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′ ? 0,f″ ? 0, and lim_u→cf(u) = ∞. Also, let χ(S) be the characteristic function of the set S. This article studies all solutions u with non-negative u_t, in the region where u < c and with continuous u_x for the problem: u_xx – u_t = ? f(u)χ({u < c}), 0 < x < a, 0 < t < ∞, subject to zero initial and first boundary conditions. For any length a larger than the critical length, it is shown that if ∫f(u) du < ∞, 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.