Global behaviour of solutions of the fast diffusion equation

Abstract

We will extend a recent result of B. Choi and P. Daskalopoulos ([CD]). For any n ≥ 3, 0 < m < n−2 n , m n−2 n+2 , β > 0 and λ > 0, we prove the higher order expansion of the radially symmetric solution v λ,β (r) of n−1As a consequence for any n ≥ 3 and 0 < m < n−2 n if u is the solution of the equation u t = n−1 m ∆u m in R n × (0, ∞) with initial value 0 ≤ u 0 ∈ L ∞ (R n ) satisfying u 0 (x) 1−m = 2(n−1)(n−2−nm) (1−m)β|x| 2 log |x| − n−2−(n+2)m 2(n−2−nm) log(log |x|) + K 1 + o(1)) as |x| → ∞ for some constants β > 0 and K 1 ∈ R, then as t → ∞ the rescaled function u(x, t) = e 2β 1−m t u(e βt x, t) converges uniformly on every compact subsets of R n to v λ 1 ,β for some constant λ 1 > 0.

Super fast vanishing solutions of the fast diffusion equation

“…We will prove that for sufficiently large ξ 1 there exists τ 0 > τ 2 such that ψ + ε (ξ, τ ) and ψ − ε (ξ, τ ) are supersolution and subsolution of (18) in the region (−∞, ξ 1 ) × (τ 0 , ∞). We first observe that sinceφ 0 (s) is a smooth strictly monotone increasing function of s, by (116) and (117) we have the following result.…”

“…Subsolution and supersolution in the domain. In this section we will construct subsolutions and supersolutions of (18) in the inner region using match asymptotic method. Since the construction is similar to section 6 of [5] we will only sketch the argument here.…”

“…for some constant β > 0, asymptotic large time behaviour of global solution of (3) was proved by S.Y. Hsu in [18]. First order symptotic behaviour of the self-similar solutions of (1) when n ≥ 3, 0 < m < n−2 n , was proved by S.Y.…”

“…In section two we will construct subsolutions and supersolutions of (8) in the outer region. In section three we will construct subsolutions and supersolutions of (18) in the inner region using match asymptotic method. In section four we will construct distributional subsolutions and supersolutions of (1) and we will use these as barriers to construct the unique solution of (3).…”

“…respectively) in the classical sense. Similarly we say that a function w is a solution (subsolution, supersolution, respectively) of (18)…”