## Invariance of attractors

(Here, we only consider global attractors, but pullback attractors are in the same case)

We know that, the continuity of semigroup is needed to prove the invariance of the corresponding global attractor.

For the reaction diffusion equation

$u_t - \Delta u + f(u) = g(x)$ with initial datum $u(0) = u_0\in L^2$,

we know that the corresponding semigroup $\{S(t)\}$ is continuous in $L^2$ and the continuity of $\{S(t)\}$ in $H^1$ is not know.

Now we can prove that $\{S(t)\}$ has an absorbing set in $H^1$ and is asymptotically compact in $H^1$, can we imply the invariance of the global attractor in $H^1$ from the invariance of the global attractor in $L^2$?

How do you think about it?