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

TL;DR

This paper investigates the ill-posedness of the forward-backward heat equation with periodic data, proving that the eigenvectors of the associated operator do not form a Riesz basis, which impacts the stability and solvability of the problem.

Contribution

It provides a rigorous proof that the eigenvectors of a certain J-self-adjoint operator do not form a Riesz basis, advancing understanding of the ill-posedness in such evolution problems.

Findings

Eigenvectors of the operator do not form a Riesz basis in L^2(-π,π).

The method applies to a broad class of PT-symmetric evolution problems.

Confirmed conjecture based on numerical evidence.

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