On approximate self-similar solutions of a~class of quasilinear heat equations with a~source

V. A. Galaktionov ; S. P. Kurdyumov ; A. A. Samarskii

Sbornik. Mathematics, Tome 52 (1985) no. 1, pp. 155-180

Résumé

Quasilinear parabolic equations of the form $$ \frac{\partial u}{\partial t}=\nabla(k(u)\nabla u)+Q(u),\qquad\nabla(\,\cdot\,) =\operatorname{grad}_x(\,\cdot\,),\quad k\geqslant0, $$ are considered; here $k(u)$ and $Q(u)$ are sufficiently smooth given functions (respectively, the coefficient of thermal conductivity and the power of heat sources depending on the temperature $u=u(t,x)\geqslant0$). A family of coefficients $\{k\}$ and corresponding functions $\{Q_k\}$ is distinguished for which the properties of the solution of the boundary value problem for the equation in question are described by invariant solutions $v_A(t,x)$ of a first-order equation of Hamilton–Jacobi type $$ \frac{\partial v}{\partial t}=\frac{k(v)}{v+1}(\nabla v)^2 +G(t)\nabla\mathbf{vx}+H(t)Q_k(v). $$ The function $u_A$ is an approximate self-similar solution of the original equation. Tables: 1. Figures: 1. Bibliography: 70 titles.