Minimal blow-up asymptotics of quasilinear heat equations.
In: Proceedings of the Royal Society of Edinburgh: Section A: Mathematics, Jg. 131 (2001-06-01), Heft 6, S. 1297-1321
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Zugriff:
We study the asymptotic properties of blow-up solutions u = u(x, t) ≥ 0 of the quasilinear heat equation , where k(u) is a smooth non-negative function, with a given blowing up regime on the boundary u(0, t) = ψ(t) > 0 for t ∈ (0, 1), where ψ(t) → ∞ 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 → 1−, in terms of the heat conductivity coefficient k(u) and of boundary data ψ(t); both are assumed to be monotone. We describe a domain, denoted by , of minimal asymptotics corresponding to the data ψ(t) with a slow growth as t → 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 ut = uxx described by its self-similar solution of the form u*(x, t) = −ln(1 − t) + g(ξ), ξ = 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 . [ABSTRACT FROM PUBLISHER]
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Titel: |
Minimal blow-up asymptotics of quasilinear heat equations.
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Autor/in / Beteiligte Person: | Chaves, M. ; Galaktionov, Victor A. |
Zeitschrift: | Proceedings of the Royal Society of Edinburgh: Section A: Mathematics, Jg. 131 (2001-06-01), Heft 6, S. 1297-1321 |
Veröffentlichung: | 2001 |
Medientyp: | academicJournal |
ISSN: | 0308-2105 (print) |
DOI: | 10.1017/S0308210500001402 |
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