  For heat flux $q$ and temperature $T$ we introduce a modified
Fourier--Cattaneo law $q_t+ l \frac{q}{t}= - kT_x .$ The consequence of it is a
non-autonomous telegraph-type equation. % $\epsilon S_{tt} + \frac{a}{t} S_t =
S_{xx}$ . This model already has a typical self-similar solution which may be
written as product of two travelling waves modulo a time-dependent factor and
might play a role of intermediate asymptotics.
