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

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.