Improved bounds for some nonstandard problems in generalized heat conduction (Q1888233)

The Maxwell-Cattaneo equations provide a model for the evolution of the temperature density \(\theta\) and accounts for finite propagation speed. Here a particular space-time boundary value problem for the cylindrical space-time domain \(\Omega\times\left[0,T\right]\subset\mathbb{R}^{3}\times\left[0,T\right]\) is considered. On \(\partial\Omega\times\left[0,T\right]\) a homogeneous Dirichlet condition is assumed, whereas on the boundary part \(\Omega\times\left\{ 0,T\right\} \) a coupled, inhomogeneous boundary condition of the form \( \theta\left(T\right)+\alpha\theta\left(0\right) = g\left(\partial_{t}\theta\right)\left(T\right)+\alpha\left(\partial_{t}\theta\right)\left(0\right) = h\) are imposed with suitably given data \(g,h\) and \(\alpha\in\mathbb{R}\setminus\left\{ 0\right\} \). The authors obtain \(L^{2}\)-bounds for solutions. In concluding remarks possible improvements are discussed in the case \(\alpha>0\) which under certain constraints on the coefficients yield exponential decay in time for the solution in \(L^{2}\left(\Omega\right)\).