arXiv subjects

Jun-cheng Wei

Publications and source records attributed to Jun-cheng Wei.

8 recordsLinked to original sources

arXiv2026

Blow-up phenomena for the $\sigma_2$-Yamabe equation

For every integer $n\ge27$, we construct a smooth metric on $\mathbb S^n$ that is invariant under the antipodal map and is not locally conformally flat. For this fixed background metric, the normalized $\sigma_2$-Yamabe equation admits a noncompact family of positive $\Gamma_2^+$-admissible solutions. The main difficulty is the possible loss of ellipticity of the linearized operator. This difficulty does not occur for the scalar Yamabe equation, whose linearization has a fixed Laplace-type principal part. In the $\sigma_2$ problem, the positive Newton tensor of a standard bubble decays in the far field, while the terms produced by the background metric need not decay at the same rate. Our construction provides the relative decay needed to keep the conformal metrics inside the ellipticity cone. A quartic profile in the finite-dimensional reduction yields the endpoint dimension $n=27$.

math.DG↗

arXiv2024

An Ohta-Kawasaki Model set on the space

We examine a non-local diffuse interface energy with Coulomb repulsion in three dimensions inspired by the Thomas-Fermi-Dirac-von Weizsäcker, and the Ohta-Kawasaki models. We consider the corresponding mass-constrained variational problem and show the existence of minimizers for small masses, and the absence of minimizers for large masses.

math.AP↗

arXiv2023

Lump type solutions: Backlund transformation and spectral properties

There are various different ways to obtain traveling waves of lump type for the KP equation. We propose a general and simple approach to derive them via a Backlund transformation. This enables us to establish an explicit connection between those low energy solutions and high energy ones. Based on this construction, spectral analysis of the degree $6$ solutions is then carried out in details. The analysis of higher energy ones can be done in an inductive way.

math.AP↗

arXiv2023

Uniqueness of lump solutions of KP-I equation

The KP-I equation has family of solutions which decay to zero at space infinity. One of these solutions is the classical lump solution. This is a traveling wave, and the KP-I equation in this case reduces to the Boussinesq equation. In this paper we classify the lump type solutions of the Boussinesq equation. Using a robust inverse scattering transform developed by Bilman-Miller, we show that the lump type solutions are rational and their tau function has to be a polynomial of degree $k(k+1)$. In particular, this implies that the lump solution is the unique ground state of the KP-I equation (as conjectured by Klein and Saut in \cite{Klein0}). Our result generalizes a theorem by Airault-McKean-Moser on the classification of rational solutions for the KdV equation.

math.AP↗

arXiv2023

Spike Solutions to the Supercritical Fractional Gierer-Meinhardt System

Localized solutions are known to arise in a variety of singularly perturbed reaction-diffusion systems. The Gierer-Meinhardt (GM) system is one such example and has been the focus of numerous rigorous and formal studies. A more recent focus has been the study of localized solutions in systems exhibiting anomalous diffusion, particularly with Lévy flights. In this paper we investigate localized solutions to a one-dimensional fractional GM system for which the inhibitor's fractional order is supercritical. Using the method of matched asymptotic expansions we reduce the construction of multi-spike solutions to solving a nonlinear algebraic system. The linear stability of the resulting multi-spike solutions is then addressed by studying a globally coupled eigenvalue problem. In addition to these formal results we also rigorously establish the existence and stability of ground-state solutions when the inhibitor's fractional order is nearly critical. The fractional Green's function, for which we present a rapidly converging series expansion, is prominently featured throughout both the formal and rigorous analysis in this paper. Moreover, we emphasize that the striking similarities between the one-dimensional supercritical GM system and the classical three-dimensional GM system can be attributed to the leading order singular behaviour of the fractional Green's function.

nlin.PS↗

arXiv2022

The Blow-up Analysis on $\mathbf{B}_2^{(1)}$ Affine Toda system: Local mass and Affine Weyl group

It has been established that the local mass of blow-up solutions to Toda systems associated with the simple Lie algebras $\mathbf{A}_n,~\mathbf{B}_n,~\mathbf{C}_n$ and $\mathbf{G}_2$ can be represented by a finite Weyl group. In particular, at each blow-up point, after a sequence of bubbling steps (via scaling) is performed, the transformation of the local mass at each step corresponds to the action of an element in the Weyl group. In this article, we present the results in the same spirit for the affine $\mathbf{B}_2^{(1)}$ Toda system with singularities. Compared with the Toda system with simple Lie algebras, the computation of local masses is more challenging due to the infinite number of elements of the {affine Weyl group of type $\mathbf{B}_{2}^{(1)}$}. In order to give an explicit expression for the local mass formula we introduce two free integers and write down all the possibilities into 8 types. This shows a striking difference to previous results on Toda systems with simple Lie algebras. The main result of this article seems to provide the first major advance in understanding the relation between the blow-up analysis of affine Toda system and the {affine Weyl group} of the associated Lie algebras.

math.AP↗

arXiv2022

Uniform bound on the number of partitions for optimal configurations of the Ohta-Kawasaki energy in 3D

We study a 3D ternary system derived as a sharp-interface limit of the Nakazawa-Ohta density functional theory of triblock copolymers, which combines an interface energy with a long range interaction term. Although both the binary case in 2D and 3D, and the ternary case in 2D, are quite well studied, very little is known about the ternary case in 3D. In particular, it is even unclear whether minimizers are made of finitely many components. In this paper we provide a positive answer to this, by proving that the number of components in a minimizer is bounded from above by some quantity depending only on the total masses and the interaction coefficients. One key difficulty is that the 3D structure prevents us from uncoupling the Coulomb-like long range interaction from the perimeter term, hence the actual shape of minimizers is unknown, not even for small masses. This is due to the lack of a quantitative isoperimetric inequality with two mass constraints in 3D, and it makes the construction of competitors significantly more delicate.

math.AP↗

arXiv2014

On a general SU(3) Toda System

We study the following generalized $SU(3)$ Toda System $$ \left\{\begin{array}{ll} -Δu=2e^u+μe^v & \hbox{ in }\R^2\\ -Δv=2e^v+μe^u & \hbox{ in }\R^2\\ \int_{\R^2}e^u<+\infty,\ \int_{\R^2}e^v<+\infty \end{array}\right. $$ where $μ>-2$. We prove the existence of radial solutions bifurcating from the radial solution $(\log \frac{64}{(2+μ) (8+|x|^2)^2}, \log \frac{64}{ (2+μ) (8+|x|^2)^2})$ at the values $μ=μ_n=2\frac{2-n-n^2}{2+n+n^2},\ n\in\N $.

math.AP↗