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, xut(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∗] ∪ [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.