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We consider the Cahn-Hilliard equation on a manifold with conical singularities. We first show the existence of bounded imaginary powers for suitable closed extensions of the bilaplacian. Combining results and methods from singular analysis with a theorem of Clément and Li we then prove the short time solvability of the Cahn-Hilliard equation in Lp-Mellin-Sobolev spaces and obtain the asymptotics of the solution near the conical points. We deduce, in particular, that regularity is preserved on the smooth part of the manifold and singularities remain confined to the conical points. We finally show how the Allen-Cahn equation can be treated by simpler considerations. Again we obtain short time solvability and the behavior near the conical points. [ABSTRACT FROM AUTHOR]

An approximation Ansatz for the operator solution, U ( z ′, z ), of a hyperbolic first-order pseudodifferential equation, ∂̸ z + a ( z , x , D x ) with Re ( a ) ≥ 0, is constructed as the composition of global Fourier integral operators with complex phases. An estimate of the operator norm in L ( H ( s ) , H ( s ) ) of these operators is provided, which yields a convergence result for the Ansatz to U ( z ′, z ) in some Sobolev space as the number of operators in the composition goes to ∞. [ABSTRACT FROM AUTHOR]