Continuous dependence result for the higher-order conduction within the second gradient of type Ⅲ

This study investigated the spatial behavior and structural stability of solutions for a higher-order heat conduction model within the second gradient theory of type Ⅲ. By constructing a tailored energy functional and employing integral-differential inequality techniques, we derived a continuous dependence estimate for the solution with respect to the elastic coefficient $ \mu $. Our results demonstrate that the energy not only decays exponentially with respect to the spatial variable $ z $, but also diminishes as the parameter $ \mu $ tends to zero. This work extends the understanding of structural stability in unbounded domains and underscores the robustness of the model under perturbations of material parameters.