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We are concerned with the asymptotic behaviour of classical solutions of systems of the form where A is a positive-definite diagonal matrix and f is a "bistable" nonlinearity satisfying conditions which guarantee the existence of a comparison principle for (1). Suppose that (1) has a travelling-front solution w with velocity c, that connects two stable equilibria of f . (There are hypotheses on f under which such a front is known to exist  .) We show that if φ is bounded, uniformlydoi:10.12775/tmna.2000.029 fatcat:d7dshvu7m5g2lf6ir5uea2yrzu