Stability and error estimates of a general modified quasi-boundary value method via a semi-linear backward parabolic equation

TL;DR

This paper analyzes a modified quasi-boundary value regularization method for a semi-linear backward parabolic PDE, establishing stability and convergence rates for approximate solutions in an ill-posed setting.

Contribution

It introduces a new regularization approach for semi-linear backward parabolic equations, extending existing methods with stability and error estimates using nonlinear spectral theory.

Findings

Proves stability of the regularized solutions.

Establishes convergence rates for the approximation.

Extends previous results to semi-linear cases.

Abstract

Regularization methods have been recently developed to construct stable approximate solutions to classical partial differential equations considered as final value problems. In this paper, we investigate the backward parabolic problem with locally Lipschitz source: $\partial_{t}u+\mu\left(t\right)\mathcal{A}u\left(t\right)=f\left(t,u\right)$ where $\mathcal{A}:\mathcal{D}\left(\mathcal{A}\right)\subset\mathcal{H}\to\mathcal{H}$ is a positive, self-adjoint and unbounded linear operator on the Hilbert space $\mathcal{H}$. The problem arises in many applications, but it is in general ill-posed. The ill-posedness is caused by catastrophic growth in the representation of solution, then independence of solution on data makes computational procedures impossible. Therefore, we contribute to this interesting field the study of the stable approximation solution via the modified quasi-boundary value method based on nonlinear spectral theory, which slightly extends the results in many works. Our main focus is to qualitatively prove the stability and convergence rate where semi-group technicalities are highly useful.