I have seen in Evans book and some papers that they call the equation \begin{equation} u_{t}-\Delta u { - \varepsilon u_{tt}}= 0, \end{equation} regularization of parabolic equation. Why do we add $\varepsilon u_{tt}$? Could you please help me with right discussion on this?
2026-03-25 11:12:23.1774437143
Regularization of heat equation
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If $\varepsilon=0$, the equation in OP amounts to $u_t - \Delta u =0$, which is the classical heat equation deduced from Fourier's law of heat conduction ($u$ is a temperature). According to this model, a temperature perturbation propagates at infinite speed. If $\varepsilon<0$, the equation in OP $u_t - \Delta u-\varepsilon u_{tt} =0$ is the hyperbolic heat equation deduced from the Maxwell-Cattaneo-Vernotte law of heat conduction. This model guarantees that a temperature perturbation propagates at finite speed.