On the applicability of the averaging method to the Cauchy problem for the equation \(u_t=(u^\sigma u_x)_x+u^\beta\) (Q1594442)

The Cauchy problem for the quasilinear parabolic equation \(u_t=(u^\sigma u_x)_x+u^\beta\) is considered. This problem attracts interest in the theory of complex systems. Earlier for the qualitative analysis of solutions the averaging method was applied. In case \(\sigma+1<\beta<\sigma+3\) the approximate solution \(u^*(x,t)=g(t)F(x/\psi(t))\), obtained by the averaging method, shows a good qualitative agreement with the numerical solution [see \textit{V. A. Belavin, S. P. Kapitsa} and \textit{S. P. Kurdyumov}, Comput. Math. Math. Phys. 38, No.~6, 849-865 (1998); translation from Zh. Vychisl. Mat. Mat. Fiz. 30 885-902 (1998; Zbl 1086.91508)]. In the present paper the case \(\beta\geq\sigma+3\) is investigated. It is shown that the averaging method is inapplicable to the analysis of the blow-up regime for \(\beta\geq\sigma+3\) and with help of numerical solutions the behavior of the averaging method's parameters is analyzed. The approximate solutions obtained by the averaging method tend to power phase curves with dimensionless exponents while numerical solutions demonstrate asymptotics with dimensional self-similar exponents.