THE BEHAVIOR OF HYPERBOLIC HEAT-EQUATIONS SOLUTIONS NEAR THEIR PARABOLIC LIMITS

Standard energy methods are used to study the relation between the sol utions of one parameter families of hyperbolic systems of equations de scribing 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 a lways a solution to the diffusion equation which after a short time st ays very close to it for all times. The separation between these solut ions depends on the square of the ratio between the assumed very short decay time appearing in Cattaneo's relation and the usual characteris tic smoothing time (initial data dependent) of the limiting diffusive equation. The techniques used in the linear case can be readily used f or 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 usuall y is much longer than the decay time of the corresponding Cattaneo rel ation. An alternative derivation of the heat theories of divergence ty pe, which are consistent with thermodynamic principles, is given as an appendix.