On the blow up of $u_ t$ at quenching

Abstract:

Let $\Omega$ be a bounded convex domain in ${{\mathbf {R}}^n}$ with smooth boundary. We consider the problems $\left ( C \right ):{u_t} = \Delta u + \varphi \left ( u \right )$ in $\Omega \times \left ( {0,T} \right )$, while $u = 0$ on $\partial \Omega \times \left ( {0,T} \right )$ and $u\left ( {x,0} \right ) = {u_0}\left ( x \right )$. Here $\varphi \left ( u \right ):\left ( { - \infty ,A} \right ) \to \left ( {0,\infty } \right )\left ( {A > 0} \right )$ satisfies $\varphi ’\left ( u \right ) \geq 0,\varphi ''\left ( u \right ) \geq 0$, and ${\lim _{u \to {A^ - }}}\varphi \left ( u \right ) = + \infty$, while ${u_0}$ satisfies $\Delta {u_0}\left ( x \right ) + \varphi \left ( {{u_0}\left ( x \right )} \right ) \geq 0$. We show that if $u$ quenches (reaches $A$ in finite time), then the quenching points are in a compact subset of $\Omega$ and ${u_t}$ blows up. We also extend the result to the third boundary value problem.