arXiv subjects

Xiaoliu Wang

Publications and source records attributed to Xiaoliu Wang.

4 recordsLinked to original sources

arXiv2023

On A Parabolic Equation in MEMS with An External Pressure

The parabolic problem $u_t-Δu=\frac{λf(x)}{(1-u)^2}+P$ on a bounded domain $Ω$ of $R^n$ with Dirichlet boundary condition models the microelectromechanical systems(MEMS) device with an external pressure term. In this paper, we classify the behavior of the solution to this equation. We first show that under certain initial conditions, there exists critical constants $P^*$ and $λ_P^*$ such that when $0\leq P\leq P^*$, $0<λ\leq λ_P^*$, there exists a global solution, while for $0\leq P\leq P^*,λ>λ_P^*$ or $P>P^*$, the solution quenches in finite time. The estimate of voltage $λ_P^*$, quenching time $T$ and pressure term $P^*$ are investigated. The quenching set $Σ$ is proved to be a compact subset of $Ω$ with an additional condition, provided $Ω\subset R^n$ is a convex bounded set. In particular, if $Ω$ is radially symmetric, then the origin is the only quenching point. Furthermore, we not only derive the two-side bound estimate for the quenching solution, but also study the asymptotic behavior of the quenching solution in finite time.

math.AP↗

arXiv2023

Dimension-Dependent Asymptotic Dynamics for Mean Curvature Flow with Robin Boundary Conditions

We consider a graphical mean curvature flow in a cylinder with Robin boundary conditions, which arises as a geometric model for interface motion in the singular limit of the Allen--Cahn equation with nonlinear boundary conditions. It was shown in [26] that, in the planar case, every solution converges to a translating Grim Reaper with a \emph{fixed profile} and \emph{finite speed}. In this paper, we investigate the radially symmetric problem in higher dimensions and reveal a completely different asymptotic dynamics caused by the spatial dimension. In contrast to the planar case, there is no fixed translating profile governing the long-time behaviour. Instead, the solution propagates with an exponentially increasing speed, while both the gradient $|Du|$ (away from the center) and the instantaneous speed $u_t$ diverge exponentially as $t\to\infty$. This reveals a fundamentally different asymptotic behaviour induced by the interaction between the Robin boundary condition and the spatial dimension, that is, the translating profile continuously degenerates and becomes asymptotically ray-like. Since the equation becomes asymptotically degenerate and no uniform-in-time $C^0$, $C^1$, or $C^2$ estimates are available, our analysis relies on a new approach based on the zero number argument.

math.DG↗

arXiv2021

Global dynamics of a parabolic type equation arising from the curvature flow

This paper studies a type of degenerate parabolic problem with nonlocal term \begin{equation*} \begin{cases} u_t=u^p(u_{xx}+u-\bar{u}) & 0 1$, $a>0$. In this paper, the classification of the finite-time blowup/global existence phenomena based on the associated energy functional and explicit expression of all nonnegative steady states are demonstrated. Moreover, we combine the applications of Lojasiewicz-Simon inequality and energy estimates to derive that any bounded solution with positive initial data converges to some steady state as $t\rightarrow +\infty$.

math.AP↗

arXiv2020

Convergence to the Grim Reaper for a Curvature Flow with Unbounded Boundary Slopes

We consider a curvature flow $V=H$ in the band domain $Ω:=[-1,1]\times \R$, where, for a graphic curve $Γ_t$, $V$ denotes its normal velocity and $H$ denotes its curvature. If $Γ_t$ contacts the two boundaries $\partial_\pm Ω$ of $Ω$ with constant slopes, in 1993, Altschular and Wu \cite{AW1} proved that $Γ_t$ converges to a {\it grim reaper} contacting $\partial_\pm Ω$ with the same prescribed slopes. In this paper we consider the case where $Γ_t$ contacts $\partial_\pm Ω$ with slopes equaling to $\pm 1$ times of its height. When the curve moves to infinity, the global gradient estimate is impossible due to the unbounded boundary slopes. We first consider a special symmetric curve and derive its uniform interior gradient estimates by using the zero number argument, and then use these estimates to present uniform interior gradient estimates for general non-symmetric curves, which lead to the convergence of the curve in $C^{2,1}_{loc} ((-1,1)\times \R)$ topology to the {\it grim reaper} with span $(-1,1)$.

math.DG↗