  This paper is concerned with parabolic gradient systems of the form \[ u_t =
-\nabla V(u) + \Delta_x u \,, \] where the space variable $x$ and the state
variable $u$ are multidimensional, and the potential $V$ is coercive at
infinity. For such systems, the asymptotic behaviour of solutions stable at
infinity, that is approaching a stable homogeneous equilibrium as $|x|$ goes to
$+\infty$, is investigated. A partial description of the global asymptotic
behaviour of such a solution is provided, depending on the mean speed of growth
of the spatial domain where the solution is not close to this equilibrium, in
relation with the asymptotic energy of the solution. If this mean speed is
zero, then the asymptotic energy is nonnegative, and the time derivative $u_t$
goes to $0$ uniformly in space. If conversely the mean speed is nonzero, then
the asymptotic energy equals $-\infty$. This result is called upon in a
companion paper where the global behaviour of radially symmetric solutions
stable at infinity is described. The proof relies mainly on energy estimates in
the laboratory frame and in frames travelling at a small nonzero velocity.

Analysis of PDEs