On the method of stationary states for quasilinear parabolic equations

A method is presented for investigating the space-time structure of unbounded nonnegative solutions of quasilinear parabolic equations of the form $u_t=\mathbf A(u)$, where $\mathbf A$ is a nonlinear elliptic operator. Three examples are considered in detail: the Cauchy problem for the equation
$$ u_t=\nabla\cdot((1+|\nabla u|^2)^{\sigma/2}\nabla u)+u^\beta, $$
where $\sigma>0$ and $\beta>1$ are constants; the boundary value problem in $\Omega=R^3\cap\{x_3>0\}$
\begin{gather*} u_t=\nabla\cdot((1+u^\sigma)\nabla u),\qquad t>0,\quad x\in\Omega; \\ -(1+u^\sigma)u_{x_3}=u^\alpha,\qquad t>0,\ x_3=0;\quad\alpha=\mathrm{const}>0; \end{gather*}
and the Cauchy problem for the system $u_t=\nabla\cdot((1+u^2)^{1/2}\nabla u)+vw$, $v_t=\nabla\cdot((1+v^2)\nabla v)+u^pw$, $w_t=\nabla\cdot((1+w^2)^{3/2}\nabla w)uw$, $p\geqslant1$. It is assumed that at the point $x=0$ the solution grows without bound as $t\to T_0^-<+\infty$. The derivation of an estimate of the solution near $t=T_0^-$, $x=0$ is based on an analysis of an appropriate family of stationary solutions $\{U_\lambda\}$: $\mathbf A(U_\lambda)=0$, $U_\lambda(0)=\lambda$, $\lambda>0$ a parameter. It is shown that the behavior of a solution as $t\to T_0^-$ depends to large extent on the structure of the “envelope” $L(x)=\sup\limits_{\lambda>0}U_\lambda(x)$. In particular, if $L(x)\equiv+\infty$, then $u(t,x)$ grows without bound as $t\to T_0^-$ at points arbitrarily far from $x=0$. If $L(x)<+\infty$ for $x\ne0$, then $L(x)$ determines a lower bound for $u(t,x)$ in a neighborhood of $t=T_0^-$, $x=0$.
Bibliography: 28 titles.