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 > 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: xqut uxx = a2S(x b)f(u(x, t)) in (0,1) x (0, T], u(:x, 0) = 0 on [0,1], u(0, t) = u(l, t) = 0 for 0 < t < T, where <5 (x) is the Dirac delta function, / is a given function such that limu^cf(u) = oo for some positive constant c, and f{u) and f'(u) are positive for 0 < u < c. It is shown that the problem has a unique continuous solution u before m&x{u(x,t) : 0 < x < 1} reaches c~, u is a strictly increasing function of t for 0 < x < 1, and if max{u(x,i) : 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.