A blow-up criterion for a degenerate parabolic problem due to a concentrated nonlinear source

Let $q$, $a$, $b$, and $T$ be real numbers with $q\geq 0$, $a>0$, $0<b<1$, and $T>0$. This article studies the following degenerate semilinear parabolic first initial-boundary value problem, \begin{gather*} x^{q}u_{t}(x,t)-u_{xx}(x,t)=a\delta (x-b)f\left ( u(x,t)\right ) \text { for }0<x<1,\text { }0<t\leq T,\\ u(x,0)=\psi (x)\text { for }0\leq x\leq 1\text {, }u(0,t)=u(1,t)=0\text { for }0<t\leq T, \end{gather*} where $\delta \left ( x\right )$ is the Dirac delta function, and $f$ and $\psi$ are given functions. It is shown that for $a$ sufficiently large, there exists a unique number $b^{\ast }\in \left ( 0,1/2\right )$ such that $u$ never blows up for $b\in \left ( 0,b^{\ast }\right ] \cup \left [ 1-b^{\ast },1\right )$, and $u$ always blows up in a finite time for $b\in (b^{\ast },1-b^{\ast })$. To illustrate our main results, two examples are given.