Super fast vanishing solutions of the fast diffusion equation

We will extend a recent result of B.Choi, P.Daskalopoulos and J.King. For any $n\ge 3$, $0<m<\frac{n-2}{n+2}$ and $\gamma>0$, we will construct subsolutions and supersolutions of the fast diffusion equation $u_t=\frac{n-1}{m}\Delta u^m$ in $\mathbb{R}^n\times (t_0,T)$, $t_0<T$, which decay at the rate $(T-t)^{\frac{1+\gamma}{1-m}}$ as $t\nearrow T$. As a consequence we obtain the existence of unique solution of the Cauchy problem $u_t=\frac{n-1}{m}\Delta u^m$ in $\mathbb{R}^n\times (t_0,T)$, $u(x,t_0)=u_0(x)$ in $\mathbb{R}^n$, which decay at the rate $(T-t)^{\frac{1+\gamma}{1-m}}$ as $t\nearrow T$ when $u_0$ satisfies appropriate decay condition.

In the recent paper [CDK] of B. Choi, P. Daskalopoulos and J. King they proved that for any n ≥ 3, m = n−2 n+2 and γ > 0, there exist finite time extinction solution of (1.3) which decay at the rate (T − t) 1+γ 1−m near the extinction time T > 0 when u 0 satisfies appropriate decay condition. They also proved the behaviour of such solutions near the extinction time and showed that such solutions have type II singularities near the extinction time.
In this paper we will extend their results. For any n ≥ 3, 0 < m < n−2 n+2 and γ > 0, we will construct subsolutions and supersolutions of (1.1) which decay at the rate (T − t) 1+γ 1−m as t ր T. As a consequence we obtain the existence of unique solution of the Cauchy problem (1.3) which decay at the rate (T − t) 1+γ 1−m as t ր T when u 0 satisfies appropriate decay condition.
We will use a modification of the technique of [CDK] to construct subsolutions and supersolutions of (1.1) using match asymptotic technique gluing some particular inner subsolutions (supersolutions respectively) and outer subsolutions (supersolutions, respectively) of (1.1). These subsolutions and supersolutions of (1.1) will then be used as barriers for constructing the unique solution of (1.3) when u 0 decays at the rate (T − t) 1+γ 1−m near the extinction time T > 0.
Unless stated otherwise we will let n and m satisfy (1.4) and m n − 2 n + 2 for the rest of the paper. Suppose u is a radially symmetric solution of (1.1) in R n × (0, T). Let w(s, t) = r 2 u(r, t) 1−m , s = log r, r = |x|, x ∈ R n . (1.5) Then w satisfies (1.7) By (1.6) and a direct computationŵ satisfies (1.11) Then φ 0 is positive in (A, ∞) and satisfies Hence φ 0 can be regarded as a limiting first order approximate solution of (1.8) as τ → ∞. Let Then Hence by assuming the boundedness ofŵ,ŵ η andŵ ηη , the term E 1 in (1.9) is negligible in the space-time region (e γτŵ (η, τ)) −1 = o(1) as τ → ∞. (1.12) This suggest that the domain is divided into the outer region given by (1.12) in which the diffusion and advection terms of the equation (1.9) are negligible and inner region given by e γτŵ (η, τ) = O(1) as τ → ∞ (1.13) in which the diffusion and advection terms of the equation (1.9) are not negligible. This suggests the transformation (1.14) Then by (1.7) and (1.14), (1.17) By (1.6), (1.16) and (1.17), Note that by (1.5) and (1.15), Let λ > 0 and v 0 be the unique radially symmetric solution of in R. Henceφ 0 may be considered as a limiting first order approximate stationary solution of (1.18) as τ → ∞. By adding some correction terms to the functions φ 0 andφ 0 we will construct subsolutions and supersolutions of (1.8) and (1.18) respectively in the outer region (1.12) and in the inner region (1.13) respectively. The plan of the paper is as follows. In section two we will construct subsolutions and supersolutions of (1.8) in the outer region. In section three we will construct subsolutions and supersolutions of (1.18) in the inner region using match asymptotic method. In section four we will construct distributional subsolutions and supersolutions of (1.1) and we will use these as barriers to construct the unique solution of (1.3).
We start with some definitions. For any open set O ∈ R n × (0, T) we say that a positive function u on O is a solution (subsolution, supersolution, respectively) of (1 (≥, ≤, respectively) in the classical sense. For any 0 ≤ u 0 ∈ L 1 loc (R n ) we say that u is a solution (subsolution, supersolution, respectively) of (1.3) if u is a solution (subsolution, supersolution, respectively) of (1.1) in R n × (t 0 , T) and satisfies for any compact subset K of R n . We say that a function u on O is a weak solution (subsolution, supersolution, respectively) of (1 (≥, ≤, respectively) for any f ∈ C ∞ 0 (O). We say that a functionŵ is a solution (subsolution, supersolution, respectively) of (1.8) in O ifŵ ∈ C 2,1 (O) satisfies L 0 (ŵ) = 0 in O (≤, ≥, respectively) in the classical sense. Similarly we say that a function w is a solution (subsolution, supersolution, respectively) of (1 Let t 2 > t 1 , R > 0, B R = {x ∈ R n : |x| < R} and Q R = B R × (t 1 , t 2 ). Let 0 ≤ g ∈ C((∂B R ×[t 1 , t 2 ))∪(B R ×{t 1 })). We say that a function ζ on Q R is a weak solution (subsolution, supersolution, respectively) of where ∂/∂n is the dervative with respect to the unit outward normal n on ∂B R .
By Lemma 2.5, Lemma 2.6 and Lemma 2.7 we have the following result.

Subsolution and supersolution in the domain
In this section we will construct subsolutions and supersolutions of (1.18) in the inner region using match asymptotic method. Since the construction is similar to section 6 of [CDK] we will only sketch the argument here. Let λ > 0 andφ 0 be given by (1.21). We first recall some results of [Hs1] and [Hs5].  [Hs5]) Let n ≥ 3, 0 < m < n−2 n andφ 0 be a solution of (1.22) given by (1.21). Thenφ 0 ∈ C ∞ (R) andφ 0,s (s) > 0 for any s > 0. Moreover if m n−2 n+2 , then the following holds: (1 − m)γA log s + K 1 + o(1) as s → ∞ for some constant K 1 = K 1 ∈ R depending on λ, m, n and A.
where L 1 is given by (1.19).
where a 0 is given by (1.11).
Then v 2 (x, t) ≥ v 1 (x, t) for any (x, t) ∈ Q R . By Theorem 2.3 of [HP], Lemma 4.2, Lemma 4.3 and an argument similar to the of proof of Theorem 1.1 of [Hs2] we have the following result.