2015/10/02 by Razvan Gabriel Iagar, Iagar, Razvan Gabriel, Philippe Laurençot +3 · 1 citation
Mathematics · Medicine · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations
paper · doi:10.48550/arxiv.1510.00500
openalex publication_date 2015/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a large class of non-negative initial data, the solutions to the quasilinear viscous Hamilton-Jacobi equation ∂_t u-Δ_p u+|∇ u|q=0 in (0,∞)×\realN are known to vanish identically after a finite time when 2N/(N+1) \textless p ≤ 2 and q∈(0,p-1). Further properties of this extinction phenomenon are established herein: instantaneous shrinking of the support is shown to take place if the initial condition u_0 decays sufficiently rapidly as |x|→∞, that is, for each t \textgreater 0, the positivity set of u(t) is a bounded subset of \realN even if u_0 \textgreater 0 in \realN. This decay condition on u_0 is also shown to be optimal by proving that the positivity set of any solution emanating from a positive initial condition decaying at a slower rate as |x|→∞ is the whole \realN for all times. The time evolution of the positivity set is also studied: on the one hand, it is included in a fixed ball for all times if it is initially bounded (localization). On the other hand, it converges to a single point at the extinction time for a class of radially symmetric initial data, a phenomenon referred to as single point extinction. This behavior is in sharp contrast with what happens when q ranges in [p-1,p/2) and p∈ (2N/(N+1),2] for which we show complete extinction. Instantaneous shrinking and single point extinction take place in particular for the semilinear viscous Hamilton-Jacobi equation when p=2 and q∈ (0,1) and seem to have remained unnoticed.