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A Poincaré–Bendixson theorem for scalar balance laws

Published online by Cambridge University Press:  14 November 2011

Athanasios N. Lyberopoulos
Athanasios N. Lyberopoulos
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, OK 73019-0315, U.S.A.
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Abstract

We study the dynamical behaviour, as t → ∞, of admissible weak solutions of the scalar balance law

with 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 .

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 3 , 1994 , pp. 589 - 607
Copyright
Copyright © Royal Society of Edinburgh 1994

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