A Poincaré–Bendixson theorem for scalar balance laws

Abstract

We study the dynamical behaviour, as t → ∞, of admissible weak solutions of the scalar balance lawwith x ∊ ≡ ℝ/Lℤ, L > 0, 0 < t < ∞, and f(·) ∊ C2, g(·) ∊ C1. We assume that f(·) is strictly convex, while g(·) is of at most linear growth, has finitely many zeros and changes sign across them. We show that, if u(·,t) stays bounded in L∞(S1), as t → ∞, then it either converges to a constant state or approaches asymptotically a rotating wave, i.e. an admissible weak solution of (1.1) of the form ũ(x − ct), c ∈ ℝ. Hence, the asymptotic state of every bounded solution of (1.1) consists precisely of either an equilibrium or one time-periodic solution. Furthermore, each one of these two alternatives is characterised by the Conley indices of the critical points of the ordinary differential equation .