Asymptotic Behavior of Discontinuous Solutions in 3-D Thermoelasticity with Second Sound

Abstract

This paper is devoted to the study of the Cauchy problem for linear and semilinear thermoelastic systems with second sound in three space dimensions with discontinuous initial data. Due to Cattaneo's law, replacing Fourier's law for heat conduction, the thermoelastic system with second sound is 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. By studying expansions with respect to the relaxation parameter of the jumps of the potential part of the system on the evolving characteristic surfaces, we obtain that the jump of the temperature goes to zero while the jumps of the heat flux and the gradient of the potential part of the elastic wave are propagated along the characteristic curves of the elastic fields when the relaxation parameter goes to zero. An interesting phenomenon is when time goes to infinity, the behavior will depend on the mean curvature of the initial surface of discontinuity. These jumps decay exponentially when time goes to infinity, more rapidly for smaller heat conductive coefficient in linear problems and in nonlinear problems when certain growth conditions are imposed on the nonlinear functions.