡ β∇ut2 .
m
t ≡ m ut 22 Δut − m ut22 Δut,
3.102
However, in order to handle the nonlinearity m
t, we need to introduce the function
1 W t ≡ ω t m ∇ut 22 ∇u t22 .
2
3.103
Hence, the corresponding energy equation for W t is
W 0 W t t
0
g τ − f τ − m τ, v τ 2 dτ −
t
0
n τ, u τ2 dτ,
3.104
where,
m t ≡ m ∇ut 22 − m ∇ut22 Δut,
n t ≡ m ∇ut 22
3.105
Δut , vt 2 Δu t.
> 0,
Notice that since the solution is uniformly bounded by K, there exists a constant K
depending on K, such that
t.
ω t ≤ W t ≤ Kω
3.106
We will prove that for any η > 0, there exists η > 0, such that
ω t ≤ η,
for every t ≥ 0, and ∈ 0, η, that is, t → ut, vt ∈ H, is uniformly continuous.
3.107
28
Abstract and Applied Analysis
For every t ≥ 0, we have one of the following two cases:
W t 1 ≤ W t,
3.108
W t 1 > W t.
3.109
From 3.109 and 3.104,
0 > W t − W t 1 t1
g τ − f τ − m τ, v τ 2 dτ −
t
t1
t
n τ, u τ2 dτ.
3.110
Notice that, by a well-known inequality,
g t, v t
2
≥ 22−λ δv tλλ .
3.111
Also, we recall the inequality
f t ≤ σrμ |ut |r−2 |ut|r−2 |u t|,
3.112
where σr 1, if r ∈ 2, 3
and σr r − 1/2, if r > 3.
From the Hölder inequality, 3.112 yields
t1
t
f τ2 dτ
2
1/2
≤ 2σrμCΩsup autr−2/2
t≥0
×
t1
t
3.113
1/2r−1
2r−1
u τ2r−1 dτ
,
where CΩ > 0, is an embedding constant in B ⊂ L2r−1 Ω.
We claim that t → ut ∈ L2r−1 Ω must be uniformly continuous. Otherwise, there
exists some η0 > 0, and sequences {n }n≥1 , {tn }n≥1 , such that n → 0, and tn → ∞, as n → ∞,
and
un tn 2r−1 > η0 ,
3.114
Abstract and Applied Analysis
29
for every n ≥ 1. By assumption, B ⊂ L2r−1 Ω is compact, then {utn n }n≥1 , {utn }n≥1 are
precompact in L2r−1 Ω, and we can extract subsequences {utn n }n≥1 , {utn }n≥1 , such
that for some fixed n0 , sufficiently big, and every n ≥ n0 ,
utn n − u tn ≤ utn n − utn0 n0 2r−1
2r−1
utn0 n0 − utn0 2r−1
utn0 − utn 2r−1
≤
3.115
η0 η0 η0
η0 .
3
3
3
This contradicts 3.114. Hence, for any η > 0 there exists some η > 0, such that for
every t ≥ 0, and every ∈ 0, η,
t1
t
1/2r−1
2r−1
u τ2r−1 dτ
≤ η4λ−1 .
3.116
Consequently, from 3.113, 3.116 and the Hölder inequality
t1
1/λ
t1 λ
4λ−1
,
f τ, v τ 2 dτ ≤ Cη
v τλ dτ
t
t
3.117
where C > 0 depends on K, μ, r, CΩ and the inclusion Lλ Ω ⊂ L2 Ω.
Now notice that,
m t2 ≤ sup m ∇ut22 ∇ut 22 − ∇ut22 Δut2
t≥0
sup m ∇ut22 |∇ut ∇ut, ∇ut − ∇ut2 |Δut2
t≥0
≤ sup m ∇ut22 Δut Δut2 Δut2 u t2
3.118
t≥0
≤ CKu t2 ,
and that
|n t, u t2 | ≤ m ∇ut 22 Δut 2 vt 2 Δu t2 u t2
≤ CKu t2 ,
where CK > 0 is a constant.
3.119
30
Abstract and Applied Analysis
Since B ⊂ L2 Ω is compact, we show like in 3.116, that
t1
t
1/2
u τ22 dτ
≤ η4λ−1 .
3.120
Consequently, from 3.118–3.120 and the Hölder inequality, we obtain
⎫
⎧
1/λ
⎬
⎨ t1
t1
4λ−1
λ
1 ,
{m τ, v τ2 n τ, u τ2 }dτ ≤ Cη
v τλ dτ
t
⎭
⎩ t
3.121
> 0 depends on K and the inclusion Lλ Ω ⊂ L2 Ω.
where C
From 3.117, 3.121, and 3.111 in 3.110, we have
t1
t
4λ−1
v τλλ dτ ≤ Cη
⎧
⎨ t1
⎩
t
1/λ
v τλλ dτ
⎫
⎬
1 ,
⎭
3.122
> 0 is a constant. Consequently, for η sufficiently small, we obtain
where C
t1
t
t1
t
v τλλ dτ ≤ η3λ ,
3.123
η2
.
5
3.124
v τ22 dτ ≤
We apply inequality 3.112 to g , and by the Hölder inequality we get
t1
t
g τ, u τ 2 dτ ≤ σλδ
t1
t
v τλλ dτ
1/λ t1
t
1/λ
u τλλ dτ
⎡
λ−2/λ t1
λ−2/λ ⎤
t1
⎦.
×⎣
vτ λλ dτ
vτλλ dτ
t
t
3.125
Notice that by 2.3 and since Et ≥ d,
∞
0
vtλλ dt ≤
E0 − d
.
δ
3.126
Abstract and Applied Analysis
31
By assumption B ⊂ Lλ Ω, then
t1
t
1/λ
u τλλ dτ
≤ 2CΩ sup utB ≤ C,
3.127
t≥0
where C > 0 depends on the embedding constant CΩ and K.
Therefore, from 3.123, 3.126, and 3.127 in 3.125, we get, for η sufficiently small
t1 E0 − d λ−2/λ 3 η2
η ≤ .
g τ, u τ 2 dτ ≤ 2σλδC
t
δ
5
3.128
By 3.113, 3.116, 3.120 and the Hölder inequality, for small η, we obtain
t1
t
η2
f τ, u τ 2 dτ ≤ Cη8λ−1 ≤ .
5
3.129
one has that
t, u t2 −m ∇ut 22 ∇u t22
m
m ∇ut 22 − m ∇ut22 Δut, u t2
3.130
≤ CKu t2 ,
where CK depends on K. Hence, by 3.120 and the Hölder inequality, and again for small
η,
t1
t
τ, u τ2 dτ ≤
m
η2
.
5
3.131
From 2.1, one has the identity
d
v t, ut 2 − v t22 Δu t22 α∇u t22 g t, u t 2
dt
m
t, u t2 f t, u t 2 .
3.132
Hence, 3.124, 3.128, 3.129, and 3.131 in 3.132 yield
t1
t
u τ2B dτ ≤ 2 sup{v t2 u t2 } t≥0
4η2
.
5
3.133
32
Abstract and Applied Analysis
For every t ≥ 0, one has
√
u t2 ≤ sup vs2 ≤ 2K,
s∈t,t1
3.134
√
v t2 ≤ 2 sup vt2 ≤ 2 2K
t≥0
then, for ∈ 0, η, with η sufficiently small, 3.133 is
t1
t
u τ2B dτ ≤ η2 .
3.135
Hence, in case 3.109, from 3.124 and 3.135, we conclude that
t1
ω τdτ ≤
3η2
.
5
3.136
W τdτ ≤
3η
.
5
3.137
t
And, from 3.106 with η small,
t1
t
From 3.104, 3.111, 3.117, 3.121 and 3.124, we have, for any s ∈ t, t 1
, t ≥ 0,
and η sufficiently small, that
W t 1 W s t1
s
m τ f τ − g τ, v τ 2 dτ t
0
n τ, u τ2 dτ
t1
t1 ≤ W s f τ, v τ 2 dτ m τ, v τ2 dτ t
t
t1
n τ, u τ2 dτ t
≤ W s 3.138
2η
.
5
Therefore, by 3.137, we obtain
W t 1 ≤
t1
W sds t
2η
≤ η.
5
3.139
Consequently, in both cases, 3.108 and 3.109,
0
1
W t 1 ≤ max η, W t ,
3.140
Abstract and Applied Analysis
33
and then
'
(
W t ≤ max η, max W s ,
s∈0,1
3.141
for any t ≥ 0.
Since the solution u, v : 0, 1
→ H, is uniformly continuous, for any η > 0 there
exists some η > 0, such that
max W s ≤ η,
s∈0,1
3.142
for any ∈ 0, η. Then, 3.141 and 3.106 imply 3.107 for any t ≥ 0, η > 0, and 0 < <
η ≡ min{
η, η}.
Next, we will prove that the orbit {ut, vt}t≥0 , is a precompact subset of H. We
start with {vt}t≥0 ⊂ L2 Ω.
Notice that because of 3.107
1 t
1 t
vτdτ ≤
vt − vτ2 dτ
vt −
t
t
2
≤ sup vt − vτ2
3.143
τ∈t,t
≤
2η.
Since {ut}t≥0 is bounded in B, then
t
1
1
vτdτ ≤ ut − utB
t
B
≤
2
sup utB
t≥0
≤
2√
2K.
3.144
2 t
Consequently, {1/ t vτdτ}t≥0 is precompact or, equivalently, totally bounded in
L2 Ω, because B ⊂ L2 Ω is compact. Hence, by 3.143, {vt}t≥0 is precompact in L2 Ω.
Like in 3.143, from 3.107, we obtain that
1 t
uτdτ ≤ sup ut − uτB ≤ 2η.
ut −
t
τ∈t,t
B
3.145
34
Abstract and Applied Analysis
2 t
If {Δ2 t uτdτ}t≥0 is precompact in B ≡ dual space of B, then precompactness of
{ut}t≥0 in B follows from 3.145, since
−1
: B −→ B,
3.146
L−1 ≡ Δ2
is a linear and continuous operator.
According to the dense and continuous inclusions
∗
B ⊂ Lλ Ω ⊂ L2 Ω ⊂ Lλ Ω ⊂ B ,
3.147
where Lλ Ω Lλ Ω , λ∗ λ/λ − 1, we extend the inner product in L2 Ω to the duality
product in B × B. Now, we integrate the wave equation, and since L : B → B is closed, we
get, in the sense of B ,
t
t 2
uτdτ vt − vt M ∇uτ22 Δuτdτ
Δ
∗
t
−
t
t
gvτdτ 3.148
t
fuτdτ.
t
t
By the Hölder inequality and 3.126,
t
t
1/λ
gvτdτ ≤ δ
E0 − dλ−1/λ .
vτλ−1
λ dτ ≤ δ
∗
t
t
3.149
λ
Boundedness of ut in B, since B ⊂ L2r−1 Ω, yields the estimate
t
t
fuτdτ ≤ μ
uτr−1
2r−1 dτ
t
t
2
≤ μCΩsuputB
3.150
t≥0
≤ μCΩCK.
Also,
t 2
M ∇uτ2 Δuτdτ ≤ sup M ∇ut22 utB ≤ CK,
t
t≥0
3.151
2
Therefore, 3.149–3.151 in 3.148 imply that
t
2
uτdτ ≤ C,
Δ
∗
t
3.152
λ
for some constant C > 0 and every t ≥ 0.
B ⊂ Lλ Ω is compact by assumption. By Schauder Theorem, see for instance Brézis
∗
27
, B ⊂ Lλ Ω is compact if and only if Lλ Ω ⊂ B is compact. Then, 3.152 implies that
2
t
{Δ2 t uτdτ}t≥0 is precompact in B .
Abstract and Applied Analysis
35
Necessity
Suppose that the solution converges, strongly in H, to the set E∞ . Since the energy is
nonincreasing, Et ≥ E∞ ≡ limt → ∞ Et ≥ d, for all t ≥ 0, and the proof is complete.
Remark 3.9. By Theorem 3.3 and 2.35, every H-neighborhood of the ground state, S ∩ U ue , 0 ∈ H : ue ∈ N∗ , is connected, through an orbit, with 0, 0. Furthermore, if E∞ d in
Theorem 3.8, the ground state attracts every solution such that ut, vt ∈ Eu, v ≥ d
, for
every t ≥ 0.
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