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Large-time behaviour of solutions of a reaction–diffusion equation

Published online by Cambridge University Press:  14 November 2011

Arturo de Pablo
Arturo de Pablo
Affiliation:
Departamento de Ingeniería Industrial, Escuela Politécnica Superior, Universidad Carlos III de Madrid, 28913 Leganés, Spain
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Abstract

We investigate the large-time behaviour of the solutions u = u(x, t) to the one-dimensional nonlinear heat equation with reaction

with exponents m > 1,p < 1. The initial function u(x, 0) is assumed to be measurable and nonnegative. In the case m + p ≧ 2 where the initial value does not uniquely determine the solution, we also fix the positivity set of the solution u(x, t) if the support of u(x, 0) is not the whole line ℝ, i.e. u(x, t) > 0 if and only if −s1(t) <x<s2(t), t ≧ 0, where 0≦si(t)≦∞ for t ≧ 0, i = 1, 2 are lower semicontinuous given functions. We prove that u converges to a self-similar function which depends only on the behaviour of u(x, 0) for |x| large or si(t) for t large. We classify the set of self-similar solutions and study the equation satisfied by their interfaces.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 2 , 1994 , pp. 389 - 398
Copyright
Copyright © Royal Society of Edinburgh 1994

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