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 >= 0, a > 0, 0 < b < 1, and T > 0. This article studies the following degenerate semilinear parabolic first initial-boundary value problem, x(q) u(t)(x,t) - u(xx)(x,t) = a delta(x - b)f (u(x,t)) for 0<x<1, 0<t <= T, u(x,0) = psi(x) for 0 <= x <= 1, u(0, t) = u(1, t) = 0 for 0 < t <= T, where 6 (x) is the Dirac delta function, and f and V) are given functions. It is shown that for a sufficiently large, there exists a unique number b* epsilon (0, 1/2) such that u never blows up for b epsilon (0, b*] U [1 - b*, 1), and u always blows up in a finite time for b epsilon (b*, 1 - b*). To illustrate our main results, two examples are given.