Nonlinear Analysis 37 (1999) 257 – 267 Uniqueness and non-uniqueness for a system of heat equations with nonlinear coupling at the boundary Carmen Cortazar a;*;1 , Manuel Elgueta a;1 , Julio D. Rossi b;2 a Facultad de Matematicas, Universidad Catolica, Casilla 306 Correo 22 Santiago, Chile b Departmento de Matem atica, F.C.E y N., UBA (1428) Buenos Aires, Argentina Received 24 March 1997; accepted 1 October 1997 1. Introduction Let  be a bounded domain in R N with smooth boundary and let p and q be two positive real numbers. In this article we are concerned with the uniqueness problem for non-negative solutions of the following system: u t =u; v t =v in  × (0;T ); @u @ = v p ; @v @ = u q on @ × (0;T ); u(x; 0) = u 0 and v(x; 0) = v 0 in : (1.1) with smooth initial data, u 0 ≥ 0 and v 0 ≥ 0. Parabolic reaction–diusion systems like Eq. (1.1) or of a more general form, allowing coupling through source terms or with dierent boundary conditions, appear in several branches of applied mathematics. They have been use to model, for exam- ple, chemical reactions, heat transfer or population dynamics and have been studied by several authors. See [13] and the references therein. Local existence in time, regularity and comparison results for solutions of several types of systems, including Eq. (1.1), have been obtained among others in [2, 6, 12]. See also the corresponding references. * Corresponding author. 1 Partially supported by Fundacion Andes and FONDECYT under grant 1971126. 2 Supported by Universidad de Buenos Aires under grant EX071, Conicet (Argentina) and Fundacion Andes. 0362-546X/99/$ – see front matter ? 1999 Elsevier Science Ltd. All rights reserved. PII: S0362-546X(98)00046-7