Short-time heat diffusion in compact domains with discontinuous transmission boundary conditions (Q2788506)

The authors deal with the heat exchange between two media, namely \(\Omega_+=\Omega\subset\mathbb{R}^n\) and \(\Omega_-=\mathbb{R}^n\setminus \overline\Omega\), prepared initially at different (normalized) temperatures, \(u_+(\cdot, 0)=1\) in \(\Omega_+\) and \(u_-(\cdot,0)=0\) in \(\Omega_-\), and separated by a partially isolating boundary, i.e. the temperature flux is related to the temperature drop at the boundary \(\partial\Omega\) due to the resistivity \(\lambda>0\). The continuity of the temperature flux across the boundary is imposed. For bounded domains \(\Omega\) that are either regular with a closed piecewise Lipschitz boundary or irregular with fractal boundary of the Hausdorff dimension, the authors establish the unique weak solvability of the problem under study associated to the equations: \(\partial_t u_\pm =D_\pm \Delta u_\pm\) in \(\Omega_\pm\times ]0,\infty[\), and the continuity of the solution \(u\) as function of \(\lambda\). The diffusion coefficients \(D_+,D_-\) are assumed to be positive constants in \(\Omega_+\) and \(\Omega_-\), respectively, such that \(D_+\not= D_-\). For a regular boundary \(\partial\Omega\in C^3\), the heat content \(N(t)=\int_\Omega\left( 1-u_+(x,t)\right)dx\) is calculated by using the approximation of solutions to one-dimensional problems, which are written in terms of the Green functions. Also the explicit calculation of the heat content is provided if \(\Omega \) is a compact bounded domain with a connected boundary of the Hausdorff dimension. In the case of a regular boundary \(\partial\Omega\in C^\infty\), the authors provide the asymptotic expansion of the heat content up to the third-order term.