Quenching for a degenerate parabolic problem due to a concentrated nonlinear source

Let q, a, T, and b be any real numbers such that q greater than or equal to 0, a > 0, T > 0, and 0 < b < 1. This article studies the following degenerate semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source situated at b: x(q)u(t) - u(xx) = a(delta)(2)(x - b) f(u(x, t)) in (0, 1) x (0, T], u(x, 0) = 0 on [0, 1], u(0, t) = u(1, t) = 0 for 0 < t less than or equal to T, where delta (x) is the Dirac delta function, f is a given function such that lim(u-->c)(-) f(u) = infinity for some positive constant c, and f (u) and f'(u) are positive for 0 less than or equal to u < c. It is shown that the problem has a unique continuous solution u before max{u(x, t) : 0 less than or equal to x less than or equal to 1} reaches c, u is a strictly increasing function of t for 0 < x < 1, and if max{u(x, t) 0 < x < 1} reaches c(-), then u attains the value c only at the point b. The problem is shown to have a unique a(*) such that a unique global solution u exists for a < a(*), and max{u(x, t) : 0 < x < 1} reaches c in a finite time for a > a(*); this a(*) is the same as that for q = 0. A formula, for computing a(*) is given, and no quenching in infinite time is deduced.