Single-point blow-up for a degenerate parabolic problem due to a concentrated nonlinear source

Let $q$ be a nonnegative real number, and $T$ be a positive real number. This article studies the following degenerate semilinear parabolic first initial-boundary value problem: \[ {x^q}{u_t}\left ( x, t \right ) - {u_{xx}}\left ( x, t \right ) = {a^2}\delta \left ( x - b \right )f\left ( u\left ( x, t \right ) \right ) \qquad for 0 < x < 1, 0 < t \le T\], \[ u\left ( x, 0 \right ) = \psi \left ( x \right ) \qquad for 0 \le x \le 1\], \[ u\left ( 0, t \right ) = u\left ( 1, t \right ) = 0 \qquad for 0 < t \le T\], where $\delta \left ( x \right )$ is the Dirac delta function, and $f$ and $\psi$ are given functions. It is shown that the problem has a unique solution before a blow-up occurs, $u$ blows up in a finite time, and the blow-up set consists of the single point $b$. A lower bound and an upper bound of the blow-up time are also given. To illustrate our main results, an example is given. A computational method is also given to determine the finite blow-up time.