arXiv subjects

Kin Ming Hui

Publications and source records attributed to Kin Ming Hui.

24 records · Page 2Linked to original sources

arXiv2010

Convergence of the Dirichlet solutions of the very fast diffusion equation

For any $-1 0$, $0\le u_0\in L^{\infty}(R)$ such that $u_0(x)\le (μ_0 |m||x|)^{\frac{1}{m}}$ for any $|x|\ge R_0$ and some constants $R_0>1$ and $0<μ_0\leq μ$, and $f,\,g \in C([0,\infty))$ such that $f(t),\, g(t) \geq μ_0$ on $[0,\infty)$ we prove that as $R\to\infty$ the solution $u^R$ of the Dirichlet problem $u_t=(u^m/m)_{xx}$ in $(-R,R)\times (0,\infty)$, $u(R,t)=(f(t)|m|R)^{1/m}$, $u(-R,t)=(g(t)|m|R)^{1/m}$ for all $t>0$, $u(x,0)=u_0(x)$ in $(-R,R)$, converges uniformly on every compact subsets of $R\times (0,T)$ to the solution of the equation $u_t=(u^m/m)_{xx}$ in $R\times (0,\infty)$, $u(x,0)=u_0(x)$ in $R$, which satisfies $\int_Ru(x,t)\,dx=\int_Ru_0dx-\int_0^t(f(s)+g(s))\,ds$ for all $0<t<T$ where $\int_0^T(f+g)\,ds=\int_Ru_0dx$. We also prove that the solution constructed is equal to the solution constructed in [Hu3] using approximation by solutions of the corresponding Neumann problem in bounded cylindrical domains.

math.AP↗

arXiv2010

Existence and dynamic properties of a parabolic nonlocal MEMS equation

Let $Ω\subset\mathbb{R}^n$ be a $C^2$ bounded domain and $χ>0$ be a constant. We will prove the existence of constants $λ_N\geλ_N^{\ast}\geλ^{\ast}(1+χ\int_Ω\frac{dx}{1-w_{\ast}})^2$ for the nonlocal MEMS equation $-Δv=\lam/(1-v)^2(1+χ\int_Ω1/(1-v)dx)^2$ in $Ω$, $v=0$ on $\1Ω$, such that a solution exists for any $0\leλ<λ_N^{\ast}$ and no solution exists for any $λ>λ_N$ where $λ^{\ast}$ is the pull-in voltage and $w_{\ast}$ is the limit of the minimal solution of $-Δv=\lam/(1-v)^2$ in $Ω$ with $v=0$ on $\1Ω$ as $λ\nearrow λ^{\ast}$. We will prove the existence, uniqueness and asymptotic behaviour of the global solution of the corresponding parabolic nonlocal MEMS equation under various boundedness conditions on $λ$. We also obtain the quenching behaviour of the solution of the parabolic nonlocal MEMS equation when $λ$ is large.

math.AP↗

arXiv2010

Quenching behaviour of a nonlocal parabolic MEMS equation

We obtain upper bounds for the quenching time of the solutions of the nonlocal parabolic MEMS equation $u_t=Δu+\lam/(1-u)^2(1+χ\int_Ω1/(1-u) dx)^2$ in $Ω\times (0,\infty)$, $u=0$ on $\1Ω\times (0,\infty)$, $u(x,0)=u_0$ in $Ω$, when $λ$ is large. We prove the compactness of the quenching set under a mild condition on the initial data. When $Ω=B_R$ and $u_0$ is radially symmetric and monotone decreasing in $0\le r\le R$, we prove that the point $x=0$ is the only possible quenching set. When $u_0$ also satisfies some strict concavity assumption, we prove that for any $β\in (2,3)$ the solution satisfies $1-u(x,t)\ge C|x|^{\frac{2}β}$ for some constant $C>0$ and we also obtain the quenching time estimate in this case.

math.AP↗

arXiv2009

Another proof for the removable singularities of the heat equation

We give two different simple proofs for the removable singularities of the heat equation in $(Ω\setminus\{x_0\})\times (0,T)$ with $n\ge 3$. We also give a necessary and sufficient condition for removable singularities of the heat equation in $(Ω\setminus\{x_0\})\times (0,T)$ for the case $n=2$.

math.AP↗

arXiv2008

Growth rate and extinction rate of a reaction diffusion equation with a singular nonlinearity

We prove the growth rate of global solutions of the equation $u_t=Δu-u^{-ν}$ in $\R^n\times (0,\infty)$, $u(x,0)=u_0>0$ in $\R^n$, where $ν>0$ is a constant. More precisely for any $0 1$. We also find various conditions on the initial value for the solution to extinct in a finite time and obtain the corresponding decay rate of the solution near the extinction time.

math.AP↗

arXiv2008

Global and touchdown behaviour of the generalized MEMS device equation

We prove the local and global existence of solutions of the generalized micro-electromechanical system (MEMS) equation $u_t =Δu+λf(x)/g(u)$, $u<1$, in $Ω\times (0,\infty)$, $u(x,t)=0$ on $\partialΩ\times (0,\infty)$, $u(x,0)=u_0$ in $Ω$, where $Ω\subset\Bbb{R}^n$ is a bounded domain, $λ>0$ is a constant, $0\le f\in C^α(\overlineΩ)$, $f\not\equiv 0$, for some constant $0<α<1$, $0 0$ such that the associated stationary problem has a solution for any $0\leλ<λ^*$ and has no solution for any $λ>λ^*$. We obtain comparison theorems for the generalized MEMS equation. Under a mild assumption on the initial value we prove the convergence of global solutions to the solution of the corresponding stationary elliptic equation as $t\to\infty$ for any $0\leλ<λ^*$. We also obtain various conditions for the existence of a touchdown time $T>0$ for the solution $u$. That is a time $T>0$ such that $\lim_{t\nearrow T}\sup_Ωu(\cdot,t)=1$.

math.AP↗