On the nature of ill-posedness of the forward-backward heat equation

Abstract

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 ${Ł}^2(-\pi ,\pi )$. Our method can be applied to a wide range of the evolutional problems given by $PT-$symmetric operators.

21 pages; Remark 5.2 added, acknowledgements added, several typos fixed

References in corpus (4)

• Spectrum of a non-self-adjoint operator associated with the periodic heat equation
• Abstract kinetic equations with positive collision operators
• On the nature of ill-posedness of the forward-backward heat equation
• Factorization of the Indefinite Convection-Diffusion Operator

Cited by in corpus (8)

• On the nature of ill-posedness of the forward-backward heat equation
• Convergence to equilibrium for a thin film equation on a cylindrical surface
• On computing the instability index of a non-selfadjoint differential operator associated with coating and rimming flows
• Correspondence of the eigenvalues of a non-self-adjoint operator to those of a self-adjoint operator
• On Factorization of a Perturbation of a J-selfadjoint Operator Arising in Fluid Dynamics
• Multi-parameter Hopf bifurcations of rimming flows
• The pseudospectrum of an operator with Bessel-type singularities
• On the stability of a forward-backward heat equation