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Juncheng Wei

Publications and source records attributed to Juncheng Wei.

At least 181 records · Page 10Linked to original sources
arXiv2020

Geometry driven Type II higher dimensional blow-up for the critical heat equation

We consider the problem v_t & = Δv+ |v|^{p-1}v \quad\hbox{in }\ Ω\times (0, T), v & =0 \quad\hbox{on } \partial Ω\times (0, T ) , v& >0 \quad\hbox{in }\ Ω\times (0, T) . In a domain $Ω\subset \mathbb R^d$, $d\ge 7$ enjoying special symmetries, we find the first example of a solution with type II blow-up for a power $p$ less than the Joseph-Lundgren exponent $$p_{JL}(d)=\infty, & \text{if $3\le d\le 10$}, 1+{4\over d-4-2\,\sqrt{d-1}}, & \text{if $d\ge11$}. $$ No type II radial blow-up is present for $p< p_{JL}(d)$. We take $p=\frac{d+1}{d-3}$, the Sobolev critical exponent in one dimension less. The solution blows up on circle contained in a negatively curved part of the boundary in the form of a sharply scaled Aubin-Talenti bubble, approaching its energy density a Dirac measure for the curve. This is a completely new phenomenon for a diffusion setting.

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arXiv2020

Stability of the saddle solutions for the Allen-Cahn equation

We are concerned with the saddle solutions of the Allen-Cahn equation constructed by Cabré and Terra \cite{C,C2} in $\mathbb{R}^{2m}% =\mathbb{R}^{m}\times\mathbb{R}^{m}$. These solutions vanish precisely on the Simons cone. The existence and uniqueness of saddle solution are shown in \cite{C,C2,C1}. Regarding the stability, Schatzman \cite{Sch} proved that the saddle solution is unstable for $m=1,$ Cabré \cite{C1} showed the instability for $m=2,3$ and stability for $m\geq7$. This has left open the case of $m=4,5,6$. In this paper we show that the saddle solutions are stable when $m=4,5,6$, thereby confirming Cabré's conjecture in \cite{C1}. The conjecture that saddle solutions in dimensions $2m\geq8$ should be global minimizers of the energy functional remains open.

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arXiv2019

The existence and stability of spike solutions for a chemotaxis system modeling crime pattern formation

This paper is a continuation of the paper Berestycki, Wei and Winter \cite{Berestycki2014}. In \cite{Berestycki2014}, the existence of multiple symmetric and asymmetric spike solutions of a chemotaxis system modeling crime pattern formation, suggested by Short, Bertozzi, and Brantingham \cite{Short2010}, has been proved in the one-dimensional case. The problem of stability of these spike solutions has been left open. In this paper, we establish the existence of a single radial symmetric spike solution for the system in the one and two-dimensional cases. The main difficulty is to deal with quasilinear elliptic problems whose diffusion coefficients vary largely near the core. We also study the linear stability of the spike solutions in both one-dimensional and two-dimensional cases which show complete different behaviors. In the one-dimensional case, we show that when the reaction time ratio $τ>0$ is small enough, or large enough, the spike solution is linearly stable. In the two-dimensional case, when $τ$ is small enough, the spike solution is linearly stable; while when $τ$ is large enough, the spike solution is linearly unstable and Hopf bifurcation occurs from the spike solution at some $τ=τ_h$.

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arXiv2019

On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program

We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, and we establish the classical gluing method of Mazzeo and Pacard for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of an ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. No traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we examine a fractional Schrödinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.

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arXiv2019

Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary

We concern $C^2$-compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are $4$, $5$ or $6$. By conducting a quantitative analysis of a linear equation associated with the problem, we prove that the trace-free second fundamental form must vanish at possible blow-up points of a sequence of blowing-up solutions. Applying this result and the positive mass theorem, we deduce the $C^2$-compactness for all $4$-manifolds (which may be non-umbilic). For the $5$-dimensional case, we also establish that a sum of the second-order derivatives of the trace-free second fundamental form is non-negative at possible blow-up points. We essentially use this fact to obtain the $C^2$-compactness for all $5$-manifolds. Finally, we show that the $C^2$-compactness on $6$-manifolds is true if the trace-free second fundamental form on the boundary never vanishes.

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arXiv2019

Finite time blow-up for the nematic liquid crystal flow in dimension two

We consider the initial-boundary value problem of a simplified nematic liquid crystal flow in a bounded, smooth domain $Ω\subset \mathbb R^2$. Given any $k$ distinct points in the domain, we develop a new {\em inner--outer gluing method} to construct solutions which blow up exactly at those $k$ points as $t$ goes to a finite time $T$. Moreover, we obtain a precise description of the blow-up.

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arXiv2019

Half space theorem for the Allen-Cahn equation and related problems

In this paper we obtain rigidity results for a bounded non-constant entire solution $u$ of the Allen-Cahn equation in $\mathbb{R}^n$, whose level set $\{u=0\}$ is contained in a half-space. If $n\leq 3$ we prove that the solution must be one-dimensional. In dimension $n\geq 4$, we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained.

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arXiv2019

Singularity formation for the two-dimensional harmonic map flow into $S^2$

We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = Δu + |\nabla u|^2 u \quad \text{in } Ω\times(0,T) \\ u &= φ\quad \text{on } \partial Ω\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } Ω, \end{align*} where $Ω$ is a bounded, smooth domain in $\mathbb{R}^2$, $u: Ω\times(0,T)\to S^2$, $u_0:\barΩ\to S^2$ is smooth, and $φ= u_0\big|_{\partialΩ}$. Given any points $q_1,\ldots, q_k$ in the domain, we find initial and boundary data so that the solution blows-up precisely at those points. The profile around each point is close to an asymptotically singular scaling of a 1-corrotational harmonic map. We build a continuation after blow-up as a $H^1$-weak solution with a finite number of discontinuities in space-time by "reverse bubbling", which preserves the homotopy class of the solution after blow-up.

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arXiv2019

Existence and stability of infinite time bubble towers in the energy critical heat equation

We consider the energy critical heat equation in $\mathbb R^n$ for $n\ge 7$ $$\left\{ \begin{aligned} u_t & = Δu+ |u|^{\frac 4{n-2}}u \hbox{ in }\ \mathbb R^n \times (0, \infty), \\ u(\cdot,0) & = u_0 \ \hbox{ in }\ \mathbb R^n, \end{aligned}\right. $$ which corresponds to the $L^2$-gradient flow of the Sobolev-critical energy $$ J(u) = \int_{\mathbb R^n} e[u] , \quad e[u] := \frac 12 |\nabla u|^2 - \frac {n-2}{2n} |u|^{\frac {2n}{n-2} }. $$ Given any $k\ge 2$ we find an initial condition $u_0$ that leads to sign-changing solutions with {\em multiple blow-up at a single point} (tower of bubbles) as $t\to +\infty$. It has the form of a superposition with alternate signs of singularly scaled {\em Aubin-Talenti solitons}, $$ u(x,t) = \sum_{j=1}^k (-1)^{j-1} {μ_j^{-\frac {n-2}2}} U \left( \frac {x}{μ_j} \right)\, +\, o(1) \quad\hbox{as } t\to +\infty $$ where $U(y)$ is the standard soliton $ U(y) = % (n(n-2))^{\frac 1{n-2}} α_n\left ( \frac 1{1+|y|^2}\right)^{\frac{n-2}2}$ and $$μ_j(t) = β_j t^{- α_j}, \quad α_j = \frac 12 \Big ( \, \left( \frac{n-2}{n-6}\right)^{j-1} -1 \Big). $$ Letting $δ_0$ the Dirac mass, we have energy concentration of the form $$ e[ u(\cdot, t)]- e[U] \rightharpoonup (k-1) S_n\,δ_{0} \quad\hbox{as } t\to +\infty $$ where $S_n=J(U)$. The initial condition can be chosen radial and compactly supported. We establish the codimension $k+ n (k-1)$ stability of this phenomenon for perturbations of the initial condition that have space decay $u_0(x) =O( |x|^{-α})$, $α> \frac {n-2}2$, which yields finite energy of the solution.

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arXiv2019

Singularity formation in the harmonic map flow with free boundary

In the past years, there has been a new light shed on the harmonic map problem with free boundary in view of its connection with nonlocal equations. Here we fully exploit this link, considering the harmonic map flow with free boundary \begin{equation}\label{e:main0} \begin{cases} u_t = Δu\text{ in }\mathbb{R}^2_+\times (0, T),\\ u(x,0,t) \in \mathbb{S}^1\text{ for all }(x,0,t)\in \partial\mathbb{R}^2_+\times (0, T),\\ \frac{du}{dy}(x,0,t)\perp T_{u(x,0,t)}\mathbb{S}^1\text{ for all }(x,0,t)\in \partial\mathbb{R}^2_+\times (0, T),\\ u(\cdot, 0) = u_0\text{ in }\mathbb{R}^2_+ \end{cases} \end{equation} for a function $u:\mathbb{R}^2_+\times [0, T)\to \mathbb{R}^2$. Here $u_0 :\mathbb{R}^2_+\to \mathbb{R}^2$ is a given smooth map and $\perp$ stands for orthogonality. We prove the existence of initial data $u_0$ such that (\ref{e:main0}) blows up at finite time with a profile being the half-harmonic map. This answers a question raised by Yunmei Chen and Fanghua Lin in Remark 4.9 of \cite{ChenLinJGA1998}.

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arXiv2019

On the general Toda system with multiple singular points

In this paper, we consider the following elliptic Toda system associated to a general simple Lie algebra with multiple singular sources \begin{equation*} \begin{cases} -Δw_i=\sum_{j=1}^na_{i,j}e^{2w_j}+2π\sum_{\ell=1}^mβ_{i,\ell}δ_{p_\ell} \quad&\mbox{in}\quad\mathbb{R}^2,\\ \\ w_i(x)=-2\log|x|+O(1)~\mbox{as}~|x|\to\infty,\quad &i=1,\cdots,n, \end{cases} \end{equation*} where $β_{i,\ell}\in[0,1)$. Under some suitable assumption on $β_{i,\ell}$ we establish the existence and non-existence results. This paper generalizes Luo-Tian's [19] and Hyder-Lin-Wei's [10] results to the general Toda system.

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arXiv2019

Fast and slow decaying solutions for $H^{1}$-supercritical quasilinear Schrödinger equations

We consider the following quasilinear Schrödinger equations of the form \begin{equation*} \triangle u-\varepsilon V(x)u+u\triangle u^2+u^{p}=0,\ u>0\ \mbox{in}\ \mathbb{R}^N\ \mbox{and}\ \underset{|x|\rightarrow \infty}{\lim} u(x)=0, \end{equation*} where $N\geq 3,$ $p>\frac{N+2}{N-2},$ $\varepsilon>0$ and $V(x)$ is a positive function. By imposing appropriate conditions on $V(x),$ we prove that, for $\varepsilon=1,$ the existence of infinity many positive solutions with slow decaying $O(|x|^{-\frac{2}{p-1}})$ at infinity if $p>\frac{N+2}{N-2}$ and, for $\varepsilon$ sufficiently small, a positive solution with fast decaying $O(|x|^{2-N})$ if $\frac{N+2}{N-2}<p<\frac{3N+2}{N-2}.$ The proofs are based on perturbative approach. To this aim, we also analyze the structure of positive solutions for the zero mass problem.

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arXiv2019

Non-uniqueness for an energy-critical heat equation on $\mathbb{R}^2$

We construct a singular solution of a stationary nonlinear Schrödinger equation on $\mathbb{R}^2$ with square-exponential nonlinearity having linear behavior around zero. In view of Trudinger-Moser inequality, this type of nonlinearity has an energy-critical growth. We use this singular solution to prove non-uniqueness of strong solutions for the Cauchy problem of the corresponding semilinear heat equation. The proof relies on explicit computation showing a regularizing effect of the heat equation in an appropriate functional space.

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arXiv2019

Ground states of Nonlinear Schrödinger System with Mixed Couplings

We consider the following $k$-coupled nonlinear Schrödinger system: \begin{align*} \begin{cases} &-Δu_j + λ_j u_j = μ_j u_j^3 + \sum_{i=1, i\not=j}^k β_{i,j} u_i^2 u_j \quad {\rm in}\ \mathbb{R}^N,\\ &u_j>0 \quad {\rm in}\ \mathbb{R}^N, \quad u_j(x) \to 0 \quad \text{as }|x|\to +\infty, \quad j=1,2,\cdots,k, \end{cases} \end{align*} where $N\leq 3$, $k\geq3$, $λ_j,μ_j>0$ are constants and $β_{i,j}=β_{j,i}\not=0$ are parameters. There have been intensive studies for the above system when $k=2$ or the system is purely attractive ($ β_{i,j}>0, \forall i \not = j$) or purely repulsive ($β_{i,j}<0, \forall i\not = j $); however very few results are available for $k\geq 3$ when the system admits {\bf mixed couplings}, i.e., there exist $(i_1,j_1)$ and $(i_2,j_2)$ such that $β_{i_1,j_1}β_{i_2,j_2}<0$. In this paper we give the first systematic and an (almost) complete study on the existence of ground states when the system admits mixed couplings. We first divide this system into {\bf repulsive-mixed} and {\bf total-mixed} cases. In the first case we prove nonexistence of ground states. In the second case we give an necessary condition for the existence of ground states and also provide estimates for the Morse index. The key idea is the {\bf block decomposition} of the system ({\bf optimal block decompositions, eventual block decompositions}), and the measure of total interaction forces between different {\bf blocks}. Finally the assumptions on the existence of ground states are shown to be {\bf optimal} in some special cases.

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arXiv2019

Non-hexagonal lattices from a two species interacting system

A two species interacting system motivated by the density functional theory for triblock copolymers contains long range interaction that affects the two species differently. In a two species periodic assembly of discs, the two species appear alternately on a lattice. A minimal two species periodic assembly is one with the least energy per lattice cell area. There is a parameter $b$ in $[0,1]$ and the type of the lattice associated with a minimal assembly varies depending on $b$. There are several thresholds defined by a number $B=0.1867...$ If $b \in [0, B)$, a minimal assembly is associated with a rectangular lattice whose ratio of the longer side and the shorter side is in $[\sqrt{3}, 1)$; if $b \in [B, 1-B]$, a minimal assembly is associated with a square lattice; if $b \in (1-B, 1]$, a minimal assembly is associated with a rhombic lattice with an acute angle in $[\fracπ{3}, \fracπ{2})$. Only when $b=1$, this rhombic lattice is a hexagonal lattice. None of the other values of $b$ yields a hexagonal lattice, a sharp contrast to the situation for one species interacting systems, where hexagonal lattices are ubiquitously observed.

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arXiv2019

On $SU(3)$ Toda system with multiple singular sources

We consider the singular $SU(3)$ Toda system with multiple singular sources \begin{align*} \left\{\begin{array}{ll}-Δw_1=2e^{2w_1}-e^{w_2}+2π\sum_{\ell=1}^mβ_{1,\ell}δ_{P_{\ell}}\quad\text{in }\mathbb{R}^2\\ \rule{0cm}{.5cm} -Δw_2=2e^{2w_2}-e^{w_1}+2π\sum_{\ell=1}^mβ_{2,\ell}δ_{P_{\ell}}\quad\text{in }\mathbb{R}^2 \\ w_i(x)=-2\log|x|+O(1)\quad\text{as }|x|\to\infty,\, i=1,2, \end{array}\right.\end{align*} with $m\geq 3$ and $β_{i,\ell}\in [0,1)$. We prove the existence and non-existence results under suitable assumptions on $β_{i,\ell}$. This generalizes Luo-Tian's \cite{Luo-Tian} result for a singular Liouville equation in $\mathbb{R}^2$. We also study existence results for a higher order singular Liouville equation in $\mathbb{R}^n$.

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