IngentaConnect Minimal blow-up asymptotics of quasilinear heat equations

Authors: Chaves M.; Galaktionov V.A.

Source: Proceedings Section A: Mathematics - Royal Society of Edinburgh, Volume 131, Number 6, 14 December 2001 , pp. 1297-1321(25)

Publisher: Royal Society of Edinburgh

Abstract:

We study the asymptotic properties of blow-up solutions u = u(x, t)

0 of the quasilinear heat equation

u_t = (k(u)u_x)_x, x > 0, t isin

(0, 1),

where k(u) is a smooth non-negative function, with a given blowing up regime on the boundary u(0, t) = (t) > 0 for t isin

(0, 1), where (t) rarr

as t

1^-, and bounded initial data u(x, 0)

0. We classify the asymptotic properties of the solutions near the blow-up time, t rarr

1^-, in terms of the heat conductivity coefficient k(u) and of boundary data (t); both are assumed to be monotone. We describe a domain, denoted by S₁₁^-, of minimal asymptotics corresponding to the data (t) with a slow growth as t rarr

1^- and a class of nonlinear coefficients k(u).

We prove that for any problem in S₁₁^-, such a blow-up singularity is asymptotically structurally equivalent to a singularity of the heat equation u_t = u_xx described by its self-similar solution of the form u_*(x, t) = -ln(1 - t) + g(), = x/(1 - t)^1/2, where g solves a linear ordinary differential equation. This particular self-similar solution is structurally stable upon perturbations of the boundary function and also upon nonlinear perturbations of the heat equation with the basin of attraction S₁₁^-.