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Classification of extinction profiles for a one-dimensional diffusive Hamilton–Jacobi equation with critical absorption

Published online by Cambridge University Press:  22 April 2018

Razvan Gabriel Iagar
Affiliation:
Instituto de Ciencias Matemáticas (ICMAT), Nicolás Cabrera, 13–15, Campus de Cantoblanco, 28049 Madrid, Spain (razvan.iagar@icmat.es)
Philippe Laurençot
Affiliation:
Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, 31062 Toulouse Cedex 9, France (laurenco@math.univ-toulouse.fr)
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Abstract

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A classification of the behaviour of the solutions f(·, a) to the ordinary differential equation (|f′|p-2f′)′ + f - |f′|p-1 = 0 in (0,) with initial condition f(0, a) = a and f′(0, a) = 0 is provided, according to the value of the parameter a > 0 when the exponent p takes values in (1, 2). There is a threshold value a* that separates different behaviours of f(·, a): if a > a*, then f(·, a) vanishes at least once in (0,) and takes negative values, while f(·, a) is positive in (0,) and decays algebraically to zero as r→∞ if a ∊ (0, a*). At the threshold value, f(·, a*) is also positive in (0,) but decays exponentially fast to zero as r→∞. The proof of these results relies on a transformation to a first-order ordinary differential equation and a monotonicity property with respect to a > 0. This classification is one step in the description of the dynamics near the extinction time of a diffusive Hamilton–Jacobi equation with critical gradient absorption and fast diffusion.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2018