Infinite-Time Quenching in a Fast Diffusion Equation with Strong Absorption

Abstract

Abstract. This work is concerned with the fast diffusion equation ut = Δu m − u κ in R n , where 0 < m < 1 and κ < 1. A global positive solution is said to quench regularly in infinite time if u(x k , t k ) → 0 for some bounded sequence (x k ) k∈N and some t k → ∞, and if sup (x,t)∈K×(0,∞) u(x, t) < ∞ for all compact K ⊂⊂ R n . It is shown that such regular quenching in infinite time occurs for a large class of initial data if κ > m, whereas it is impossible in one space dimension when κ < −m and the solution is radially symmetric and nondecreasing for x > 0. Mathematics Subject Classification (2000). 35K55, 35K65, 35B40.

Non-self-similar dead-core rate for the fast diffusion equation with strong absorption

“…The fast diffusion equation with strong absorption in (1.1) was studied before from the point of view of decay and extinction [7,8,34,35,41], and of regularity and behaviour of interfaces [1,12,13]. We note that the porous medium or slow diffusion case m > 1 > p was also studied from the latter point of view [4,[12][13][14][15] and that the existence of finite-time dead-core was proved in [2].…”