The article under review is devoted to studying regularity of interfaces and solutions to the Cauchy problem for the diffusion-absorption model $$ u_t = (u^m)_{xx} - u^p,\qquad u\geq 0$$ in the range of parameters $m>1$, $0< p <1$, $m + p \geq 2$. The authors expose a new approach for the study of the smoothness of the interfaces of the solutions, the smoothness of the solutions near the interfaces and the relationship between both. The method is based on the idea that the local analysis of the solutions of degenerate parabolic equations near an interface can be done by means of intersection comparison with the family of travelling-wave solutions. The authors obtain the formula which expresses the interface velocity in terms of the solution profile, the so-called interface equation, valid for both expanding and receding waves of general solutions and the interface equation has the following form $$ \eta' = \pi_x - \frac{1}{w_x}, $$ where the functions in the right-hand member are defined as $\pi = (m/(m-1))u^{m-1}$, $w = (1/(1-p))u^{1-p}$ and $\eta'$ stands for the right derivative of $\eta(t)$ at time $t$. It is proven that the interface function $\eta (t)$ is actually left- and right-differentiable at every instant $t\in (0,T_e)$ and the solutions of the Cauchy problem have Lipschitz continuous interfaces in any compact time interval $0<t_1\leq t \leq t_2<T_e$. The application of the method to the class of nonlinear evolution equations of the form $$ u_t = \varphi(u)_{xx} + \psi(u,u_x) $$ is also discussed under suitable assumptions on $\varphi$ and $\psi$. The authors observe that previous methods have failed to provide an adequate analysis of the interface motion and regularity.