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Asymptotic Behavior of Discontinuous Solutions to Thermoelastic Systems with Second Sound

We consider the Cauchy problem for linear and semilinear thermoelastic systems with second sound in one space dimension with discontinuous initial data. Due to Cattaneo’s law, replacing Fourier’s law for heat conduction, the system is strictly hyperbolic. We investigate the behavior of discontinuous solutions as the relaxation parameter tends to zero, which corresponds to a formal convergence of the system to the hyperbolic-parabolic type of classical thermoelasticity. We obtain that the jump of the temperature goes to zero while the jumps of the gradient of the displacement and the spatial derivative of the temperature are propagated along the characteristic curves of the elastic fields when the relaxation parameter vanishes. Moreover, when certain growth conditions are imposed on the nonlinear functions, we deduce that these jumps decay exponentially when the time goes to infinity, more rapidly for small heat conduction coefficient.