  This article addresses the issue of global convergence towards pushed
travelling fronts for solutions of parabolic systems of the form \[ u_t = -
\nabla V(u) + u_{xx} \,, \] where the potential $V$ is coercive at infinity. It
is proved that, if an initial condition $x\mapsto u(x,t=0)$ approaches, rapidly
enough, a critical point $e$ of $V$ to the right end of space, and if, for some
speed $c_0$ greater than the linear spreading speed associated with $e$, the
energy of this initial condition in a frame travelling at the speed $c_0$ is
negative $\unicode{x2013}$ with symbols, \[ \int_{\mathbb{R}} e^{c_0
x}\left(\frac{1}{2} u_x(x,0)^2 + V\bigl(u(x,0)\bigr)- V(e)\right)\, dx < 0 \,,
\] then the corresponding solution invades $e$ at a speed $c$ greater than
$c_0$, and approaches, around the leading edge and as time goes to $+\infty$,
profiles of pushed fronts (in most cases a single one) travelling at the speed
$c$. A necessary and sufficient condition for the existence of pushed fronts
invading a critical point at a speed greater than its linear spreading speed
follows as a corollary. In the absence of maximum principle, the arguments are
purely variational. The key ingredient is a Poincar\'e inequality showing that,
in frames travelling at speeds exceeding the linear spreading speed, the
variational landscape does not differ much from the case where the invaded
equilibrium $e$ is stable. The proof is notably inspired by ideas and
techniques introduced by Th. Gallay and R. Joly, and subsequently used by C.
Luo, in the setting of nonlinear damped wave equations.
