Existence of L¹-connections between equilibria of a semilinear parabolic equation

We give a necessary and sufficient condition for the existence of L ¹-connections between equilibria of a semilinear parabolic equation. By an L¹-connection from an equilibrium φ^- to an equilibrium φ⁺ we mean a function u(·, t) which is a classical solution on the interval (-∞, T) for some T ∈ ℝ and blows up at t = T but continues to exist in the space L¹ for t ∈ [T, ∞) and satisfies u(·, t) → φ^± (in a suitable sense) as t → ± ∞. The main tool in our analysis is the zero number.