We consider nonlinear parabolic systems of the form u t = −∇V (u) + u xx , where u ∈ R n , n 1, x ∈ R, and the potential V is coercive at infinity. For such systems, we prove a result of global convergence toward bistable fronts which states that invasion of a stable homogeneous equilibrium (a local minimum of the potential) necessarily occurs via a traveling front connecting to another (lower) equilibrium. This provides, for instance, a generalization of the global convergence result obtained by Fife and McLeod [P. Fife, J.B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rat. Mech. Anal. 65 (1977) 335-361] in the case n = 1. The proof is based purely on energy methods, it does not make use of comparison principles, which do not hold any more when n > 1.

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This paper is concerned with spatially extended gradient systems of the formwhere spatial domain is the whole real line, state-parameter u is multidimensional, D denotes a fixed diffusion matrix, and the potential V is coercive at infinity. Bistable solutions, that is solutions close at both ends of space to stable homogeneous equilibria, are considered. For a solution of this kind, it is proved that, if the homogeneous equilibria approached at both ends belong to the same level set of the potential and if an appropriate (localized in space) energy remains bounded from below when time increases, then the solution approaches, when time approaches infinity, a pattern of stationary solutions homoclinic or heteroclinic to homogeneous equilibria. This result provides a step towards a complete description of the global behaviour of all bistable solutions that is pursued in a companion paper. Some consequences are derived, and applications to some examples are given.

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