  We consider the Swift-Hohenberg equation on manifolds with conical
singularities and show existence, uniqueness and maximal regularity of the
short time solution in terms of Mellin-Sobolev spaces. Moreover, we give a
necessary and sufficient condition so that the above solution exists for all
times. Space asymptotic expansion of the solution near the singularity is also
provided and its relation to the local geometry is shown. The same problem is
considered on closed manifolds and similar results are obtained by using the
above singular analysis theory.
