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

A rather general approach is presented to the investigation of the asymptotic behavior of solutions to boundary value problems for quasilinear parabolic equations
$$ \frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\biggl(k(u)\frac{\partial u}{\partial x}\biggr) $$
with arbitrary coefficients $k(u)>0$, $u>0$, and arbitrary boundary regimes $u(t,0)=\psi(t)$ (the problem is considered in the half space $x \in(0,+\infty)$). The investigation is carried out by constructing so-called approximate self-similar solutions which do not satisfy the equation but to which the solution of the problem converges asymptotically in special norms. In this paper the case $[k(u)/k'(u)]'-1/\sigma$ as $u\to+\infty$, $\sigma=\operatorname{const}t>0$, is considered.
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