The behavior of hyperbolic heat equations’ solutions near their parabolic limits

Standard energy methods are used to study the relation between the solutions of one parameter families of hyperbolic systems of equations describing heat propagation near their parabolic limits, which for these cases are the usual diffusive heat equation. In the linear case it is proven that given any solution to the hyperbolic equations there is always a solution to the diffusion equation which after a short time stays very close to it for all times. The separation between these solutions depends on the square of the ratio between the assumed very short decay time appearing in Cattaneo’s relation and the usual characteristic smoothing time (initial data dependent) of the limiting diffusive equation. The techniques used in the linear case can be readily used for nonlinear equations. As an example we consider the theories of heat propagation introduced by Coleman, Fabrizio, and Owen, and prove that near a solution to the limiting diffusive equation there is always a solution to the nonlinear hyperbolic equations for a time which usually is much longer than the decay time of the corresponding Cattaneo relation. An alternative derivation of the heat theories of divergence type, which are consistent with thermodynamic principles, is given as an appendix.