Bounded solutions 365
where f
n
∈ L
2
J, H . Assume that u
0n
⇀ u
weak in V
1
, v
0n
⇀ v weak in H ,
f
n
⇀ f
weak in L
2
J, H . Then, for each t
∈ J , u
n
t ⇀ ut weak in V
1
, ˙u
n
t ⇀ ˙ut weak in H .
The following lemma is useful to construct Lyapunov functions. It follows from the similar result for the Galerkin approximations, and a limit process.
L
EMMA
2. Let ut be a solution of Eq. 3 and define η
t = c
2
|ut|
2
+ 2c ut, ˙ut + 2 | ˙ut|
2
+ 2|ut|
2 1
. Then η
∈ W
1,1
J ; and
˙ηt = −2c | ˙ut|
2
+ |ut|
2 1
− f t ,
2 c
˙ut + ut in the sense of distributions on J .
Remark that the derivative ˙ηt can be understood in the classical sense and η ∈ C
1
J as soon as f t is continuous.
Let us consider the initial value problem ¨u + c ˙u + Au + gt, u = 0 ,
t ∈ J ,
ut = u
, ˙ut
= v ,
9 where J is a bounded interval in
, t ∈ J , u
∈ V
1
and v ∈ H. Let us assume that A and
gt, u satisfy the hypotheses in Section 2. Under these conditions, the problem 9 possesses a unique solution which is defined in J see e.g. [14]. The following proposition shows its
continuous dependence in the weak topology.
L
EMMA
3. Let ut be the solution of Eq. 9 and u
n
be the solution of the same equation with initial conditions u
n
t = u
0n
, ˙u
n
t = v
0n
. Assume that u
0n
⇀ u
weak in V
1
, v
0n
⇀ v weak in H ,
Then, for each t ∈ J ,
u
n
t ⇀ ut weak in V
1
, ˙u
n
t ⇀ ˙ut weak in H .
4. Dissipativeness
Let us consider the evolution equation 2 and assume that all the hypotheses stated in Section 2 on the operator A and the function g hold. The dissipativeness of 2 will follow from a
semi-coercivity condition upon g.
366 J. Mawhin
T
HEOREM
1. Assume that there exists α, β, γ 0 such that gt, u, u
≥ α|Pu| − β|I − Pu| − γ , 10
for all t, u ∈
× H. Then Eq. 2 is dissipative. Moreover, there exists ρ 0 such that if ut is a solution of Eq. 2 and
|ut |
2 1
+ |Put |
2
+ | ˙ut |
2
ρ
2
for some t ∈ , then
|ut|
2 1
+ |Put|
2
+ | ˙ut|
2
ρ
2
for all t ≥ t
. Proof. The expression
ku, vk := c
2
|u|
2
+ 2cu, v + 2|v|
2
+ 2|u|
2 1
12
, defines a norm in V
1
× H equivalent to the usual one, and can be used in Definition 3. The function
η t :
= kut, ˙utk
2
is differentiable see Lemma 2 and ˙ηt = −2c
| ˙ut|
2
+ |ut|
2 1
+ 2
c gt, ut ,
˙ut + gt, ut, ut .
From the boundedness of g and from inequality 6, we obtain that ˙ηt ≤ −2c
| ˙ut|
2
+ |ut|
2 1
− 2 e
M c
| ˙ut| + α|Put| − β R|ut|
1
− γ .
11 Using the fact that
lim
| x |+|y|+|z|→∞
x
2
+ y
2
− 2 e
M c
|x| + α|z| − β R|y| − γ = +∞ ,
it follows that there exist ρ , δ 0 such that η
t ≥ ρ
2
H⇒ ˙ηt −δ .
12 We deduce from 12 that there exists τ
≥ t such that
τ ≤ t
+ max n
0, δ
− 1
η t
− ρ
2
o ,
and kuτ , ˙uτ k ρ .
Now, we assert that kut, ˙utk ρ ,
for all t τ . Otherwise there must exist t
∗
τ such that
ut
∗
, ˙ut
∗ 2
= ρ
2
and kut, ˙utk
2
ρ
2
, for all t
∈ [τ, t
∗
. In consequence, ˙ηt
∗
≥ 0, a contradiction with 12.
Bounded solutions 367
5. Bounded solutions
We use the results obtained in Section 4 to prove the existence of a solution of Eq. 2 that is bounded on the whole line.
T
HEOREM
2. If Eq. 2 is dissipative, it has a solution ut such that u,
˙u ∈ BC , V
1
× H . 13
Proof. Let u
n
t be the solution of Eq. 2 with initial conditions u
n
−n = 0 , ˙u
n
−n = 0 . By definition, there exists T, ρ 0 such that
|u
n
t |
2 1
+ |Pu
n
t |
2
+ | ˙u
n
t |
2
ρ
2
, 14
for all t ≥ T − n. We can assume, without loss of generality, that there exists u
∈ V
1
and v
∈ H such that u
n
0 ⇀ u weak in V
1
, ˙u
n
0 ⇀ v weak in H .
Let ut be the solution of 2 with initial conditions u0
= u ,
˙u0 = v .
Lemma 3 applies and we obtain for each t ∈
u
n
t ⇀ ut weak in V
1
, ˙u
n
t ⇀ ˙ut weak in H.
Moreover it follows from 14 that |ut|
2 1
+ |Put|
2
+ | ˙ut|
2
≤ ρ
2
, for all t
∈ , and therefore 13 holds. We make a first use of Theorem 2 to prove the existence of a bounded solution of the linear
equation 3 where f ∈ BC , H, a problem studied in [6, 14] when λ
1
0. In the resonant case λ
1
= 0, an additional hypothesis is required. We treat both cases in the following Theo- rem 3, whose proof requires a result of Ortega [12] on second order linear ordinary differential
equations, that we include here for completeness. L
EMMA
4. Let p : →
be continuous and c 6= 0. Then the equation
y
′′
t + cy
′
t = pt
15 has a bounded solution if and only if p
∈ B P , . Proof. Necessity. Let y be a bounded solution of Eq. 15 i.e. y and y
′
are bounded on , and
set Pt
= Z
t
ps ds . 16
368 J. Mawhin
Then y
′
t − y
′
+ c[yt − y0] = Pt , and so P is bounded.
Sufficiency. Let p ∈ B P , and consider the equation
u
′
t + cut = Pt ,
17 where P is defined in 16. By a classical result see e.g. [4], equation 17 has a unique bounded
solution U . From the equation, we see immediately that U
′
is also bounded. As P ∈ C
1
, U
∈ C
1
, and satisfies the differential equation 15. T
HEOREM
3. If λ
1
0, all solutions of Eq. 3 are bounded in the future and Eq. 3 has a solution ut which satisfies u,
˙u ∈ BC , V
1
× H. If λ
1
= 0, the same statement is valid if and only if
P f ∈ B P , ker A .
18 Proof. If λ
1
0, condition 10 with P = 0 holds for gt, u = − f t taking α = 1, β =
sup
t ∈
| f t|, γ = 1. Consequently, Theorems 1 and 2 apply. If λ
1
= 0, and m = dim ker A, let e H :
= span{ϕ
m+1
, ϕ
m+2
, . . . } be the orthogonal comple-
ment of ker A. The restriction e A of the operator A to e
H ∩ DA is positive definite and we can
apply the first assertion to deduce that the equation ¨u + c ˙u + e
Au = I − P f t
has a bounded solution e
ut which satisfies e
u, ˙ e
u ∈ BC
, e V
1
× e H
where e V
1
= V
1
∩ e H is endowed with the norm
| · |
1
. On the other hand, by Lemma 4, the equation
¨u + c ˙u = P f t 19
in the finite dimensional space ker A has a bounded solution, denoted by u t , if and only if
P f ∈ B P , ker A. Now, the function ut = u
t + ˜ut is a solution of Eq. 3 which
satisfies 13. In addition, all the solutions of the autonomous equation ¨u + c ˙u + Au = 0
are bounded in the future and this implies that all the solutions of Eq. 3 are also bounded in the future. Conversely, if Eq. 3 has a bounded solution ut , then Put is a bounded solution of
Eq. 19. Because condition 18 is both necessary and sufficient for the existence of a bounded solutions of Eq. 19, we deduce that 18 holds.
6. Nonlinear telegraph equation
