We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by a J-self-adjoint linear operator L depending on a small parameter. The problem originates from the lubrication approximation of a viscous fluid film on the inner surface of a rotating cylinder. For a certain range of the parameter we rigorously prove the conjecture, based on numerical evidence, that the complete set of eigenvectors of the operator L does not form a Riesz basis in L 2 (−π, π). Our method can be applied to a wide range of evolution problems given by P T -symmetric operators.
Mathematics Subject Classification (2000). Primary 35P10; Secondary 35Q35, 35K15, 76A20. Keywords. Viscous fluid film, forward-backward diffusion, parabolic equation of mixed type, highly non-self-adjoint differential operator, Riesz basis property, completeness.