Continuous dependence on spatial geometry for the generalized Maxwell-Cattaneo system (Q2769916)

Let \(D\) be a bounded domain in \(\mathbb{R}^3\). The authors study heat conduction at low temperatures in \(D\). Let \((u_1, u_2, u_3)\) denote the heat flux, and \(T\) the temperature. They consider the Maxwell-Cattaneo system NEWLINE\[NEWLINE\tau \partial_t u_i = -u_i -K \partial_i T + \mu \Delta u_i + \nu \Sigma_j \partial_j \partial_i u_j, \qquad C\partial_t T= - \Sigma_i \partial_i u_i NEWLINE\]NEWLINE on \(D \times \{t > 0 \}\), subject to the boundary conditions \(\Sigma_{j,k} \varepsilon_{ijk} u_j n_k = 0\) and \(T=0\) on \(\partial D \times \{t>0\}\), and the initial conditions \(u_i (\cdot, 0) = h_i\) and \(T(\cdot, 0) = f\) on \(D\). In fact, they consider a pair of such problems, on domains \(D_1\) and \(D_2\) which are star-shaped with respect to a point in \(D_1 \cap D_2\), with initial data \((h_i^{\beta}, f^{\beta})\) for \(\beta = 1,2\). They prove that the temperatures \(T^{\beta}\) satisfy, for all \(t > 0\), the estimate NEWLINE\[NEWLINE\begin{multlined} \int^t_0 \int_{D_1 \cap D_2} (T^1 - T^2)^2 dx dt \\ \leq a_1 \int_{D_1 \cap D_2} ((f^1 - f^2)^2 + (\Sigma_i \partial_i (h_i^1 - h^2_i))^2 + (\Delta(f^1 -f^2))^2) dx \\ + \delta a(t) \Sigma_\beta \int_{D_\beta} ((f^\beta)^2 + (\Sigma_i \partial_i h_i^\beta)^2 + \Sigma_i (\partial_i f^\beta)^2) dx, \end{multlined}NEWLINE\]NEWLINE where \(a_1\) is a computable constant, \(a(t)\) is a computable function which is bounded for bounded \(t\), and \(\delta \) is the maximum distance along a ray between the boundaries of \(D_1\) and \(D_2\).