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Analytic solutions for cylindrical thermal waves in solid medium is given based on the nonlinear hyperbolic system of heat flux relaxation and energy conservation equations. The Fourier-Cattaneo phenomenological law is generalized where the relaxation time and heat propagation coefficient have a general power law temperature dependence. From such laws one cannot form a second order parabolic or telegraph-type equation. We consider the original non-linear hyperbolic system itself with the self-similar Ansatz for the temperature distribution and for the heat flux. As results continuous and shock-wave solutions are presented. For physical establishment numerous materials with various temperature dependent heat conduction coefficients are mentioned., Comment: 5 pages, 2 figures, the manuscript is submitted to Eur. Phys. Letters

We study properties of the solutions of two semilinear parabolic equations, the singular Zhang equation and its hydrodynamic version, $$ u_t=u_{xx}+\ln|u_x|\quad {\rm and} \,\,\,v_t = v_{xx} + {v_x}/v , $$ which exhibit strong singularities on the sets $\{u_x=0\}$ and $\{v = 0\}$ respectively. Using the concept of maximal solutions constructed by a regular monotone approximation, we prove existence and boundedness of such solutions. We show that for some classes of initial data, the solutions become instantly entirely singular: $v(t) \equiv 0$ and $u(t)\equiv -\infty$ for any arbitrarily small $t>0$. In general, we establish that for both equations the solutions cannot satisfy the initial condition in any weak sense; i.e., the corresponding semigroups are not continuous at $t=0$. Such discontinuous semigroups are exhibited by a wide class of singular nonlinear parabolic equations.