  We consider a four-elastic-constant Landau-de Gennes energy characterizing
nematic liquid crystal configurations described using the $Q$-tensor formalism.
The energy contains a cubic term and is unbounded from below. We study
dynamical effects produced by the presence of this cubic term by considering an
$L^2$ gradient flow generated by this energy. We work in two dimensions and
concentrate on understanding the relations between the physicality of the
initial data and the global well-posedness of the system.
