364 J. Mawhin
D
EFINITION
1. We say that ut is a solution of Eq. 2 if u
∈ C , V
1
∩ C
1
, H
and for each w ∈ V
1
one has d
2
dt
2
ut , w + c
d dt
ut , w + ut, w
1
+ gt, ut, w = 0 7
in the sense of distributions, or, equivalently d
2
dt
2
ut , w + c
d dt
ut , w +
A
12
ut , A
12
w + gt, ut, w = 0 .
D
EFINITION
2. We say that a solution ut of Eq. 2 is bounded or bounded on the whole line if u,
˙u ∈ BC , V
1
× H. We say that a solution ut of Eq. 2 is bounded in the future if, for each t
∈ , one has sup
t ≥t
h |ut|
2 1
+ |Put|
2
+ | ˙ut|
2
i +∞ .
A case in which all the solution of Eq. 2 are bounded in the future is when the equation is dissipative. Among the various notions of dissipativeness which exist for evolution equations
see [3, 7, 8, 9, 16], we use the following one. D
EFINITION
3. The equation 2 is called dissipative if there exists a constant ρ 0 and a map T :
+
→
+
such that, for each M 0, each t ∈ , and each solution ut of 2 with
|ut |
2 1
+ |Put |
2
+ | ˙ut |
2
≤ M , one has
|ut|
2 1
+ |Put|
2
+ | ˙ut|
2
≤ ρ , for all t
≥ T M + t .
3. The Cauchy problem
Under the assumptions of Section 2, let us consider the initial value problem ¨u + c ˙u + Au = f t ,
t ∈ J ,
ut = u
, ˙ut
= v ,
8 where J is a bounded interval in
, f ∈ L
2
J, H , t ∈ J , u
∈ V
1
and v ∈ H. It is well
known that the problem 8 has a unique solution see [14]. A proof can be based on Galerkin’s method, from which, using the classical theory of ordinary differential equations and Gronwall’s
Lemma, one can deduce not only the existence of a unique solution u, ˙u ∈ CJ, V
1
× H of Eq. 8 and its continuous dependence on u
, v and f in the strong topologies of V , H and
L
2
J, H , but also its continuous dependence in the weak topologies. L
EMMA
1. Let ut be the solution of Eq. 8 and let u
n
t be the solution of ¨u + c ˙u + Au = f
n
t , t
∈ J , ut
= u
0n
, ˙ut
= v
0n
,
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
