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 → c f ( 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 ∫   0 cf ( u ) d u < ∞, 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.