Exact local estimates for the supports of solutions in problems for non-linear parabolic equations

The phenomenon of instantaneous shrinking of the support in the Cauchy problem for non-linear parabolic equations with a positive initial function that is infinitesimal as $|x|\to\infty$ is considered. Exact local estimates for the boundary of the support of the solutions are proved. For example, the exact asymptotic formula
$$ u_0\bigl(\eta^\pm(t)\bigr)\sim\bigl[(1-\beta)t\bigr]^{1/(1-\beta)}, \qquad t\to 0, $$
holds for the solution of the equation $u_t=(u^nu_x)_x-u^\beta$, $0<\beta<1$, $n\geqslant 1-\beta$, where $\eta^+(t)=\sup\bigl\{x:u(x,t)>0\bigr\}$ and $\eta^-(t)=\inf\bigl\{x:u(x,t)>0\bigr\}$.