  In this paper, we study self-expanding solutions for mean curvature flows and
their relationship to minimal cones in Euclidean space. In [18], Ilmanen proved
the existence of self-expanding hypersurfaces with prescribed tangent cones at
infinity. If the cone is $C^{3,{\alpha}}$-regular and mean convex (but not
area-minimizing), we can prove that the corresponding self-expanding
hypersurfaces are smooth, embedded, and have positive mean curvature everywhere
(see Theorem 1.1). As a result, for regular minimal but not area-minimizing
cones we can give an affirmative answer to a problem arisen by Lawson [4].
