Apologize in advance for the length of this question, I need some help in clearing some things up that I haven't quite got my head around yet. It seems to be easy to find things out about finite dimensional gradient systems but as we move into the infinite dimensional case all I can find are cryptic and occasionally contradictory scraps.
If a parabolic PDE \begin{equation} u_t = F(x,u,\nabla u,\Delta u,...) \,\, \mbox{in} \,\,\mathbb{R}^N \end{equation} is a gradient system (i.e. it possesses a Lyapunov Function $E$ such that $\frac{\mathrm{d}}{\mathrm{d}t}E \leq 0$ on bounded orbits, which I believe is the correct definition in the PDE setting), then we know that the attractor consists of stationary states corresponding to solutions of the elliptic equation $F(x,u,...)=0$. Is this correct?
The only explicit example I've found in a textbook is the semilinear heat equation \begin{equation} u_t = \Delta u + u^p \end{equation} which has Lyapunov Function \begin{equation} E(t) = ||\nabla u ||_{L^2}^2 - \frac{1}{p+1}||u||_{L^{p+1}}^{p+1} \end{equation} So that $\frac{\mathrm{d}}{\mathrm{d}t}E = -||u_t||_{L^2}^2\leq0$.
However, I believe this is considered on bounded domains. Is this significant? In at least one paper I've read recently it's been explicitly stated that this equation is not a gradient system in $\mathbb{R}^N$ (which would tally with other things I've heard, specifically that the semilinear heat equation in higher dimensions can have nonstationary asymptotic behaviour). In contrast, the fourth order equation \begin{equation} u_t = \Delta^2 u -\Delta u^p \end{equation} is a gradient system with the same Lyapunov as above, and this time we have $\frac{\mathrm{d}}{\mathrm{d}t}E = -||u_t(-\Delta^{-1})u_t||_{L^2}=||u_t||_{H^{-1}}^2\leq0$.
It seems the topology in which the Lyapunov 'acts' is significant, and perhaps my uneasiness lies in not really properly understanding $H^{-1}(\mathbb{R}^N)$. What is the significance of the space in regards to the behaviour of gradient systems?
If no one can be bothered to explain (I wouldn't blame you), I would very much appreciate being pointed in the direction of any books or papers that handle this sort of thing.
Many thanks.