Quasi-Linear Equation of Thermoconductivity with a Source. LS-Mode. Averaging Method and Automodelness

In the article the applicability of the averaging method is investigated, for the solving of quasi-linear equations of thermoconductivity with the power functions in the thermoconductivity coefficient and in the source, in the case of blow-up LS-regime. Solutions, constructed by means of the averaging method, are compared with the numerical solutions of the partial differential problem. The averaging method applicability for the constructing of approximate solutions of the investigated problem appears to depend on the initial profile shape of the solution and its further evolution, as well as on the definite correlation of the exponents in the thermoconductivity coefficient and in the source. In particular, there is a region of the exponents values, where the averaging method gives the qualitatively incorrect behaviour of approximate solutions, predicting the non-automodel asymptotic of solutions nearby the blow-up moment, while the analysis of the given problem shows, that the automodelness at the asymptotic stage is preserved within this parameters region, as well.