  We consider bistable reaction-diffusion equations in funnel-shaped domains of
R N made up of straight parts and conical parts with positive opening angles.
We study the large time dynamics of entire solutions emanating from a planar
front in the straight part of such a domain and moving into the conical part.
We show a dichotomy between blocking and spreading, by proving especially some
new Liouville type results on stable solutions of semilinear elliptic equations
in the whole space R N. We also show that any spreading solution is a
transition front having a global mean speed, which is the unique speed of
planar fronts, and that it converges at large time in the conical part of the
domain to a well-formed front whose position is approximated by expanding
spheres. Moreover, we provide sufficient conditions on the size R of the
straight part of the domain and on the opening angle $\alpha$ of the conical
part, under which the solution emanating from a planar front is blocked or
spreads completely in the conical part. We finally show the openness of the set
of parameters (R, $\alpha$) for which the propagation is complete.
