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

V. A. Galaktionov ; A. A. Samarskii

Sbornik. Mathematics, Tome 48 (1984) no. 1, pp. 1-18

Résumé

A rather general approach to the investigation of the asymptotic behavior of solutions of quasilinear parabolic heat equations $$ \frac{\partial u}{\partial t}=\frac\partial{\partial x}\biggl(k(u)\frac{\partial u}{\partial x}\biggr);\qquad k(u)>0,\quad u>0. $$ is proposed. The investigation is carried out by constructing so-called approximate self-similar solutions (ap.s-s.s's.) which do not satisfy the equation but to which solutions of the problems considered converge asymptotically. A system of ap.s-s.s's. which is complete in a particular sense is constructed for the case where the coefficient $k(u)$ satisfies the condition $[k(u)/k'(u)]'\to0$ as $u\to+\infty$ (for example, $k(u)=\exp(u^\lambda)$, $\lambda>0$; $k(u)=\exp(\exp u)$, etc.). Bibliography: 4 titles.