2006/09/19 by Panagiota Daskalopoulos, Daskalopoulos, Panagiota, Nataša Šešum +1
Mathematics · Medicine · #35B40 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.math/0609513
openalex publication_date 2006/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the extinction behavior of solutions to the fast diffusion equation ut = Δum on \RN× (0,T), in the range of exponents m ∈ (0, (N-2)/(N)), N > 2. We show that if the initial data u0 is trapped in between two Barenblatt solutions vanishing at time T, then the vanishing behaviour of u at T is given by a Barenblatt solution. We also give an example showing that for such a behavior the bound from above by a Barenblatt solution B (vanishing at T) is crucial: we construct a class of solutions u with initial data u0 = B (1 + o(1)), near |x| >> 1, which live longer than B and change behaviour at T. The behavior of such solutions is governed by B(⋅,t) up to T, while for t >T the solutions become integrable and exhibit a different vanishing profile. For the Yamabe flow (m = (N-2)/(N+2)) the above means that these solutions u develop a singularity at time T, when the Barenblatt solution disappears, and at t >T they immediately smoothen up and exhibit the vanishing profile of a sphere. In the appendix we show how to remove the assumption on the bound on u0 from below by a Barenblatt.