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

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.
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