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 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.

Shooting methods for a PT-symmetric periodic eigenvalue problem

“…The differences reported in the second last column suggest that our methods can tackle this more general case too. However the rigorous error analysis which we presented here does not immediately generalize to the case Table 2 Comparison of our results with those provided by the WKB method and the method in [6] for f (x) = sin(x) and = 0.13 k λ k,4 λ k,6 λ k,WKB μ k |λ k,4 − λ k,6 | |λ k,WKB − λ k,6 | |μ k − λ k,6 | Table 3 Comparison of our results with respect to those provided by the method in [6] for problem (7) for f (x) = sin(x), g(x) = − cos(x), = 0.13, and a = 0.4 a = 0, with general functions g and f satisfying the properties specified in the proof of Theorem 1, because the logarithmic semi-norm μ 1 (J(x)) is no longer bounded (see Proposition 4). We shall defer consideration of this case to future work.…”

“…Coupled with the fact that the eigenvalues are all real, this immediately establishes that the eigen-and associated functions cannot form a Riesz basis. This argument appears in Chugunova et al [7] for f (x) = sin(x) and can clearly be generalized 1 to a much wider class of coefficients. The failure of Riesz basisness of the eigen-and associated functions is generally associated with ill-conditioning of the spectrum and wild pseudospectral behaviour, due to the angles between eigenfunctions not being bounded away from zero.…”

“…Proof Most of the results in this theorem have appeared in several recent articles including [4] and [7]. A proof is included here for the convenience of the reader, and applies equally well to the more general equation…”

Generalized quasi-reversibility method for a backward heat equation with a fractional Laplacian

“…Fu-Xiong-Qian ( [11]), Tuan and Trong ( [21]) studied the backward heat equation by using the Fourier transform. In order to deal with the backward heat system, Chugunova-Karabash-Pyatkov ( [7]), Cheng-Fu-Qin ( [6]), Boulton-Marletta-Rule ( [4]) applied their spectrum methods. Lu and Maubach ([15]), Ternat-Orellana-Daripa ( [20]), Douglas and Gallie ( [9]) used numerical methods to analyze the backward heat equation.…”

On the Stability of a Forward–Backward Heat Equation

“…Forward-backward heat equation originated in applications from hydrodynamics [1], and it has recently attracted some interest due to various unusual stability and symmetry properties. Spectral properties of ε were examined simultaneously in various works by Chugunova, Karabash, Pelinovsky and Pyatkov [4,5], and Davies and Weir [8,9,14,15,16]. Remarkably it was noted that the associated closed operator L ε : Dom(L ε ) −→ L 2 (−π, π), defined on a suitable domain reproducing the singularities and boundary conditions, has a purely discrete spectrum comprising conjugate pairs lying on the imaginary axis and accumulating only at ±i∞.…”