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Kin Ming Hui

Publications and source records attributed to Kin Ming Hui.

At least 19 recordsLinked to original sources
arXiv2025

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

Let $n\ge 3$, $0 0$, for the fast diffusion equation $u_t=\Delta (u^m/m)$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$, where $f$ satisfies \begin{equation*} \Delta (f^m/m) + \alpha f + \beta x \cdot \nabla f =0 \quad \text{in} \; \mathbb{R}^n\setminus\{0\} \end{equation*} with $\lim_{|x| \to 0} |x|^{ \frac{\alpha}{\beta}}f(x)=A$ and $\lim_{|x| \to \infty}f(x) = D_A$ for some constants $A>0$, $D_A > 0$. We also obtain an asymptotic expansion of such singular radially symmetric solution $f$ near the origin. We will also prove the asymptotic large time behaviour of the singular solutions of the fast diffusion equation $u_t= \Delta (u^m/m)$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$, $u(x,0)=u_0(x)$ in $\mathbb{R}^n\setminus\{0\}$, satisfying the condition $A_1|x|^{-\gamma}\leq u_0(x)\leq A_2|x|^{-\gamma}$ in $\mathbb{R}^n\setminus\{0\}$, for some constants $A_2>A_1>0$ and $n\le\gamma<\frac{n-2}{m}$.

math.AP↗
arXiv2025

A matching construction of self-similar profiles for the fast diffusion equation

Let $n\ge 3$, $0 0$, $\eta_0>0$, $\rho_1>0$, $\beta_-(\rho_1)=-\frac{\rho_1}{2}$, $\beta_+(\rho_1)=\frac{m\rho_1}{n-2-nm}$ and $\alpha=\frac{2\beta+\rho_1}{1-m}$. For any $\beta_-(\rho_1)\le\beta\le\beta_+(\rho_1)$, we construct the unique maximal positive radial branch of \[ \Delta(f^m/m)+\alpha f+\beta x\cdot\nabla f=0 \] issuing from prescribed origin data $f(0)=\eta_0$ and $f_r(0)=0$. For any $\beta\le\beta_+(\rho_1)$, we construct the unique maximal positive radial branch at infinity satisfying \[ \lim_{|x|\to\infty}|x|^{\frac{n-2}{m}}f(x)=\eta. \] The unmatched branches may reach zero at a finite radius. We formulate the shooting construction through the slope--amplitude equations of the increasing and decreasing half-branches before their first turning points. As a consequence we obtain a new proof of the existence result of Peletier and Zhang \cite{PeZ}: there exists $\beta\in (\beta_-(\rho_1),\beta_+(\rho_1))$ for which the equation $\Delta(f^m/m)+\alpha f+\beta x\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n$ has a positive radial solution $f$ satisfying \[ f(0)=\eta _0,\qquad f_r(0)=0, \qquad \lim_{r\to\infty}r^{\frac{n-2}{m}}f(r) =C_*\rho _1^{-\frac{n-2}{2m}} \eta _0^{-\frac{n-2-nm}{2m}}. \] for some constant $C_*>0$ depending on $n$, $m$, and is independent of $\rho_1$ and $\eta_0$. For every selected matching value of $\beta$, this solution is unique among positive radial solutions with the prescribed origin data. When $\rho _1=1$, the function $V(x,t)=(T-t)^\alpha f((T-t)^\beta x)$ is a backward self-similar solution of $u_t=\Delta(u^m/m)$ in $\mathbb{R}^n\times (-\infty,T)$.

math.AP↗
arXiv2024

Existence and uniqueness of the singular self-similar solutions of the fast diffusion equation and logarithmic diffusion equation

Let $n\ge 3$, $0 0$, $η>0$, $β>\frac{mρ_1}{n-2-nm}$, $α=α_m=\frac{2β+ρ_1}{1-m}$, $β_0>0$ and $α_0=2β_0+1$. We use fixed point argument to give a new proof for the existence and uniqueness of radially symmetric singular solution $f=f^{(m)}$ of the elliptic equation $Δ(f^m/m)+αf+βx\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{α/β}f(x)=η$. We also prove the existence and uniqueness of radially symmetric singular solution $g$ of the equation $Δ\log g+α_0 g+β_0x\cdot\nabla g=0$, $g>0$, in $\mathbb{R}^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{α_0/β_0}g(x)=η$. Such equations arises from the study of backward singular self-similar solution of the fast diffusion equation $u_t=Δu^m$ and the logarithmic diffusion equation $u_t=Δ\log u$ respectively. We will also prove the asymptotic decay rate of the function $f$ as $|x|\to\infty$.

math.AP↗
arXiv2024

Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions

By using fixed point argument we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric $g=\frac{da^2}{h(a^2)}+a^2g_{S^n}$ for some function $h$ where $g_{S^n}$ is the standard metric on the unit sphere $S^n$ in $\mathbb{R}^n$ for any $n\ge 2$. More precisely for any $λ\ge 0$ and $c_0>0$, we prove that there exist infinitely many solutions $h\in C^2((0,\infty);\mathbb{R}^+)$ for the equation $2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-λr-(n-1))$, $h(r)>0$, in $(0,\infty)$ satisfying $\underset{\substack{r\to 0}}{\lim}\,r^{\sqrt{n}-1}h(r)=c_0$ and prove the higher order asymptotic behaviour of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behaviour near the origin.

math.DG↗
arXiv2021

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

For $n\ge 3$, $0 0$. As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation $u_t=\frac{n-1}{m}Δu^m$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$ with initial value $u_0$ satisfying $f_{λ_1}(x)\le u_0(x)\le f_{λ_2}(x)$, $\forall x\in\mathbb{R}^n\setminus\{0\}$, which satisfies $U_{λ_1}(x,t)\le u(x,t)\le U_{λ_2}(x,t)$, $\forall x\in \mathbb{R}^n\setminus\{0\}, t\ge 0$, for some constants $λ_1>λ_2>0$. We also prove the asymptotic behaviour of such singular solution $u$ of the fast diffusion equation as $t\to\infty$ when $n=3,4$ and $\frac{n-2}{n+2}\le m<\frac{n-2}{n}$ holds. Asymptotic behaviour of such singular solution $u$ of the fast diffusion equation as $t\to\infty$ is also obtained when $3\le n<8$, $1-\sqrt{2/n}\le m<\min\left(\frac{2(n-2)}{3n},\frac{n-2}{n+2}\right)$, and $u(x,t)$ is radially symmetric in $x\in\mathbb{R}^n\setminus\{0\}$ for any $t>0$ under appropriate conditions on the initial value $u_0$.

math.AP↗
arXiv2018

Vanishing time behavior of solutions to the fast diffusion equation

Let $n\geq 3$, $0< m<\frac{n-2}{n}$ and $T>0$. We construct positive solutions to the fast diffusion equation $u_t=Δu^m$ in $\mathbb{R}^n\times(0,T)$, which vanish at time $T$. By introducing a scaling parameter $β$ inspired by \cite{DKS}, we study the second-order asymptotics of the self-similar solutions associated with $β$ at spatial infinity. We also investigate the asymptotic behavior of the solutions to the fast diffusion equation near the vanishing time $T$, provided that the initial value of the solution is close to the initial value of some self-similar solution and satisfies some proper decay condition at infinity. Depending on the range of the parameter $β$, we prove that the rescaled solution converges either to a self-similar profile or to zero as $t\nearrow T$. The former implies asymptotic stabilization towards a self-similar solution, and the latter is a new vanishing phenomenon even for the case $n\ge3$ and $m=\frac{n-2}{n+2}\,$ which corresponds to the Yamabe flow on $\mathbb{R}^n$ with metric $g=u^{\frac{4}{n+2}}dx^2$.

math.AP↗
arXiv2018

Uniqueness and time oscillating behaviour of finite points blow-up solutions of the fast diffusion equation

Let $n\ge 3$ and $0<m<\frac{n-2}{n}$. We will extend the results of J.L. Vazquez and M. Winkler and prove the uniqueness of finite points blow-up solutions of the fast diffusion equation $u_t=Δu^m$ in both bounded domains and $\mathbb{R}^n\times (0,\infty)$. We will also construct initial data such that the corresponding solution of the fast diffusion equation in bounded domain oscillate between infinity and some positive constant as $t\to\infty$.

math.AP↗
arXiv2018

Existence and large time behaviour of finite points blow-up solutions of the fast diffusion equation

Let $Ω\subset\R^n$ be a smooth bounded domain and let $a_1,a_2,\dots,a_{i_0}\inΩ$, $\widehatΩ=Ω\setminus\{a_1,a_2,\dots,a_{i_0}\}$ and $\widehat{R^n}=\R^n\setminus\{a_1,a_2,\dots,a_{i_0}\}$. We prove the existence of solution $u$ of the fast diffusion equation $u_t=Δu^m$, $u>0$, in $\widehatΩ\times (0,\infty)$ ($\widehat{R^n}\times (0,\infty)$ respectively) which satisfies $u(x,t)\to\infty$ as $x\to a_i$ for any $t>0$ and $i=1,\cdots,i_0$, when $0 \frac{n(1-m)}{2}$ and $u_0(x)\ge λ_i|x-a_i|^{-γ_i}$ for $x\approx a_i$ and some constants $γ_i>\frac{2}{1-m},λ_i>0$, for all $i=1,2,\dots,i_0$. We also find the blow-up rate of such solutions near the blow-up points $a_1,a_2,\dots,a_{i_0}$, and obtain the asymptotic large time behaviour of such singular solutions. More precisely we prove that if $u_0\geμ_0$ on $\widehatΩ$ ($\widehat{R^n}$, respectively) for some constant $μ_0>0$ and $γ_1>\frac{n-2}{m}$, then the singular solution $u$ converges locally uniformly on every compact subset of $\widehatΩ$ (or $\widehat{R^n}$ respectively) to infinity as $t\to\infty$. If $u_0\geμ_0$ on $\widehatΩ$ ($\widehat{R^n}$, respectively) for some constant $μ_0>0$ and satisfies $λ_i|x-a_i|^{-γ_i}\le u_0(x)\le λ_i'|x-a_i|^{-γ_i'}$ for $x\approx a_i$ and some constants $\frac{2}{1-m}<γ_i\leγ_i'<\frac{n-2}{m}$, $λ_i>0$, $λ_i'>0$, $i=1,2,\dots,i_0$, we prove that $u$ converges in $C^2(K)$ for any compact subset $K$ of $\2Ω\setminus\{a_1,a_2,\dots,a_{i_0}\}$ (or $\widehat{R^n}$ respectively) to a harmonic function as $t\to\infty$.

math.AP↗
arXiv2016

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

Let $n\geq 3$, $0\le m<\frac{n-2}{n}$, $ρ_1>0$, $β>β_0^{(m)}=\frac{mρ_1}{n-2-nm}$, $α_m=\frac{2β+ρ_1}{1-m}$ and $α=2β+ρ_1$. For any $λ>0$, we prove the uniqueness of radially symmetric solution $v^{(m)}$ of $\La(v^m/m)+α_m v+βx\cdot\nabla v=0$, $v>0$, in $\R^n\setminus\{0\}$ which satisfies $\lim_{|x|\to 0}|x|^{\frac{α_m}β}v^{(m)}(x)=λ^{-\frac{ρ_1}{(1-m)β}}$ and obtain higher order estimates of $v^{(m)}$ near the blow-up point $x=0$. We prove that as $m\to 0^+$, $v^{(m)}$ converges uniformly in $C^2(K)$ for any compact subset $K$ of $\R^n\setminus\{0\}$ to the solution $v$ of $\La\log v+αv+βx\cdot\nabla v=0$, $v>0$, in $\R^n\bs\{0\}$, which satisfies $\lim_{|x|\to 0}|x|^{\fracαβ}v(x)=λ^{-\frac{ρ_1}β}$. We also prove that if the solution $u^{(m)}$ of $u_t=Δ(u^m/m)$, $u>0$, in $(\R^n\setminus\{0\})\times (0,T)$ which blows up near $\{0\}\times (0,T)$ at the rate $|x|^{-\frac{α_m}β}$ satisfies some mild growth condition on $(\R^n\setminus\{0\})\times (0,T)$, then as $m\to 0^+$, $u^{(m)}$ converges uniformly in $C^{2+θ,1+\fracθ{2}}(K)$ for some constant $θ\in (0,1)$ and any compact subset $K$ of $(\R^n\setminus\{0\})\times (0,T)$ to the solution of $u_t=\La\log u$, $u>0$, in $(\R^n\setminus\{0\})\times (0,T)$. As a consequence of the proof we obtain existence of a unique radially symmetric solution $v^{(0)}$ of $\La \log v+αv+βx\cdot\nabla v=0$, $v>0$, in $\R^n\setminus\{0\}$, which satisfies $\lim_{|x|\to 0}|x|^{\fracαβ}v(x)=λ^{-\frac{ρ_1}β}$.

math.AP↗
arXiv2015

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

We study the asymptotic large time behavior of singular solutions of the fast diffusion equation $u_t=Δu^m$ in $({\mathbb R}^n\setminus\{0\})\times(0,\infty)$ in the subcritical case $0 A_1>0$ and $\frac{2}{1-m}<γ<\frac{n-2}{m}$, where $β:=\frac{1}{2-γ(1-m)}$, $α:=\frac{2β-1}{1-m},$ 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 $\lim_{|x|\to0}|x|^{\frac{ α}{ β}}f_i(x)=A_i$ and $\lim_{|x|\to\infty}|x|^{\frac{n-2}{m}}{f_i}(x)= D_{A_i} $ for some constants $D_{A_i}>0$. When $\frac{2}{1-m}<γ<n$, under an integrability condition on the initial value $u_0$ of the singular solution $u$, we prove that the rescaled function $$ \tilde u(y,τ):= t^{\,α} u(t^{\,β} y,t),\quad{ τ:=\log t}, $$ converges to some self-similar profile $f$ as $τ\to\infty$.

math.AP↗
arXiv2015

Singular limit of the generalized Burgers equation with absorption

We prove the convergence of the solutions $u_{m,p}$ of the equation $u_t+(u^m)_x=-u^p$ in $\R\times (0,\infty)$, $u(x,0)=u_0(x)\ge 0$ in $\R$, as $m\to\infty$ for any $p>1$ and $u_0\in L^1(\R)\cap L^{\infty}(\R)$ or as $p\to\infty$ for any $m>1$ and $u_0\in L^{\infty}(\R)$ . We also show that in general $\underset{p\to\infty}\lim\underset{m\to\infty}\lim u_{m,p}\ne\underset{m\to\infty}\lim\underset{p\to\infty}\lim u_{m,p}$.

math.AP↗
arXiv2014

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

Let $n\ge 3$, $0 0$, $β\ge\frac{mρ_1}{n-2-nm}$ 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 $T>0$.

math.AP↗
arXiv2012

Extinction profile of the logarithmic diffusion equation

Let $u$ be the solution of $u_t=Δ\log u$ in $\R^N\times (0,T)$, N=3 or $N\ge 5$, with initial value $u_0$ satisfying $B_{k_1}(x,0)\le u_0\le B_{k_2}(x,0)$ for some constants $k_1>k_2>0$ where $B_k(x,t) =2(N-2)(T-t)_+^{N/(N-2)}/(k+(T-t)_+^{2/(N-2)}|x|^2)$ is the Barenblatt solution for the equation. We prove that the rescaled function $\4{u}(x,s)=(T-t)^{-N/(N-2)}u(x/(T-t)^{-1/(N-2)},t)$, $s=-\log (T-t)$, converges uniformly on $\R^N$ to the rescaled Barenblatt solution $\4{B}_{k_0}(x)=2(N-2)/(k_0+|x|^2)$ for some $k_0>0$ as $s\to\infty$. We also obtain convergence of the rescaled solution $\4{u}(x,s)$ as $s\to\infty$ when the initial data satisfies $0\le u_0(x)\le B_{k_0}(x,0)$ in $\R^N$ and $|u_0(x)-B_{k_0}(x,0)|\le f(|x|)\in L^1(\R^N)$ for some constant $k_0>0$ and some radially symmetric function $f$.

math.AP↗
arXiv2012

Decay rate and radial symmetry of the exponential elliptic equation

Let $n\geq 3$, $α$, $β\in\mathbb{R}$, and let $v$ be a solution $Δv+αe^v+βx\cdot\nabla e^v=0$ in $\mathbb{R}^n$, which satisfies the conditions $\lim_{R\to\infty}\frac{1}{\log R}\int_{1}^{R}ρ^{1-n} (\int_{B_ρ}e^v\,dx)dρ\in (0,\infty)$ and $|x|^2e^{v(x)}\le A_1$ in $\R^n$. We prove that $\frac{v(x)}{\log |x|}\to -2$ as $|x|\to\infty$ and $α>2β$. As a consequence under a mild condition on $v$ we prove that the solution is radially symmetric about the origin.

math.AP↗
arXiv2011

Large time behaviour of higher dimensional logarithmic diffusion equation

Let $n\ge 3$ and $ψ_{λ_0}$ be the radially symmetric solution of $Δ\logψ+2βψ+βx\cdot\nablaψ=0$ in $R^n$, $ψ(0)=λ_0$, for some constants $λ_0>0$, $β>0$. Suppose $u_0\ge 0$ satisfies $u_0-ψ_{λ_0}\in L^1(R^n)$ and $u_0(x)\approx\frac{2(n-2)}β\frac{\log |x|}{|x|^2}$ as $|x|\to\infty$. We prove that the rescaled solution $\widetilde{u}(x,t)=e^{2βt}u(e^{βt}x,t)$ of the maximal global solution $u$ of the equation $u_t=Δ\log u$ in $R^n\times (0,\infty)$, $u(x,0)=u_0(x)$ in $R^n$, converges uniformly on every compact subset of $R^n$ and in $L^1(R^n)$ to $ψ_{λ_0}$ as $t\to\infty$. Moreover $\|\widetilde{u}(\cdot,t)-ψ_{λ_0}\|_{L^1(R^n)} \le e^{-(n-2)βt}\|u_0-ψ_{λ_0}\|_{L^1(R^n)}$ for all $t\ge 0$.

math.AP↗
arXiv2011

Collapsing behaviour of a singular diffusion equation

Let $0\le u_0(x)\in L^1(\R^2)\cap L^{\infty}(\R^2)$ be such that $u_0(x) =u_0(|x|)$ for all $|x|\ge r_1$ and is monotone decreasing for all $|x|\ge r_1$ for some constant $r_1>0$ and ${ess}\inf_{\2{B}_{r_1}(0)}u_0\ge{ess} \sup_{\R^2\setminus B_{r_2}(0)}u_0$ for some constant $r_2>r_1$. Then under some mild decay conditions at infinity on the initial value $u_0$ we will extend the result of P. Daskalopoulos, M.A. del Pino and N. Sesum \cite{DP2}, \cite{DS}, and prove the collapsing behaviour of the maximal solution of the equation $u_t=Δ\log u$ in $\R^2\times (0,T)$, $u(x,0)=u_0(x)$ in $\R^2$, near its extinction time $T=\int_{R^2}u_0dx/4π$.

math.AP↗