Realizing vector fields without loss of derivatives
On admissibility for parabolic equations in ℝⁿ
We consider the parabolic equation (P) , (t,x) ∈ ℝ₊ × ℝⁿ, and the corresponding semiflow π in the phase space H¹. We give conditions on the nonlinearity F(x,u), ensuring that all bounded sets of H¹ are π-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...
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