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