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

Abstract

We study the asymptotic large time behavior of singular solutions of the fast diffusion equation ut = ∆u m in (R n \ {0}) × (0, ∞) in the subcritical case 0 < m < n−2 n , n ≥ 3. Firstly, we prove the existence of the singular solution u of the above equation that is trapped in between selfsimilar solutions of the form of t −α f i (t −β x), i = 1, 2, with the initial value u 0 satisfying A 1 |x| −γ ≤ u 0 ≤ A 2 |x| −γ for some constants A 2 > A 1 > 0 and

Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space

“…More precisely, if f is a solution of (5), then as proved by K.M. Hui and Soojung Kim [20] the function g given by…”

“…We note that for 0 < m < n−2 n and n ≥ 3, forward self-similar solutions of (1) which blow up at the origin for t > 0 are constructed by K.M. Hui and Soojung Kim in [20]. The self-similar solutions constructed in [20] are of the form,…”

“…In the study of the L 1 -contraction, we found that if we let |x| −µ with 0 < µ ≤ n − 2/(1 − m) as the weight function and estimate the difference of two solutions by a self-similar solution (cf. [20]), then the growth rates of the solutions at the origin are too high so that one does not have an appropriate L 1 -contraction for the uniqueness and large time asymptotic behaviour.…”

“…It is proved in [20] that such function f with α, β satisfying ( 9) satisfies lim r→0 r α/β f (r) = A and lim…”

“…In section 2, we will use the inversion technique of K.M. Hui and Soojung Kim [20] to study the existence and uniqueness of the solution of (5) with α, β satisfying (6). More precisely, if f is a solution of (5), then as proved by K.M.…”