Quenching for a degenerate parabolic problem due to a concentrated nonlinear source

Let $q$, $a$, $T$, and $b$ be any real numbers such that $q \ge 0$, $a > 0$, $T > 0$, and $0 < b < 1$. This article studies the following degenerate semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source situated at $b$: \[ {x^q}{u_t} - {u_{xx}} = {a^2}\delta \left ( x - b \right )f\left ( u\left ( x, t \right ) \right ) in \left ( 0, 1 \right ) \times \left ( 0, T \right ],\] \[ u\left ( x, 0 \right ) = 0 on \left [ 0, 1 \right ], u\left ( 0, t \right ) = u\left ( 1, t \right ) = 0 for \; 0 < t \le T,\] where $\delta \left ( x \right )$ is the Dirac delta function, $f$ is a given function such that ${\lim _{u \to {c^ - }}}f\left ( u \right ) = \infty$ for some positive constant $c$, and $f\left ( u \right )$ and $f’\left ( u \right )$ are positive for $0 \le u < c$. It is shown that the problem has a unique continuous solution $u$ before $max\left \{ {u\left ( x, t \right ) : 0 \le x \le 1} \right \}$ reaches ${c^ - }$, $u$ is a strictly increasing function of $t$ for $0 < x < 1$, and if $max\left \{ {u\left ( x, t \right ):0 \le x \le 1} \right \}$ reaches ${c^ - }$, then $u$ attains the value $c$ only at the point $b$. The problem is shown to have a unique ${a^*}$ such that a unique global solution $u$ exists for $a \le {a^*}$, and $max\left \{ {u\left ( x, t \right ):0 \le x \le 1} \right \}$ reaches ${c^ - }$ in a finite time for $a > {a^*}$; this ${a^*}$ is the same as that for $q = 0$. A formula for computing ${a^*}$ is given, and no quenching in infinite time is deduced.