The qualitative properties of the problem Cauchy for the equation of nonlinear heat conductivity a with gradient nonlinearity at presence of an absorption is investigated. It is proved, that if the coefficient of heat conductivity includes, time-dependent function that occurs localization of the soluision even an absience of absorption. The Kalashnikov A.S. , Knerr B.F. , Kershner R.. type estimation of a weak solution of the problem Cauchy and an estimation of a free boundary are received..Based on qualitative properties the numerical analysys of solutons carried out
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