For heat flux $q$ and temperature $T$ we introduce a modifiedFourier--Cattaneo law $q_t+ l \frac{q}{t}= - kT_x .$ The consequence of it is anon-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 bewritten as product of two travelling waves modulo a time-dependent factor andmight play a role of intermediate asymptotics.
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