Methods of constructing approximate self-similar solutions of nonlinear heat equations. IV

V. A. Galaktionov   ; A. A. Samarskii

Sbornik. Mathematics, Tome 49 (1984) no. 1, pp. 125-149

Résumé

Asymptotic properties of nonnegative solutions of quasilinear parabolic equations $$ \frac{\partial u}{\partial t}=\frac\partial{\partial x}\bigg(k(u)\frac{\partial u}{\partial x}\bigg);\qquad k(u)>0\quad\text{for}\quad u>0 $$ with coefficients $k(u)$ of rather general form are studied in the paper. The investigation is carried out by constructing approximate self-similar solutions which do not satisfy the original equation but nevertheless correctly describe the asymptotic behavior of solutions of the boundary value or Cauchy problems considered. On the basis of a unified method “transformation laws” are established for well-known self-similar solutions of an equation with a power nonlinearity $\dfrac{\partial u}{\partial t}=\dfrac\partial{\partial x}\biggl(u^\sigma\dfrac{\partial u}{\partial x}\biggr)$ (the cases $\sigma=0$ and $\sigma>0$ are considered separately) which result from small changes of the coefficient $u^\sigma\to k(u)$ (for example, transformations of the form $u^\sigma\to u^\sigma\ln(1+u)$, $ u^\sigma\to u^\sigma\exp[|\ln u|^{1/2}]$, etc.). Figures: 1. Bibliography: 24 titles.

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