CLASSIFICATION OF EXTINCTION PROFILES FOR A ONE-DIMENSIONAL DIFFUSIVE HAMILTON-JACOBI EQUATION WITH CRITICAL ABSORPTION

Razvan Gabriel Iagar (1) , Philippe Laurençot (2)

1 ICMAT - Instituto de Ciencias Matematicas
2 IMT - Institut de Mathématiques de Toulouse UMR5219

A classification of the behavior of the solutions f (·, a) to the ordinary differential equation (|f ′ |^{p−2} f ′) ′ + 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, the exponent p ranging in (1, 2). There is a threshold value a_* which separates different behaviors 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.

Mots clés: self-similar solutions; singular diffusion; diffusive Hamilton-Jacobi equation; decay at infinity; comparison

Domaines: Equations aux dérivées partielles [math.AP]

Razvan Gabriel Iagar, Philippe Laurençot. CLASSIFICATION OF EXTINCTION PROFILES FOR A ONE-DIMENSIONAL DIFFUSIVE HAMILTON-JACOBI EQUATION WITH CRITICAL ABSORPTION. Proceedings of the Royal Society of Edinburgh: Section A, Mathematics, 2018, 148, pp.559--574. ⟨10.1017/S030821051700049X⟩. ⟨hal-01324506⟩