  We are concerned with the asymptotic behaviour of classical solutions of
systems of the form u_t = Au_xx + f(u, u_x), x in R, t>0, u(x,t) a vector in
RN, with u(x,0)= U(x), where A is a positive-definite diagonal matrix and f is
a 'bistable' nonlinearity satisfying conditions which guarantee the existence
of a comparison principle. Suppose that there is a travelling-front solution w
with velocity c, that connects two stable equilibria of f. We show that if U is
bounded, uniformly continuously differentiable and such that w(x) - U(x) is
small when the modulus of x is large, then there exists y in R such that u(.,
t) converges to w(.+y-ct) in the C1 norm at an exponential rate as t tends to
infinity. Our approach extends an idea developed by Roquejoffre, Terman and
Volpert in the convectionless case, where f is independent of u_x.
