arXiv 1407.26962014-07math.AP

快扩散方程解在消亡时附近的渐近性

Asymptotic behaviour of solutions of the fast diffusion equation near its extinction time

Kin Ming Hui

备注：: 22 pages, the proof of Theorem 1.4 is completely re-written, Lemma 5.2 is added and other typos corrected

摘要：

令$n\ge 3$、$0<m<\frac{n-2}{n}$、$ρ_1>0$、$β\ge\frac{mρ_1}{n-2-nm}$和$α=\frac{2β+ρ_1}{1-m}$。对于任意的$λ>0$，我们证明了椭圆方程$Δv^m+αv+βx\cdot\nabla v=0$，$v>0$在$R^n\setminus\{0\}$中满足$\displaystyle\lim_{|x|\to 0}|x|^{α/β}g_λ(x)=λ^{-\frac{ρ_1}{(1-m)β}}$的径向对称奇异解$g_λ\in C^{\infty}(R^n\setminus\{0\})$的存在性和唯一性（对于$β\ge\frac{ρ_1}{n-2-nm}$）.当$β$足够大时，我们证明了上述椭圆型方程的径向对称解的高阶渐近性态为$|x|\to\infty$。我们还得到了上述方程的径向对称解的反演公式。作为结果，我们将证明快速扩散方程$u_t=Δu^m$在$R^n\times (0,T)$中的解$u$在消亡时$T>0$附近的消失行为。

英文摘要：

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $ρ_1>0$, $β\ge\frac{mρ_1}{n-2-nm}$ and $α=\frac{2β+ρ_1}{1-m}$. For any $λ>0$, we will prove the existence and uniqueness (for $β\ge\frac{ρ_1}{n-2-nm}$) of radially symmetric singular solution $g_λ\in C^{\infty}(R^n\setminus\{0\})$ of the elliptic equation $Δv^m+αv+βx\cdot\nabla v=0$, $v>0$, in $R^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{α/β}g_λ(x)=λ^{-\frac{ρ_1}{(1-m)β}}$. When $β$ is sufficiently large, we prove the higher order asymptotic behaviour of radially symmetric solutions of the above elliptic equation as $|x|\to\infty$. We also obtain an inversion formula for the radially symmetric solution of the above equation. As a consequence we will prove the extinction behaviour of the solution $u$ of the fast diffusion equation $u_t=Δu^m$ in $R^n\times (0,T)$ near the extinction time $T>0$.

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英文摘要：

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