196
Since ~ is Frechet differentiable, this yields
dy
D~(y) dt + A~(y) = QR(y + ~(y)),
(2.39)
and, with (1.25),
D~(y)( -Ay + PRey + ~(y))) + A~(y) = QR(y + ~(y)).
(2.40)
Equation (2.40) is valid for y = yet}, V t ~ O. In particular for t = 0,
D~(yo)( -Ayo + PR(yo + ~(Yo)) + A~(yo) = QR(yo + ~(Yo)).
(2.41)
Since Yo E P E is arbitrary, (2.41) is in fact a semilinear hyperbolic equation (in infinite
dimension, with n variables), satisfied by ~.(1)
Now let u be any solution of (1.25) and let y = Pu, z = Qu j Uo does not belong
necessarily to M. We show that this orbit is attracted by M by comparing it to the
companion curve t -+ yet) + ~(y(t)) lying on M (see Section 2.2). We observe that y
and z satisfy (1.16) and (1.17) and we write
d
dt(z - ~(y)) = -Az + QR(y + z) - D~(y)(-Ay + PRey + z))
= (with (2.41))
= -A(z - ~(y)) + QR(y + z) - QR(y +
By the variation of constants formula,
z(t) - ~(y(t)) = e-At(zo - ~(Yo)) + 1t e-A(t-'>Q[R(y(s) + z(s)) - R(y(s) + ~(y(s)))
- D~(y(s))(PR(y(s) + z(s)) - PR(y(s) + ~(y(s))))Jds.
We use Dcp E gt, (1.12) (at t = 0), and (1.4) :
ID
:::; kll,x°IR(y(s) + z(s)) - R(y(s) +
:::; k1M1l((t - s)-O + AO)lz(s) - ~(y(S))IE'
(1) This equation called Sacker's equation has been used also to construct inertial
manifolds. Here we use it only to show that M is exponentially attracting.
Since ~ is Frechet differentiable, this yields
dy
D~(y) dt + A~(y) = QR(y + ~(y)),
(2.39)
and, with (1.25),
D~(y)( -Ay + PRey + ~(y))) + A~(y) = QR(y + ~(y)).
(2.40)
Equation (2.40) is valid for y = yet}, V t ~ O. In particular for t = 0,
D~(yo)( -Ayo + PR(yo + ~(Yo)) + A~(yo) = QR(yo + ~(Yo)).
(2.41)
Since Yo E P E is arbitrary, (2.41) is in fact a semilinear hyperbolic equation (in infinite
dimension, with n variables), satisfied by ~.(1)
Now let u be any solution of (1.25) and let y = Pu, z = Qu j Uo does not belong
necessarily to M. We show that this orbit is attracted by M by comparing it to the
companion curve t -+ yet) + ~(y(t)) lying on M (see Section 2.2). We observe that y
and z satisfy (1.16) and (1.17) and we write
d
dt(z - ~(y)) = -Az + QR(y + z) - D~(y)(-Ay + PRey + z))
= (with (2.41))
= -A(z - ~(y)) + QR(y + z) - QR(y +
By the variation of constants formula,
z(t) - ~(y(t)) = e-At(zo - ~(Yo)) + 1t e-A(t-'>Q[R(y(s) + z(s)) - R(y(s) + ~(y(s)))
- D~(y(s))(PR(y(s) + z(s)) - PR(y(s) + ~(y(s))))Jds.
We use Dcp E gt, (1.12) (at t = 0), and (1.4) :
ID
:::; kll,x°IR(y(s) + z(s)) - R(y(s) +
:::; k1M1l((t - s)-O + AO)lz(s) - ~(y(S))IE'
(1) This equation called Sacker's equation has been used also to construct inertial
manifolds. Here we use it only to show that M is exponentially attracting.