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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 statevariable $u$ are multidimensional, and the potential $V$ is coercive atinfinity. For such systems, the asymptotic behaviour of solutions stable atinfinity, that is approaching a stable homogeneous equilibrium as $|x|$ goes to$+\infty$, is investigated. A partial description of the global asymptoticbehaviour of such a solution is provided, depending on the mean speed of growthof the spatial domain where the solution is not close to this equilibrium, inrelation with the asymptotic energy of the solution. If this mean speed iszero, 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, thenthe asymptotic energy equals $-\infty$. This result is called upon in acompanion paper where the global behaviour of radially symmetric solutionsstable at infinity is described. The proof relies mainly on energy estimates inthe laboratory frame and in frames travelling at a small nonzero velocity.

Global behaviour of solutions stable at infinity for gradient systems in higher space dimension: the no invasion case