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 $u_t=\Delta u^m$ in $({\mathbb{R}}^n\setminus \{0\})\times (0,\infty )$ in the subcritical case $0<m<\frac{n-2}{n}$, $n\ge 3$. Firstly, we prove the existence of singular solution $u$ of the above equation that is trapped in between self-similar solutions of the form of $t^{-\alpha }{f}_i(t^{-\beta }x)$, $i=1,2$, with initial value $u_0$ satisfying $A_1|x{|}^{-\gamma }\le u_0\le A_2|x{|}^{-\gamma }$ for some constants $A_2>A_1>0$ and $\frac{2}{1-m}<\gamma <\frac{n-2}{m}$, where $\beta :=\frac{1}{2-\gamma (1-m)}$, $\alpha :=\frac{2\beta -1}{1-m},$ and the self-similar profile $f_i$ satisfies the elliptic equation $$ Δf^m+αf+βx\cdot \nabla f=0\quad \mbox{in ${\mathbb R}^n\setminus\{0\}$} $$ with ${\lim }_{|x|\to 0}|x{|}^{\frac{\alpha }{\beta }}{f}_i(x)=A_i$ and ${\lim }_{|x|\to \infty }|x{|}^{\frac{n-2}{m}}{f}_i(x)=D_{A_i}$ for some constants $D_{A_i}>0$. When $\frac{2}{1-m}<\gamma <n$, under an integrability condition on the initial value $u_0$ of the singular solution $u$, we prove that the rescaled function $$\tilde{u}(y,\tau ):=t^{\alpha }u(t^{\beta }y,t),\tau :=\log t,$$ converges to some self-similar profile $f$ as $\tau \to \infty$.

37 pages

References in corpus (2)

• Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities
• Type II ancient compact solutions to the Yamabe flow