The quenching behavior of a quasilinear parabolic equation with double singular sources (Q721027)

Let $u$ be a weak solution to the initial-boundary value problem \begin{align*} \partial_t u & = \partial_x \left( \left|\partial_ x u\right|^{p-2} \partial_x u \right) - u^{-r}, \quad t>0\ , \ x\in (0,1)\ , \\ \partial_x u(t,0) & = u^{-q}(t,0)\ , \ \partial_x u(t,1)=0\ , \quad t>0\ , \\ u(0,x) & = u_0(x)\ , \quad x\in (0,1), \end{align*} where the parameters satisfy $p>1$, $q>0$ and $r>0$, and the initial condition is positive, non-decreasing and concave with $\partial_x \left( \left|\partial_ x u_0\right|^{p-2} \partial_x u_0 \right) - u_0^{-r} \le 0$ in $(0,1)$ and not identically zero. It is shown that there is a time $T>0$ such that \[ \lim_{t\to T} \min_{[0,1]} u(t,x)=0 \] and that $x=0$ is the only point where this so-called quenching phenomenon occurs. Optimal temporal upper and lower bounds for $u(t,0)$ at the quenching time $T>0$ are derived when $r\ge q$, the rate being different when $r\in [q,1+pq]$ and $r>pq+1$.