On admissibility for parabolic equations in R^n

Details On admissibility for parabolic equations in R^n On admissibility for parabolic equations in R^n

2003-03-06

arxiv.org/abs/math/0303080v1

Mathematics

Analysis of PDEs

16 pages; to appear in "Fundamenta Mathematicae"

Scientific paper

We consider the parabolic equation $$u_t-\Delta u=F(x,u),\quad (t,x)\in\R_+\times\R^n\tag{P}$$ 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 $\R^n$.

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