Convergence analysis of a variational quasi-reversibility approach for an inverse hyperbolic heat conduction problem

TL;DR

This paper extends a variational quasi-reversibility method to a hyperbolic heat conduction problem based on the Maxwell--Cattaneo model, establishing well-posedness and convergence rates for the regularized inverse problem.

Contribution

It adapts a previous method for classical heat problems to hyperbolic models with thermal memory, providing a generic regularization scheme and convergence analysis.

Findings

Proves well-posedness of the regularized scheme

Establishes H"older convergence rate in mixed spaces

Demonstrates adaptability of the method to hyperbolic heat models

Abstract

We study a time-reversed hyperbolic heat conduction problem based upon the Maxwell--Cattaneo model of non-Fourier heat law. This heat and mass diffusion problem is a hyperbolic type equation for thermodynamics systems with thermal memory or with finite time-delayed heat flux, where the Fourier or Fick law is proven to be unsuccessful with experimental data. In this work, we show that our recent variational quasi-reversibility method for the classical time-reversed heat conduction problem, which obeys the Fourier or Fick law, can be adapted to cope with this hyperbolic scenario. We establish a generic regularization scheme in the sense that we perturb both spatial operators involved in the PDE. Driven by a Carleman weight function, we exploit the natural energy method to prove the well-posedness of this regularized scheme. Moreover, we prove the H\"older rate of convergence in the mixed $L^2$--$H^1$ spaces.