arXiv subjects

Juncheng Wei

Publications and source records attributed to Juncheng Wei.

At least 55 records · Page 3Linked to original sources

arXiv2026

Collapsing-tube type II blow-up for the energy-supercritical heat equation

We construct a new type II finite-time blow-up mechanism for the energy-supercritical heat equation \[ u_t=Δu+u^3, \qquad n\geq 5. \] The solution is positive and blows up only at the origin, but in a highly anisotropic fashion. As $t\nearrow T$, the solution concentrates in a thin tubular region around an $(n-4)$-dimensional sphere whose radius shrinks to zero at the self-similar scale \[ ξ_r(t)\sim \sqrt{2(n-4)(T-t)}. \] At the same time, concentration takes place transversely to the sphere at the much smaller scale \[ λ(t)\sim κ_* \frac{T-t}{|\log(T-t)|^{\frac n{n-2}}}, \] for some $κ_*>0$. More precisely, in cylindrical coordinates $r=|x'|$, $z\in\mathbb R^3$, the leading profile is \[ u(x,t) \sim \frac{1}{λ(t)} U\left( \frac{r-ξ_r(t)}{λ(t)}, \frac{z}{λ(t)} \right), \] where $U$ is the Aubin--Talenti bubble in $\mathbb R^4$. The construction reveals a two-scale singularity mechanism in which a critical transverse bubble concentrates around a geometric set that itself collapses. The concentration tube evolves at the parabolic scale $\sqrt{T-t}$, whereas its transverse thickness is governed by the much smaller type II scale $λ(t)$. The logarithmic blow-up law is determined by a nonlocal modulation equation arising from the interaction between the four-dimensional critical bubble and the axisymmetric heat kernel. To our knowledge, this seems to be the first Type II blowup that quantifies the effect of a self-similar collapsing tube. The exponent $p=3$ is energy-supercritical in dimensions $n\geq5$, but lies below the Joseph--Lundgren exponent for $5\leq n\leq 12$, in a regime where positive radial type II blow-up is ruled out. The present result provides the first example of a positive type II, single-point blow-up through a collapsing thin-tube geometry.

math.AP↗

arXiv2026

Giga-Kohn-type results for the fully fractional heat equation

We consider the semilinear fully fractional heat equation \[ (\partial_t-Δ)^σu = |u|^{p-1}u \quad \text{in } \mathbb{R}^n \times \mathbb{R}_{-}, \qquad 0 < σ< 1. \] For $n\leq 2σ$ or $1<p\leq \frac{n+2σ}{n-2σ}$, we generalize the monotonicity formula and Liouville-type theorem when $σ=1$ proved by Giga and Kohn. In order to overcome the difficulty that this equation is nonlocal, we give a new interpretation of the classical Giga-Kohn's Pohozaev identity in terms of Hermite expansion. This insight is new and interesting even for $σ=1$. We further establish a space-time nonlocal monotonicity formula for the self-similar equation. As far as we are concerned, this is the first monotonicity formula for space-time nonlocal equations without using an extension by Stinga and Torrea.

math.AP↗

arXiv2026

A refined blow-up analysis of the Brezis-Nirenberg equation and its application: The one-bubble case for $N\geq4$

In this paper, we consider the famous Brezis-Nirenberg equation \begin{eqnarray*} \left\{ \aligned &-Δu=λu+|u|^{\frac{4}{N-2}}u,\quad&\mbox{in}\,\, Ω,\\ &u=0,\quad&\mbox{on}\,\, \partialΩ, \endaligned \right. \end{eqnarray*} where $N\geq3$ is the dimension, $Ω\subset\mathbb{R}^N$ is a bounded domain with smooth boundary $\partialΩ$ and $λ>0$ is a parameter. By developing a refined blow-up analysis based on the inverse reduction argument developed in \cite{WW2019,WW2019-2}, we classify, for the fist time, the Struwe decomposition of the Brezis-Nirenberg equation in the one-bubble case as the parameter $λ$ varies for $N\geq4$. As applications, we prove that the $4d$ Brezis-Nirenberg equation has a nontrivial solution (least energy solution) for $λ\inσ(-Δ)$ in general bounded domains, where $σ(-Δ)$ is the spectrum of $-Δ$ in $H^1_0(Ω)$. Our result completes the existence theory of the Brezis-Nirenberg equation for $N\geq4$ in \cite{AP2025,CFP1985,CFS,CSS1986,CW2005,CSZ2012,SWW2009,TYZ2022} since 1984.

math.AP↗

arXiv2026

Complete Classification and Nondegeneracy of $N$-Component Cubic Nonlinear Schrödinger System in ${\mathbb R}$

We study the one-dimensional cubic nonlinear Schrödinger system \[ u_i''+2\left(\sum_{k=1}^N u_k^2\right)u_i=-μ_i u_i \quad \mbox{in } \ \mathbb R,\ \ i=1,2,\cdots,N, \] where $u=(u_1,\cdots,u_N)\in \big(H^1(\mathbb{R})\big)^N$, $μ_1\leqμ_2\leq\cdots\leqμ_N<0$, and $N\geq 2$ is arbitrary. In this paper, we prove the following results for any $N\ge 2$: (i). All nontrivial solutions of the system can be completely classified; (ii). The linearized operator at any nontrivial solution of the system is non-degenerate; (iii). For all $i=1, 2,\cdots, N$, the exact $L^2$-mass identity of $u_i$ is derived in terms of $2\sqrt {|μ_i|}$, which yields a complete characterization of normalized solutions satisfying $\int_{\mathbb{R}}u_i^2dx=1$. These settle some conjectures of [R. Frank, D. Gontier and M. Lewin, CMP, 2021] and [Y. Guo, Y. Luo and J. Wei, APDE, 2026], where the system was addressed specially for $N=2$ and $N=3$, respectively.

math.AP↗

arXiv2026

On Brezis Open Problem 3.1

Let $B_1$ be the unit disk in ${\mathbb R}^2$. We consider the harmonic map equation $$ -Δu=|\nabla u|^2u,$$ subject to the Dirichlet boundary condition $ u(e^{iθ})=(R\cosθ,R\sinθ,\sqrt{1-R^2}):=g_R$, where $0<R<1$ and $u: B_1\to {\mathbb S}^2$ is understood in the weak harmonic-map sense. In 1983, Brezis and Coron proved the existence of two explicit solutions of this nonlinear Dirichlet problem and showed that they are the unique minimizers in their respective relative homotopy classes. In this paper, we resolve a long-standing open question originally posed in their work, later posed as Open Problem 3.1 in Brezis Favorite Open Problems List. Specifically, we prove that these two explicit maps are the only weak harmonic maps with boundary trace $g_{R}$, thereby providing a definitive affirmative answer to Brezis open problem. The proof is based on a boundary rigidity argument. An auxiliary potential $X$ associated with $u$, the Pohozaev identity for the Hopf differential, and the planar isoperimetric inequality imply $$|u_r|\equiv R, \qquad u_r\cdot u_θ\equiv0 \qquad\text{on }\partial B_1. $$ Thus the Hopf differential vanishes on the boundary and hence, by holomorphicity, on the whole disk. The problem is then reduced to the conformal case, where a stereographic-coordinate classification gives exactly the two Brezis--Coron maps.

math.AP↗

arXiv2026

Infinite time bubbling for the $SU(2)$ Yang-Mills heat flow on $\mathbb{R}^4$

We investigate the long time behaviour of the Yang-Mills heat flow on the bundle $\mathbb{R}^4\times SU(2)$. Waldron \cite{Waldron2019} proved global existence and smoothness of the flow on closed $4-$manifolds, leaving open the issue of the behaviour in infinite time. We exhibit two types of long-time bubbling: first we construct an initial data and a globally defined solution which {\sl blows-up} in infinite time at a given point in $\mathbb R^4$. Second, we prove the existence of {\sl bubble-tower} solutions, also in infinite time. This answers the basic dynamical properties of the heat flow of Yang-Mills connection in the critical dimension $4$ and shows in particular that in general one cannot expect that this gradient flow converges to a Yang-Mills connection. We emphasize that we do not assume for the first result any symmetry assumption; whereas the second result on the existence of the bubble-tower is in the $SO(4)$-equivariant class, but nevertheless new.

math.AP↗

arXiv2026

On ratios of theta functions

Motivated by the average partition function of c free bosons $($Afhkami-Jeddi et al. \cite{Afhk2021}$)$ and the average of the genus 1 partition function over the Narain moduli space $($Maloney-Witten \cite{Witten2020}$)$, we investigate ratios of theta functions. In this paper, we completely classify the minimizers (or maximizers) for ratios of theta and Epstein zeta functions. We find that the hexagonal lattice plays a pivotal role there. These results have direct applications in conformal and Liouville field theory via partition functions. Additionally, they yield the minima of differences of theta and Epstein zeta functions, which have implications for the mathematics of crystallization and interacting particle theory (\cite{Bet2016,Bet2019AMP}).

math.CA↗

arXiv2026

Ground States of One-Dimensional Fermionic Schrödinger Systems Near a Critical Exponent

We study ground states of the fermionic nonlinear Schrödinger system $J_2(p)$ in $\R$, where $p>1$ denotes a polynomial exponent of the nonlinear term. It is known that the system $J_2(p)$ admits ground states for any $1<p<2$, while there is no ground state for $J_2(2)$. We prove that there is no ground state of $J_2(p)$ as $p\searrow 2$, which addresses the special case of Conjecture 5 in [D. Gontier, M. Lewin and F. Q. Nazar, ARMA, 2021]. The refined limiting profile of ground states for $J_2(p)$ is also analyzed as $p\nearrow 2$, which shows that the corresponding density admits exactly two bumps whose distance goes up to infinity as $p\nearrow 2$.

math.AP↗

arXiv2026

Classification and Nondegeneracy of Cubic Nonlinear Schrödinger System in $\mathbb{R}$

We study the following one-dimensional cubic nonlinear Schrödinger system: \[ u_i''+2\Big(\sum_{k=1}^Nu_k^2\Big)u_i=-μ_iu_i \ \,\ \mbox{in}\, \ \mathbb{R} , \ \ i=1, 2, \cdots, N, \] where $μ_1\leqμ_2\leq\cdots\leqμ_N<0$ and $N\ge 2$. In this paper, we mainly focus on the case $N=3$ and prove the following results: (i). The solutions of the system can be completely classified; (ii). Depending on the explicit values of $μ_1\leqμ_2\leqμ_3<0$, there exist two different classes of normalized solutions $u=(u_1, u_2, u_3)$ satisfying $\int _{R}u_i^2dx=1$ for all $i=1, 2, 3$, which are completely different from the case $N=2$; (iii). The linearized operator at any nontrivial solution of the system is non-degenerate. The conjectures on the explicit classification and nondegeneracy of solutions for the system are also given for the case $N>3$. These address the questions of [R. Frank, D. Gontier and M. Lewin, CMP, 2021], where the complete classification and uniqueness results for the system were already proved for the case $N=2$.

math.AP↗

arXiv2026

Sharp spectral estimates for free boundary problems arising in plasma physics

We derive a sharp spectral estimate for a superlinear free boundary problem arising in plasma physics. The semilinear equation is coupled with a constraint, which forces the analysis of a non-local eigenvalue equation. Consequently the corresponding first eigenvalue, say $σ_1$, is not a standard one and it is shown that it cannot satisfy a general isoperimetric property of Faber-Krahn type. This motivates a careful analysis of the problem on balls in any dimension $N\geq 2$, where we prove that in fact $σ_1$ is always positive. The implications about the uniqueness problem for the Emden equation are also discussed.

math.AP↗

arXiv2026

Existence and multiplicity of solutions to the mean-field games model with mixed interactions

In this paper, we consider the stationary version of the Mean-Field Games (MFG) models. Inspired by \cite{Albuquerque-Silva2020, Bieganowski-Mederski2021, Lin-Wei05, Mederski-Schino2021}, we develop the minimization method on the Pohozaev manifold introduced in \cite{Soave20JDE, Soave20JFA} for the existence theory of the stationary version of the Mean-Field Games (MFG) models with $2$-homogeneous hamiltonians and mixed interactions. As applications, we prove the existence and multiplicity of radial solutions of the Mean-Field Games (MFG) models with general $p$-homogeneous hamiltonians and mixed interactions under more general conditions, some of which are even new for $2$-homogeneous hamiltonians. We hope that our techniques and ideas introduced in this paper would be helpful in understanding the optimal value of the total mass in the existence theory of radial solutions to the Mean-Field Games (MFG) models with general $p$-homogeneous hamiltonians and mixed interactions, as well as that of other models.

math.AP↗

arXiv2026

On the Classification of blow-up solutions of a singular Liouville equation on the disk

We study the blow-up behavior of solutions to the singular Liouville equation \[ Δ\tilde u+λe^{\tilde u}=4παδ_0 \quad\text{in }B,\quad \tilde u=0 \quad\text{on }\partial B, \] where $α>0$, $λ>0$ and $B\subset\mathbb R^2$ is the unit disk. Our main results give a complete classification of all blow-up solutions and determine the exact number of solutions to the above equation. More precisely, for fixed $α>0$ and $λ\in(0,λ_α)$, the singular Liouville equation has exactly $\lceil α\rceil+2$ solutions (up to rotation): a unique minimal energy solution; a unique singular sequence blowing up at the origin; and for each $1\le m\le\lceil α\rceil$, a unique $m$-peak sequence whose blow-up points are the vertices of a regular $m$-gon centered at the origin. This result answers the questions raised in Bartolucci-Montefusco \cite{Bartolucci-Montefusco06} and Bartolucci \cite{Bartolucci10}. We also prove the non-degeneracy of these solutions. Thus we provide a full description of the blow-up structure for the singular Liouville equation on the disk.

math.AP↗

arXiv2026

The Rabinowitz continuum of subcritical Gelfand problems and free boundary-type equations arising in plasma physics

The qualitative behavior of the Rabinowitz unbounded continuum of subcritical Gelfand problems is well known on balls in any dimension. We don't know of any such sharp and detailed description otherwise, which is our motivation to look for a new approach to the problem. The underlying idea is to describe solutions of Gelfand problems via suitably defined constrained problems of free boundary-type arising in plasma physics and to replace the usual $L^\infty$ norm of the solution with the energy of the plasma. Toward this goal, we first solve a long standing open problem of independent interest about the uniqueness of solutions of Grad-Shafranov type equations. Thus, we exploit these unique solutions to detect a curve containing both minimal and non minimal solutions of the associated Gelfand problem. In other words we come up with a new global parametrization of the Rabinowitz continuum, the monotonicity of the energy along the branch providing a meaningful generalization of the classical pointwise monotonicity property of minimal solutions, suitable to describe non minimal solutions as well. On a ball in any dimension, we come up as expected with a bell-shaped profile of the full branch of solutions of the Gelfand problem.

math.AP↗

arXiv2026

Second order classification for singular Liouville equations with a coefficient function

In this article we are concerned with the existence of blow-up solutions to the following boundary value problem $$-Δv= λV(x) |x|^2e^v\;\mbox{in}\quad B_1,\quad v=0 \;\mbox{ on }\quad \partial B_1,$$ where $B_1$ is the unit ball in $\mathbb R^2$ centered at the origin, $V(x)$ is a positive smooth potential, and $λ>0$ is a small parameter. We find necessary and sufficient conditions on the potential $V$ for the existence of a blow-up sequence of solutions tending to infinity near the origin as $λ\to 0^+$. In particular, we obtain a second-order classification of the coefficient function $V$ for which (simple) blow-up occurs at the origin.

math.AP↗

arXiv2026

Global Convergence of the Gursky-Malchiodi $Q$-curvature Flow

In their seminal work, Gursky and Malchiodi introduced a non-local conformal flow in dimensions $n \geq 5$ to resolve the constant $Q$-curvature problem. They proved sequential convergence of the flow for initial metrics with positive scalar curvature and $Q$-curvature, provided the energy was sufficiently small. In this paper, we prove the global convergence of the flow for arbitrary initial energy under the same positivity assumptions by establishing a non-local version of the Łojasiewicz-Simon inequality for the Paneitz-Sobolev quotient along the flow. We construct test bubbles and estimate their Paneitz-Sobolev quotients, a strategy that was carried out in the celebrated work of Brendle in the context of the Yamabe flow. We develop a more geometric and systematic proof that addresses the algebraic and computational complexity inherent in the $Q$-curvature and the Paneitz operator. Along the way, we derive a stability inequality for the Paneitz-Sobolev quotient using a higher-order Koiso-Bochner formula established in recent work of Bahuaud, Guenther, Isenberg, and Mazzeo.

math.DG↗

arXiv2026

Blow-up suppression for the nematic liquid crystal flow via Couette flow on $\mathbb{R}^2$

As is well known, for the harmonic heat flow or liquid crystal flow in two-dimension, the solution may blow up when the initial energy is greater than $8π$. Motivated by Lai--Lin--Wang--Wei--Zhou (CPAM, 2022), where singular solutions were constructed in the presence of small-scale velocity fields, it is natural to ask whether large-scale velocities may play a stabilizing role, preventing the concentration of blow-up. Here we show that the blow-up phenomenon can be suppressed by a Couette flow whose amplitude is large enough under a weak assumption on the anisotropic norm of the initial data. In particular, we construct examples with initial energy exceeding $8π$ that satisfy our assumptions.

math.AP↗

arXiv2026

Global oscillatory solutions for the Yang-Mills heat flow

We investigate the long-time dynamics for the global solution of the $SO(4)$-equivariant Yang-Mills heat flow (YMHF) with structure group $SU(2)$ in space dimension $4$. For a class of initial data with specific decay at spatial infinity, we prove that the long-time dynamics of YMHF can be described by the initial data in a unified manner. As a consequence, the global solutions can exhibit blow-up, blow-down, and more exotically, {\it oscillatory} asymptotic behavior at time infinity. This seems to be the first example of Yang-Mills heat flows with oscillatory behavior as $t\to \infty$.

math.AP↗

arXiv2025

Infinitely many positive solutions to nonlinear scalar field equation with nonsmooth nonlinearity

This paper investigates the existence of infinitely many positive solutions for the logarithmic scalar field equation \begin{equation} \tag{$P$} \label{equ1} -Δu+ V(x) u= u\log u^2, \quad u\in H^1(\mathbb{R}^N), \end{equation} and its counterpart with prescribed $L^2$-norms \begin{align}\label{equ2} \tag{$P_N$} & -Δu+ V(x) u +λu= u\log u^2, \quad u\in H^1(\mathbb{R}^N), &\int_{\mathbb{R}^N} u^2 ~\mathrm{d}x=a^2>0, \end{align} which come from physically relevant situations. Here, $N\geq 2$, $V:\mathbb{R}^N\to \mathbb{R}$ is a non-symmetric and non-periodic potential satisfying certain decay conditions, $ a $ is prescribed constant, and $λ$ arises as an unknown Lagrange multipliers. For problem \eqref{equ1}, using purely variational methods, we establish the existence of multi-bump positive solutions with either finitely or infinitely many bumps. For normalized problem \eqref{equ2}, we prove the existence of normalized multi-bump positive solutions with a finite number of bumps. The main difficulty comes from the nonsmooth nature of logarithmic nonlinearity, which introduces some challenges to the variational framework. In particular, the corresponding energy functional is not of class $C^1$ on $H^1(\mathbb{R}^N)$, which prevents the direct application of standard critical point theory for $C^1$ functional or any reduction methods for $C^{1+σ}$ nonlinearity. The main ingredients in this paper are nonsmooth critical point theory, localized variational methods and a max-min argument. To the best of our knowledge, this paper appears to be the first successful application of the localized variational method to nonsmooth functionals.

math.AP↗