Single-point blow-up for a degenerate parabolic problem with a nonlinear source of local and nonlocal features

Let q, m and T be any real numbers such that q ≥ 0, m > 1, and T > 0. This article studies the following degenerate semilinear parabolic first initial-boundary value problem: xqut(x, t) - uxx(x, t) = amq+2δ(x - b)f(u(x, t))Um(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 U(t) 10 xq|u(x, t)|dx, δ(x) is the Dirac delta function, and f and ψ are given functions such that f(0) ≥ 0, f(u) and f'(u) are positive for u > 0, and ψ is nontrivial, nonnegative and continuous such that ψ(0) = 0 = ψ(1), and ψ + amq+2δ(x - b)f(ψ(x))Um(0) ≥ 0 for 0 < x < 1. It is shown that it has a unique solution before a blow-up occurs. A criterion for u to blow-up in a finite time is given. If u blows up, then the blow-up set consists of the single-point b.