On admissibility for parabolic equations in R^n

We consider the parabolic equation $$\begin{array}{cccc}& u_t-\mathrm{\Delta }u=F(x,u),(t,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}^n& & \text{(P)}\end{array}$$ and the corresponding semiflow $\pi$ in the phase space $H^1$. We give conditions on the nonlinearity $F(x,u)$, ensuring that all bounded sets of $H^1$ are $\pi$-admissibile in the sense of Rybakowski. If $F(x,u)$ is asymptotically linear, under appropriate non-resonance conditions, we use Conley’s index theory to prove the existence of nontrivial equilibria of (P) and of heteroclinic trajectories joining some of these equilibria. The results obtained in this paper extend earlier results of Rybakowski concerning parabolic equations on {\it bounded} open subsets of ${\mathbb{R}}^n$.