Convergence analysis of a variational quasi-reversibility approach for an inverse hyperbolic heat conduction problem
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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 Hölder rate of convergence in the mixed L^2–H^1 spaces. Under some certain choice of the perturbations and stabilizations, we thereupon obtain the Lipschitz rate in L^2. We also show that under a weaker conditional estimate, it is sufficient to perturb only the highest order differential operator to gain the Hölder convergence.
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