  The level-set curvature G-equation, a well-known model in turbulent
combustion, has the following form
  $G_t + \left(1-d\,
\mathrm{dvi}\left({\frac{DG}{|DG|}}\right)\right)_+|DG|+V(X)\cdot DG=0.$
  Here the cutoff correction $()_+$ is imposed to avoid non-physical negative
local burning velocity. The existence of the effective burning velocity has
been established for a large class of physically relevant incompressible flows
$V$ in two dimensions [13] via game theory dynamics. In this paper, we show
that the effective burning velocity associated with shear flows in dimensions
three or higher ceases to exist when the flow intensity surpasses a bifurcation
point. The characterization of the bifurcation point in three dimensions is
closely related to the regularity theory of two-dimensional minimal surface
type equations due to [29]. As a consequence, a bifurcation also exists for the
validity of full homogenization of the curvature G-equation associated with
shear flows.
