Instantaneous shrinking and single point extinction for viscous Hamilton-Jacobi equations with fast diffusion

For a large class of non-negative initial data, the solutions to the quasilinear viscous Hamilton-Jacobi equation ∂tu−∆pu+ |∇u| = 0 in (0,∞)×R are known to vanish identically after a finite time when 2N/(N + 1) < 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 u0 decays sufficiently rapidly as |x| → ∞, that is, for each t > 0, the positivity set of u(t) is a bounded subset of R even if u0 > 0 in R N . This decay condition on u0 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 R 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. AMS Subject Classification: 35K59, 35K67, 35K92, 35B33, 35B40.