  We address the question of existence of nonconstant stable stationary
solution to the heat equation on a class of convex domains subject to nonlinear
boundary flux involving a positive parameter. Such solutions which were known
to exist in dumbbell-type domains and not to exist in $N$-dimensional balls are
shown to exist in some convex domains obtained by smoothing out, in a
convenient way, the edges and corners of a cube. Therefore convexity of the
domain is not a necessary condition for nonexistence of this type of solutions.
If the parameter is small enough we prove that the only equilibria are the
constant ones by using the Implicit Function Theorem in a special setting.
Symmetry inheritance by local minimizers from symmetry properties of the domain
is also addressed.
