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 Holder rate of convergence in the mixed $L^2$--$H^1$ spaces.

中文翻译：

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反双曲热传导问题的变分准可逆方法的收敛性分析

我们研究了基于非傅立叶热定律的 Maxwell-Cattaneo 模型的时间反转双曲热传导问题。此热量和质量扩散问题是具有热记忆或有限时间延迟热通量的热力学系统的双曲型方程，其中傅立叶或菲克定律已被实验数据证明不成功。在这项工作中，我们展示了我们最近用于经典时间反向热传导问题的变分准可逆性方法，它遵循傅立叶或菲克定律，可以适应这种双曲线情况。在我们扰乱 PDE 中涉及的两个空间算子的意义上，我们建立了一个通用的正则化方案。在卡尔曼权重函数的驱动下，我们利用自然能量方法来证明这个正则化方案的适定性。而且，