Very Weak Solution of the Discrete Heat Equation with Irregular Time-Dependent Thermal Conductivity

In this paper, we investigate the semi-classical version of the heat equation with irregular time-dependent coefficients. When regular coefficients are taken into account, we show that the Cauchy problem is well-posed in \(\ell ^{2}(\hbar \mathbb {Z}^{n})\) . In the case of irregular coefficients, we analyse how the notion of a very weak solution adapts to our consideration and show that the Cauchy problem is “weakly well-possed” and that the very weak solution converges approximates (in a suitable norm sense) the classical solution. This presentation is based on the paper Chatzakou et al. (Proc R Soc Edinb A: Math, 1–24, 2023. https://doi.org/10.1017/prm.2023.84 ), to which we refer for further details.