BEYOND QUENCHING FOR SINGULAR REACTION-DIFFUSION PROBLEMS

Let f(u) be twice continuously differentiable on [0, c) for some constant c such that f(0) > 0,f' greater-than-or-equal-to 0,f'' greater-than-or-equal-to 0, and lim(u-->c)f(u) = infinity. Also, let chi(S) be the characteristic function of the set S. This article studies all solutions u with non-negative u, in the region where u < c and with continuous u(x) for the problem: u(xx) - u(t) = -f(u)chi({u < c}), 0 < x < a, 0 < t < infinity , subject to zero initial and first boundary conditions. For any length a larger than the critical length, it is shown that if integral-c/0f(u)du < infinity, then as t tends to infinity, all solutions tend to the unique steady-state profile U(x), which can be computed by a derived formula, furthermore, increasing the length a increases the interval where U(x) = c by the same amount. For illustration, examples are given.