Continuous Dependence on the Relaxation Time and Modeling, and Unbounded Growth, in Theories of Heat Conduction with Finite Propagation Speeds

The problem is studied of how the solution varies as the relaxation time is altered for hyperbolic heat conduction theories of Maxwell-Cattaneo type. We also investigate the behaviour of the Maxwell-Cattaneo models as the relaxation time, τ, tends to zero. For the improperly posed backward in time problem we show that the solution depends Hölder continuously on changes in the model. Finally, a model of heat propagation is studied which allows for three relaxation times; we show that under a large class of conditions for the coefficients, this theory always allows choices of initial data for which solutions are unstable.