Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Abstract

Let n ≥ 3, 0 < m ≤ (n − 2)/n, p > max(1, (1 − m)n/2), and 0 ≤ u 0 ∈ L p loc (R n ) satisfy lim inf R→∞ R −n+ 2 1−m |x|≤R u 0 dx = ∞. We prove the existence of unique global classical solution of u t = n−1 m ∆u m , u > 0, in R n × (0, ∞), u(x, 0) = u 0 (x) in R n . If in addition 0 < m < (n − 2)/n and u 0 (x) ≈ A|x| −q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t α u(t β x, t) converges uniformly on every compact subset of R n to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x| −q as t → ∞. Note that when m = (n − 2)/(n + 2), n ≥ 3, if g ij = u 4 n+2 δ ij is a metric on R n that evolves by the Yamabe flow ∂g ij /∂t = −Rg ij with u(x, 0) = u 0 (x) in R n where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.

Global behaviour of solutions of the fast diffusion equation

“…holds and φ satisfies (1.12) for some constant K 1 ∈ R where c 1 = 2(n − 1)(n − 2 − nm)/(1 − m). If u is the unique solution of (1.1) in R n × (0, ∞) given by Theorem 1.1 of [Hs2], then as t → ∞, the rescaled function u(x, t) given by (1.6) converges to v λ 1 ,β uniformly in C 2,1 (E) for any compact subset E ⊂ R n with λ 1 = e 2(K 1 −K 0 ) /β 1 1−m where the constant K 0 is given by Theorem 1.1. Moreover lim sup |x|→∞ |x| 2 u(x, t) 1−m − c 1 β log |x| − n − 2 − (n + 2)m 2(n − 2 − nm) log(log |x|) ≤K 2 − 2(n − 1)(n − 2 − nm) (1 − m) t ∀t ≥ 0.…”

“…On the other hand if (1.3) holds and 0 ≤ u 0 ∈ L p loc (R n ) for some constant p > then S.Y. Hsu [Hs2] proved the existence and uniqueness of solutions of          u t = n − 1 m ∆u m in R n × (0, ∞)…”

“…for some constant K 1 ∈ R. If u is the unique solution of (1.1) in R n × (0, ∞) given by Theorem 1.1 of [Hs2], then as t → ∞, the rescaled function u(x, t) given by (1.6) converges to v λ 1 ,β in L 1 loc (R n ) with λ 1 = e 2K 1 /K 0 /β 1 1−m where the constant K 0 is given by Theorem 1.1.…”

Singular limits and properties of solutions of some degenerate elliptic and parabolic equations

“…Recently there is a lot of study on the equation [DGL], [DS1], [FVWY], , [Hs3], [KL], [PS], [VW1], [VW2], u t = △φ m (u), u > 0, (1.1)…”

Asymptotic large time behavior of singular solutions of the fast diffusion equation

“…King, M. del Pino, N. Sesum, M. Sáez, [6,7,8,9,24], S.Y. Hsu [15,16,17], K.M. Hui [18,19,20], M. Fila, J.L.…”