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

• Published in: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 148; no. 3; pp. 559 - 574
• Main Authors: Iagar, Razvan Gabriel, Laurençot, Philippe
• Format: Journal Article
• Language: English
• Published: Edinburgh, UK Royal Society of Edinburgh Scotland Foundation 01.06.2018; Cambridge University Press; Royal Society of Edinburgh
• Subjects: Absorption; Analysis of PDEs; Classification; Decay rate; Differential equations; Diffusion rate; Extinction; Hamilton-Jacobi equation; Mathematical analysis; Mathematics; Ordinary differential equations; Primary 35C06; decay at infinity; comparison; 35K92; singular diffusion; 35B33; self-similar solutions; diffusive Hamilton–Jacobi equation; 35K67; Secondary 34D05; diffusive Hamilton-Jacobi equation
• ISSN: 0308-2105; 1473-7124
• DOI: 10.1017/S030821051700049X

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.