Beyond quenching for singular reaction-diffusion problems

✍ Scribed by C. Y. Chan; Lan Ke

• Publisher: John Wiley and Sons
• Year: 1994
• Tongue: English
• Weight: 425 KB
• Volume: 17
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1670170102

✦ Synopsis

Abstract

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 ∫f(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.