  We study the Cauchy problem with periodic initial data for the
forward-backward heat equation defined by the J-self-adjoint linear operator L
depending on a small parameter. The problem has been originated from the
lubrication approximation of a viscous fluid film on the inner surface of the
rotating cylinder. For a certain range of the parameter we rigorously prove the
conjecture, based on the numerical evidence, that the set of eigenvectors of
the operator $L$ does not form a Riesz basis in $\L^2 (-\pi,\pi)$. Our method
can be applied to a wide range of the evolutional problems given by
$PT-$symmetric operators.
