On admissibility for parabolic equations in ${\Bbb R}^n$

Martino Prizzi¹

¹ Dipartimento di Scienze Matematiche Università degli Studi di Trieste via Valerio 12 34127 Trieste, Italy

Fundamenta Mathematicae, Tome 176 (2003) no. 3, pp. 261-275

Résumé

We consider the parabolic equation $$ u_t-{\mit \Delta } u=F(x,u), \hskip 1em \ (t,x)\in {{\mathbb R}}_+\times {{\mathbb R}}^n, \tag{P} $$ 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 $-admissible 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 extend earlier results of Rybakowski concerning parabolic equations on bounded open subsets of ${{\mathbb R}}^n$.

Détail

DOI : 10.4064/fm176-3-5

Keywords: consider parabolic equation t mit delta u hskip mathbb times mathbb tag corresponding semiflow phase space conditions nonlinearity ensuring bounded sets admissible sense rybakowski asymptotically linear under appropriate non resonance conditions conleys index theory prove existence nontrivial equilibria heteroclinic trajectories joining these equilibria results obtained extend earlier results rybakowski concerning parabolic equations bounded subsets mathbb